Lightning Rod Improvement Studies

8/12/2019 Lightning Rod Improvement Studies

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VOLUME 39 MAY 2000J O U R N A L O F A P P L I E D M E T E O R O L O G Y

2000 American Meteorological Society 593

Lightning Rod Improvement StudiesC. B. M OORE , WILLIAM RISON , JAMES MATHIS , AND GRAYDON AULICH

Langmuir Laboratory for Atmospheric Research, New Mexico Institute of Mining and Technology,Socorro, New Mexico

(Manuscript received 15 December 1997, in nal form 10 April 1999)

ABSTRACT

Although lightning rods have long been used to limit damage from lightning, there are currently no Americanstandards for the shape and form of these devices. Following tradition, however, sharp-tipped Franklin rods arewidely installed despite evidence that, on occasion, lightning strikes objects in their vicinity. In recent tests of various tip congurations to determine which were preferentially struck by lightning, several hemisphericallytipped, blunt rods were struck but none of the nearby, sharper rods were hit by lightning.

Measurements of the currents from the tips of lightning rods exposed to strong electric elds under negativelycharged cloud bases show that the emissions consist of periodic ion charge bursts that act to reduce the strengthof the local elds. After a burst of charge, no further emissions occur until that charge has moved away fromthe tip. Laboratory measurements of the emissions from a wide range of electrodes exposed to strong, normal-polarity thunderstorm electric elds show that positive ions are formed and move more readily over sharp-tippedelectrodes than over blunter ones. From these ndings, it appears that the electric eld rates of intensicationover sharp rods must be much greater than those over similarly exposed blunt rods for the initiation of upward-going leaders.

Calculations of the relative strengths of the electric elds above similarly exposed sharp and blunt rods showthat although the elds, prior to any emissions, are much stronger at the tip of a sharp rod, they decrease morerapidly with distance. As a result, at a few centimeters above the tip of a 20-mm-diameter blunt rod, the strengthof the eld is greater than that over an otherwise similar, sharper rod at the same height. Since the eld strengthat the tip of a sharpened rod tends to be limited by the easy formation of ions in the surrounding air, the eldstrengths over blunt rods can be much stronger than those at distances greater than 1 cm over sharper ones.

The results of this study suggest that moderately blunt metal rods (with tip heighttotip radius of curvatureratios of about 680:1) are better lightning strike receptors than are sharper rods or very blunt ones.

1. Introduction

In 1747, Benjamin Franklin observed that he coulddischarge electried objects in his parlor silently by ap-proaching them with a sharp, metal needle in his hand.This discovery led him to suggest that thunderstormssimilarly might be discharged by erecting sharp, metalrods beneath them, thus preventing lightning. When hetried this idea out under a thundercloud, however, oneof his rst rods was struck by lightning. He then pro-posed a different use for the rod; if it did not preventlightning (which, demonstrably, it never does), an ele-vated, grounded rod might provide a preferred path tothe earth for the strikes to a building. However, Franklin(1774) continued to advocate that the tips of his light-ning rods be sharp, a practice that has continued tomodern times. In the years since Franklin invented them,

Corresponding author address: C. B. Moore, Langmuir Laboratoryfor Atmospheric Research, New Mexico Institute of Mining and Tech-nology, Socorro, NM 87801.E-mail: [email protected]

many of his rods have reduced lightning damage tostructures, but it is recognized widely that objects intheir vicinity sometimes are struck and that there stillis no clear understanding of how a rod connects to anapproaching lightning discharge.

As Franklin found, the application of strong electricelds to an exposed, sharp electrode such as a lightningrod causes an electric current to ow into the air; wenow know that this current is a result of ionization pro-cesses in the air around the tip. The space charge formedby the ions created around the tip of a rod, however,acts to weaken the applied electric eld. This weakening

causes a problem for lightning protection effortsbecausevery strong electric elds are required above a lightningrod to establish the conditions necessary for it to connectto approaching lightning. In the early 1930s, Schonlandand Collens (1934) found that an upward-going returnstroke usually is launched from an exposed object inthe path of lightning approaching the earth. The con-ditions required for the conversion of the eld-weak-ening emissions of ions from the air around the tip of a lightning rod into return strokes have not been estab-lished, and at the current time there is no consensus as


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594 VOLUME 39J O U R N A L O F A P P L I E D M E T E O R O L O G Y

to the optimal conguration of a lightning rod. In fact,the current National Fire Protection Association (NFPA)standard for the installation of lightning protection sys-tems, NFPA 780 (1997), does not specify the form orthe shape of a lightning rod.

2. Investigations into lightning rod behavior

In an effort to understand the function of lightningrods, several different approaches were used to inves-tigate their behavior. These investigations, which arediscussed below, included

1) arranging a competition between sharp and bluntrods exposed on a mountain ridge over which thun-derclouds frequently formed to determine whichwould be struck preferentially by lightning,

2) measuring the currents that owed from the earth tothe tips of various lightning rods having differentsizes and shapes,

3) high-speed digitizing and recording of these currentsto selected rods,

4) making laboratory measurements of the dischargesfrom ellipsoidal electrodes,

5) analyzing theoretically the electric elds around el-lipsoidal electrodes prior to discharge onset,

6) examining the point discharge and return-stroke ini-tiation processes, and

7) applying the ndings toward the design of improvedlightning rods.

The efforts thus far have been directed toward a studyof protection against normal-polarity lightning in whicha negatively charged, stepped leader descends from the

negatively charged lower regions of a thundercloud andprovokes the emission of a positively charged return-stroke leader from exposed objects on the earth below.Thus far in the measurements, none of the less frequent,positive lightning discharges such as those that some-times occur in the dissipating stages of a thunderstormand in tornadic storms have been encountered. As aresult, no study of positive lightning has been made.

The eld portions of this study were carried out nearthe 3287-m-high summit of South Baldy Peak in theMagdalena Mountains of central New Mexico. The re-cording instruments used in the study were installed inKiva II, a steel-walled room buried near the summit. Asummary describing each of these investigations fol-

lows.

a. A lightning-reception competition between sharpand blunt lighting rods

During the past ve years, a number of differentlyshaped rods were exposed to the atmospheric electricelds over South Baldy in an effort to determine whichshape of lightning rod would be preferentially struck bylightning produced by the summer thunderstorms. Inthese tests, sharply tipped Franklin rods were exposed,

with blunter rods mounted nearby (with distances rang-ing from 5 to 20 m). Five blunt rods were used in 1994,seven in 1995, and eight in 1996, 1997, and 1998.

1) D ESCRIPTION OF THE LIGHTNING RODINSTALLATIONS

Each of the lightning rods used in this effort wasmounted on a nylon insulator inserted in the top of a6.1-m-high metal mast constructed of nominal 44-mm-diameter electrical conduit. Each rod was connected tothe earth by a copper-wire down conductor in whicheither a 2-A fuse or a lightning ash counter was in-serted to provide indications of any direct lightningstrike. (A subsequent blown fuse was sufcient ev-idence of a strike.)

The Franklin rods used in this study were standard,Underwriters Laboratory, Inc., (UL)approved, light-ning rods obtained from the East Coast Lightning Equip-

ment Co. of Winsted, Connecticut. Most of the rodswere made of aluminum round stock; a few of themwere copper. The rods were 12.7 mm in diameter and305 mm in length. The upper section of each rod hadbeen tapered toward its tip, with the tapers starting about76 mm from the top and continuing until the diameterwas reduced to about 3.5 mm. The top of each rod thenhad been roughly machined into a cone with an includedangle of approximately 45 . Although the conical tipsinitially were very sharp, after exposure to the weatherand to strong electric elds beneath thunderclouds, thesetips generally became somewhat less sharp. The radiiof curvature for several rods observed after exposureduring this study were measured and found to be about

0.1 mm, a value that is used later in some calculations.The blunter rods that were used in this competitionwere made from round stock with heights of 305 mmand with diameters ranging from 9.5 to 51 mm. Mostof the rods were made of aluminum but others made of brass and of stainless steel also were used. The upperends of all the blunter rods were machined such thatthe top became a hemisphere with the same diameteras the parent round stock. After machining, the hemi-spheres and the upper-cylindrical portions of each rodwere polished to a mirror nish to eliminate any pro-trusions that would concentrate the electric elds lo-cally. When these rods were installed on top of the6.1-m-high masts, all of their tips were at heights of 6.4

m above ground level.

2) L IGHTNING STRIKES TO THE COMPETINGLIGHTNING RODS

In 1994, a 19-mm-diameter aluminum, blunt rod ona 6.1-m-high mast located between two similarly mount-ed, sharply tipped, proprietary air terminals wasstruck by lightning initiated aloft by a small rocket tow-ing an ungrounded, 90-m length of ne wire. A dis-charge propagated both above and below the ends of


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MAY 2000 595M O O R E E T A L .

FIG . 1. Photograph of two blunt rods that were struck by lightningwhile they were mounted on top of 6.1-m-high masts. The left-handrod was 12.7 mm in diameter; the right-hand one was 19 mm indiameter.

the wire, creating a large lightning discharge that con-nected to the blunt rod.

During the 1996 summer thunderstorms, four alu-minum rods with blunt tips adjacent to sharper Franklinrods participated in cloud-to-ground discharges at dif-ferent times. Two of these instances were strikes caused

by natural lightning that blew fuses and produced weldmarks on the hemispherical tips of two different 19-mm-diameter rods. The third instance was a dischargefrom another 19-mm-diameter rod to a rocket aloft trail-ing a short length of ungrounded wire. An unaffectedsharp rod was exposed on a similar mast located at adistance of 5.5 m away. The fourth incident involvinga blunt rod was the activation of a strike counter in thedown conductor from a 25.4-mm-diameter rod duringa thunderstorm that occurred when no observers werein the Kiva area.

In 1997, four blunt rods again participated in light-ning discharges that missed the sharp rods nearby. A12.7-mm-diameter aluminum rod was struck twice and

a 19-mm rod was struck once when lightning was ini-tiated by rockets that injected grounded wires into theair beneath thunderclouds; the lightning traveling downthe grounded wires transferred over to the blunt rodsand produced weld marks on their hemispherical tips.A natural strike hit a 12.7-mm-diameter rod on 10 Sep-tember 1997. Sharp-tipped, aluminum Franklin rodswere located within a few meters of the blunt rods butnone of them participated in any of these strikes. Figure1 is a photograph of two of the blunt rods that werestruck during this competition.

b. Measurements of the currents owing to the tips of lightning rods

In this study, eight rods with different congurationswere investigated during three different summer thun-derstorm seasons. Each lightning rod was connected to

the earth through a 10-k resistor and a 2-A fuse by anumber-10 wire. A blown fuse provided proof of a light-ning strike to the rod. The voltage drop across the re-sistor, used to measure the current ows from the earthto the rod, was transferred into the Kiva through a dif-ferential amplier and was recorded digitally at the rateof 5 measurements per second. As may be expected, thegreater currents owed to the rods with the sharper tips.The currents to the UL-approved, sharp-tipped Franklinrods commenced when the ambient electric eld becamestronger than 2 kV m 1 and increased to values of about15 A under eld strengths in excess of 10 kV m 1 .The blunt rods that were exposed also emitted discharg-es under strong elds, but the eld strengths requiredfor the current onsets usually were in excess of 5 kVm 1 ; the emissions under strong electric elds were onthe order of 5 A or less.

Although these currents may seem trivial, the amountof charge that they release into the air around the tip isequivalent to that necessary for canceling the ambientelectric eld over a surface area at the rate of more than100 m 2 s 1 . The effect of the concentrated charges emit-ted around the tip of one of these rods obviously limitsthe strength of the local electric eld.

c. Digitized measurements of the currents owing tolightning rods

1) INSTRUMENTATION

Because the time resolution of the current-measuringrecorder was poor, a high-speed digitizer was obtainedand used to measure the currents owing to the tips of three differently shaped rods during lightning strikesnear Kiva II. The digitizer, an Innovative Integrationmodel PC 31, contained two analog-to-digital channels.The inputs to each of these channels were multiplexedso that the system allowed digitization of four inputsignals, each at a 2-MHz rate. The three lightning rodsused in this study were placed around the periphery of Kiva II and were arranged so as to be at the vertices of an equilateral triangle with legs 5.5-m long. During the

1996 study, one of these rods was a 12.7-mm-diameter,UL-approved Franklin rod with a tip sharpened into a45 cone. The second rod was a 19-mm-diameter, cy-lindrical, brass rod with its upper tip rounded into ahemisphere. The third rod was made of stainless steel,was 50.8 mm in diameter, and its tip again was roundedinto a hemisphere.

The lower end of each of these three rods was ttedwith set screws to capture the upper end of a wire thatwas used as a coaxial down conductor and was passeddown inside the steel conduit mast. Each of these down


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FIG . 2. The circuit diagram for the connection between each lightning rod and the digitizer. (Note: the resistance values are given inohms.)

conductors extended below the mounting socket to aspark-gap box in the space beneath the skirt sur-rounding Kiva II. The purpose of the coaxial conduitsand the spark-gap boxes was to allow sensitive mea-surements of the currents that owed to the tips in theintervals just before a lightning strike to one of the

terminals to be made without suffering major damageto the measuring instrumentation once a strike occurred.From the Kiva-entrance box, the signals were led to avoltage-dividing, 50- termination and a voltage-fol-lower buffer circuit that was connected to the four-chan-nel digitizer. The schematic diagram for the current-measuring circuit is shown in Fig. 2.

The change in ambient electric eld outside Kiva IIwas sensed by an isolated, circular electrode that wasmounted in an inverted housing about 1 m above theearth. This electrode was connected to the earth throughan operational amplier that measured the displacementcurrents caused by changes in the local electric eld.The signal from the eld-change sensor also was ter-

minated in 50 at the input of the fourth buffer. Theoutput from this buffer fed the fourth channel of thedigitizer and a trigger generator consisting of a differ-entiator, a rectier, and a comparator with an adjustablethreshold.

Whenever the time rate of change in the electric eldsignal exceeded the preset threshold, the generator pro-vided a trigger to the digitizer that then stored fourchannels of digitized data for a 1-s period around thetime of the trigger. In an effort to capture a record of the initiation of each discharge, the system was set up

to record about 130 ms of data from each channel priorto the trigger and about 900 ms afterward. The systemthus operated automatically in capturing data on thenearby lightning strikes that produced electric-eld ratesof change greater than the preset threshold.

2) CURRENT MEASUREMENTS

During the late summer of 1996, more than 50 da-tasets were recorded from lightning within about 2 kmof the summit but none of these discharges struck anyof the air terminals or the area directly around the sum-mit. Examples of some of the data collected are shownin Figs. 3, 4, and 5. The currents that were measuredincluded both the displacement currents caused bychanges in the ambient electric elds and the chargeemissions associated with the creation of ions aroundthe tips. As shown by these and other recordings, thesharp Franklin rod emitted strong bursts of positive

charge during the close approach of an initiating steppedleader that descended from the thundercloud overhead.On the other hand, there were no similar pulsed emis-sions from the blunter rods during the approach of thesenegative leaders, although large displacement currentsto the blunt rods were recorded during the initiation of the strikes. The current excursions shown in Fig. 3 forthe 19-mm and the 51-mm rods just prior to the strikeare due largely to displacement currents induced by rap-id changes in the electric eld associated with the ap-proaching lightning.


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MAY 2000 597M O O R E E T A L .

FIG . 3. Plots of the change in the atmospheric electric eld strengthon 7 Sep 1996 at the start of a lightning strike near South Baldy Peak and of the currents that owed to the tips of three lightning rodsexposed on top of 6.1-m-high masts. The height-totip radius ratiofor the Franklin rod was about 64 000:1; the ratio for the 19-mm-diameter rod was 680:1 and that for the 51-mm-diameter rod was250:1. The strength of the electric eld at the tip of the Franklin rodwas calculated to be enhanced over the ambient eld by a factor k eof about 12 250; over the 19-mm rod, k e was calculated to be 230,

and over the 51-mm rod, k e was 102.

FIG . 4. Plots of the change in the atmospheric electric eld at 1730MST 25 Aug 1996 at the start of a lightning strike near South BaldyPeak and of the currents that owed to the tips of two lightning rodsexposed on 6.1-m-high masts.

d. Laboratory studies of discharges from ellipsoidalelectrodes subjected to strong electric elds

In an effort to make quantitative determinations of the relations between the strength of the applied electricelds and the emitted point discharge currents withoutthe perturbations caused by variable winds blowing onthe electrodes, the investigation was transferred to theSocorro laboratory. The laboratory study was limited toan investigation of positive discharges, because mostlightning return strokes develop from the emissions of positive charges from the ground strike point. Further,

we chose to use prolate, semiellipsoidal electrodes forthis part of the study because, as is discussed later (insection 5), the potential function for this shape is known(Smythe 1950, 168169) and the strength of the electricelds around their tips could be calculated.

Four such brass electrodes, with tip radii of 0.0625,0.125, 0.25, and 0.5 mm, respectively, were turned ona computer-controlled lathe. Further, in an effort to ob-tain an electrode with an even smaller tip radius, a steelneedle also was used; its tip was ground and polisheduntil it appeared to be that of a prolate ellipsoid with a

radius of curvature of approximately 0.01 mm, as de-termined with a microscope. The height of each elec-

trode used in this study was 141 mm.Each electrode was placed, in turn, upright on, butisolated with a thin Mylar sheet from, a horizontal, 1-m 2aluminum plate that was connected to the earth. A sec-ond, 1-m 2 parallel plate was mounted on insulators 295mm above the lower plate. The electrode under test wasconnected to the earth through a 9875- resistor; thevoltage drops across this resistor were measured with adigital voltmeter and with a digital oscilloscope to ob-tain information on the current ows from the electrode.Negative voltages V of up to 30 kV then were appliedto the upper plate, creating positive (upwardly directed)electric elds in the space between the plates withstrengths that could be controlled between zero and 100

kV m1

. The electric elds thus created, V applied dividedby the plate separation, are the applied electric eldsthat are discussed below.

A typical plot of the currents emitted from the ellip-soid with the 0.0625-mm-radius tip is shown in Fig. 6in which it can be seen that, above a threshold, thecurrent varies with the square of the applied electriceld strength. This square-law dependence was rst not-ed by Warburg (1899), who found a relation betweenthe voltages V he applied to sharpened electrodes in hislaboratory to create the controlling electric elds and


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MAY 2000 599M O O R E E T A L .

TABLE 1. Characteristics of the I / E vs E data in Fig. 7.

Radius of tip (mm)

Field strength at onsetof steady current

(kV m 1)

Slope of I /[( E 0 E intercept ) E 0]

[fA (V m 21) ]

E intercept with E axis (kV m 1)

Peak frequencyof bursts (s 1)

Approximate minperiod between

bursts (ms)

0.010.06250.1250.250.50

25.4740.2449.0061.3482.23

1.1341.3081.3621.3841.414

24.233.542.253.572.2

2900100

5sporadic

0

0.3510

200

ly. No currents above the detection threshold of 0.01A were detected from any of the electrodes until the

applied eld became very strong. When the eld appliedto the sharp, steel needle exceeded 25 kV m 1 , sporadic,short-duration bursts of current (with peak amplitudesof about 1 A and durations of less than 0.5 s), weredetected by the oscilloscope; these discharges increasedin amplitude and frequency as the eld strength was

increased slowly. The periods between the pulses de-creased to about 0.35 ms and the mean currents attainedlevels of about 0.2 A when the applied eld strengthexceeded 30 kV m 1 . Then, as the eld strength wasincreased still further, the voltage drop across the re-sistor increased, indicating increased current emissions,but the bursts had ceased. When the needle tip wasexamined in the dark, a faint luminosity could be seenaround the tip that indicated that the discharge regimehad changed from the charge-burst mode to theglow regime described by Hermstein (1960).

In the glow regime, ionization of the air occurs con-tinuously at the electrode tip under the inuence of thevery strong, local, electric elds. Electrons liberated

from neutral molecules fall into the tip while the re-sulting positive ions migrate away under the inuenceof the elds without causing further ionization. (Thepositive ions are so massive, about 60 000 times themass of an electron, that they are not accelerated suf-ciently for ionization under the inuence of the electricelds around an electrode in the glow regime.) In thisregime, as the eld strength was increased further, theratio of emitted current to the applied eld strength in-creased linearly with the eld strength excess above theglow-regime onset threshold, as shown in Fig. 7.

The blunter, brass electrodes showed a somewhat sim-ilar behavior with increasing eld strength except thatthe burst regimes occurred over increasingly narrower

ranges of eld strength as the tip radius increased; withthe 0.5-mm-radius electrode, the mean current increasefrom 0.01 to 1.1 A took place with a eld strengthchange of about 1 kV m 1 . The current ows from allof these electrodes became continuous at eld strengthsgreater than the E onset level. Characteristics of the linearplots shown in Fig. 7 are listed in Table 1. The slopefor each of these plots was determined by a least-squarest to the data. Note that no emissions were detected ateld strengths weaker than those at which the linearplots would intercept the E axis in Fig. 7; E intercept for a

given electrode appears to be the strength of the ambientelectric eld below which existing discharges extin-guish. These laboratory measurements suggested thatan examination of the discharge processes would beworthwhile, starting with the earlier studies of positivepoint discharges done by Kip and Loeb.

3. Sudden current onsets and Kips studies of positive point discharges

The sudden onset phenomenon was discovered by Kip(1938, 1939), who also provided an explanation for theinitiation of positive point discharges. He demonstratedthat the onset of positive emissions depended on theavailability of free electrons in the air around the elec-trode tip. When a free electron appeared in the air abovean electrode subjected to a strong electric eld, it wouldbe accelerated toward the tip, liberating electrons fromneutral molecules in its path. These new electrons, inturn, accelerated toward the tip, liberating even moreelectrons and resulting in an avalanche of electrons thatleft the newly electron-decient gas molecules behind

as a column of more slowly moving, positive ions. Kipfound that the free electrons necessary to initiate anelectron avalanche could arise from cosmic rays, fromradioactive emissions in the air, or, more readily, by theaction of strong electric elds, which could extract elec-trons from existing negative ions in the vicinity.

Loeb (1935) earlier had found that negative oxygenions gave up free electrons whenever the strength of thelocal electric eld exceeded 6.8 MV m 1 under a pres-sure of 1 atm and that the critical eld strength variedwith the atmospheric pressure. Kip established that thecritical ratio of eld strength E to air pressure p for theonset of positive discharges was an E / p of 90 V cm 1torr 1 , in the units used by the early investigators into

point discharges.These ndings provide an explanation for the periodicbursts of charge from a sharp lightning rod shown inFig. 4 and for the point discharge current dependenceon the excess of the ambient eld strength above somethreshold value. Both of these features arise because theinitiation of an electron avalanche requires much stron-ger elds than those required to continue the process.The continuing electron avalanches in this phase leavea positive ion space charge in the air around the tip thatincreases until it is approximately equivalent to the dif-


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ference between the initial displacement charge on thetip at the onset and that left on the tip at the time whenthe avalanches cease (because of the weakening of theeld by the space charge accumulation). No furtheremissions occur until this space charge moves away

from the tip; it migrates away under the inuences of the residual electric elds and any wind moving pastthe tip, and as a result of the ions repulsive forces awayfrom each other. After enough of this charge has beenremoved so that it no longer shields the tip, the eldstrength there can rise again above the E / p of 90 V cm 1torr 1 , whereupon a new electron avalanche may occur,and a new burst of shielding positive ions is created. Inthis mode, the electrodeair system under a strong elec-tric eld acts as a relaxation oscillator.

After an electron avalanche is initiated, it makes manyfree electrons available. (During the individual ava-lanche episodes observed at the current onsets shownin Fig. 5, charge bursts on the order of 10 nC were

measured, which indicates that about 6 10 10 electronshad arrived at the tip following the liberation of thesingle, avalanche-initiating, seed electron.) Oncestarted, the avalanche process can continue until, ac-cording to Kip (1938), the strength of the electric eldat the tip is decreased below an E / p ratio of about 30V cm 1 torr 1 (about 22 V m 1 Pa 1 ), equivalent to aeld strength of less than 2.3 MV m 1 at an electrodetip under 1 atm. Harrison and Geballe (1953) later foundthat 30 V cm 1 torr 1 was the level at which ionizationand electron attachment in air become about equallyprobable.

If, however, the ambient eld around the tip has in-tensied rapidly enough, additional electron avalanchescan form in the air ahead of the positive space charge,creating more positive ions, which, in turn, create newavalanches at ever greater distances from the tip. Thisprocess can result in the formation of a positive streamerthat can propagate on its own under the inuence of theambient electric elds. Phelps (1971, 1974) found that,after initiation between parallel plates, a positive dis-charge will propagate in the direction of the eld when-ever the local eld is stronger than 400 kV m 1 (at 1atm pressure). In weaker elds, the streamers he intro-duced quickly extinguished. Allen and Ghaffar (1995)have rened Phelpss measurement by eliminating theeld contribution around the initiating electrode; they

determined the critical eld strength for streamer prop-agation in the regions remote from the source to be 440kV m 1 , which is equivalent to an E / p of 5.8 V cm 1torr 1 (4.3 V m 1 Pa 1 ).

These measurements aid in understanding why thenimbus of light around a point in discharge is limitedto the space within a few millimeters of the tip. Theend of the luminosity marks the region where the eldstrength has decreased below the level required for cre-ating electron avalanches; the luminosity itself occursin the region where avalanche-produced electrons are

exciting the air molecules and recombining with positiveions.

The ndings by Loeb, Phelps, Allen, and others thatthere are critical ratios of electric eld strength to at-mospheric pressure raise questions as to what factorsdetermine a given ratio and about what energies areinvolved. We now turn to a consideration of the sig-nicance of the E / p values that have been reported.

4. The energy equivalents of E / p values

The dependence of the critical electric eld strengthson the atmospheric pressure is determined by the lengthof the mean free path that a charged particle such as anion can travel under the inuence of the electric eldbetween collisions with air molecules. From this per-spective, values of E / p can be considered as measuresof the mean potential difference traversed by an ion inone mean free path. Consider an E / p value of 1 V cm 1

torr1

. Conversion of this value into mks units yields E / p 1 V cm 1 torr 1 0.75 V m 1 (N m 2 ) 1 . (3)

From the gas law, the atmospheric pressure p is a func-tion of the gas molecule concentration n and the Kelvintemperature T ; p is dened by

p nkT, (4)

where k is Boltzmanns constant: 1.38 10 23 J K 1 .Multiplying Eq. (3) through by p gives

0.75nkT V m N 1 . E 1 1@1V cm torr (5)

Now, from Cobine (1958, p. 36), the mean free path Lof a rapidly moving oxygen ion is

1 L , (6)

n

where is the collision cross section for a rapidly mov-ing oxygen ion [about 4.3 10 19 m 2 , calculated fromLoebs data given in Cobine (1958, p. 23)]. The nega-tively charged ions from which electrons may be lib-erated are probably those of oxygen and water vaporbecause they compose most of the negative ion popu-lation in the atmosphere; oxygen and water vapor havea greater afnity for electrons than do nitrogen mole-cules.

Elimination of n between Eqs. (5) and (6) gives

kT 1 E L 0.75 V m N . (7)1 1@1Vcm torr h ighveloci ty ion For an E / p of 1 V cm 1 torr 1 at 293 K (20 C), thetemperature of the air during the AllenGhaffar mea-surements,

Lhigh velocity ion 7.0 mV. E 1 1@1V cm torr (8)At an E / p of 90 V cm 1 torr 1 , an ion moving a

distance of one mean free path between collisions withair molecules under the inuence of the local electric


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MAY 2000 601M O O R E E T A L .

FIG . 8. An example of confocal ellipsoidal and hyperboloidal coor-dinates and .

FIG . 9. A plot of the calculated electric eld lines of force arounda conducting, vertical prolate semiellipsoid exposed to a uniformvertically directed electric eld prior to the onset of any discharges.The height-totip radius ratio for this ellipsoid is 500; k e of the electriceld at the tip of the semiellipsoid has been calculated to be about180.eld would drop across a potential difference of about

0.63 V. In so doing, it could acquire a kinetic energycontribution equal to the potential given up times theionic charge (1.6 10 19 C). This energy transfer canbe written alternatively as that of 0.63 eV (electron volt).Because the energy required for the ionization of a ni-trogen molecule is about 14.5 eV and that for an oxygenmolecule is about 13.6 eV, it is obvious that the energies

required for electron liberation are far less than the en-ergy required to extract an electron from either a nitro-gen or an oxygen molecule. Kip concluded, therefore,that his positive discharges began after electrons wereliberated from negatively charged ions in the vicinityof his electrodes tip at the E / p equivalent of about 0.6eV.

With this information, it becomes of interest to in-vestigate how the strength of the electric eld varieswith distance from the tip of a lightning rod.

5. The variation of electric eld strength aroundthe tip of an air terminal

When a curved conductor is exposed to an externallyuniform electric eld E 0 the eld strength at the con-ductor tip is intensied greatly over that at locations farremoved from the tip. The amount of intensicationrelative to that of the undisturbed eld ( E tip / E 0 ) is knownas the eld enhancement factor k e . It is possible to cal-culate k e for some simple, symmetric conductor shapessuch as a conducting hemispherical boss protrudingabove a conducting plane (for which k e 3.0), butsimple analytic solutions do not exist for vertical sec-tions of cylinders. As a result, the lightning rods of

interest cannot be treated analytically, but their geom-etry can be approximated by simulating a lightning rodwith a conducting, vertical, prolate semiellipsoid (ashape for which an analytic solution does exist) havingthe same radius of curvature for the tip and the sametip height. The coordinate system used is shown in Fig.8, and the calculated electric eld lines of force aroundsuch an ellipsoidal conductor are shown in Fig. 9.

With this assumption of the equivalent shape for theconductor tip, an approximate value for k e can be cal-culated by differentiating Smythes (1950, 168169)prolate semiellipsoid potential function with the use of the proper metrics. This function, in confocal ellipsoidaland hyperboloidal coordinates, is

1 1coth

V E z 1 , (9) 0 1

1 coth 0 0where E 0 is the strength of the uniform, external, axiallydirected electric eld; z is the height of the point of interest above a plane earth; is the location of thepoint of interest in ellipsoidal coordinates; 0 is theellipsoidal coordinate of the surface of the ellipsoid sim-ulating the lightning rod; and

1 11coth log . (10)e 2 1

The calculated ellipsoidal ( E ) and hyperboloidal ( E )


10/17


FIG . 10. Plot of the height-to-tip ratio c:a vs k e at the tip of con-ducting, prolate semiellipsoids exposed to an axially directed, ambientelectric eld prior to the onset of discharges. The dashed rectangleidenties the conguration bounds that are recommended here forlightning rods.

components of the electric eld in the air around thesimulated lightning rod are given by

11coth

2 1 E E 1 0 2 2

11coth 0 0

1 (11)

12 1 ( 1) coth 0 0

and

1 1coth

21 E E 1 . 0 2 2 1

1 coth 0 0(12)

The 0 coordinate for the surface of the prolate ellipsoidis given by

0 [1 (a / c)] 1/2 , (13)

where a is the radius of curvature of the tip, and c isthe length of the semimajor axis of the simulating pro-late semiellipsoid (i.e., the height of the lightning rodtip).

The electric eld at the tip of a semiellipsoid is con-centrated as a result of the curvature and height of theconductor. At the top of the ellipsoid where the hyper-boloidal coordinate equals 1.0 and equals 0 , k e canbe calculated from Eq. (11) by taking the ratio of theeld strength at the tip E tip to that of the undisturbedambient eld E 0 :

1 E 1tip 1k ( 1) coth . (14)e 0 0 0 [ ] E 0 0

A plot of this relation is shown in Fig. 10.To calculate the eld strength at any point outside the

ellipsoid, one rst must determine the focal height of the ellipsoid and the value of at the desired point. Theheight of the focus of the ellipsoid h f is given by

h f c / 0 . (15)

The coordinate for any point outside the ellipsoid isdened by

2 2 2 2 2 2 2 2 2 1/2 x z h [( x z h ) 4h z ] f f f 1/2

2 2h , f (16)

where x is the horizontal distance from the axis of thevertical semiellipsoid. The coordinate can vary from

0 to innity (which denotes a at plane, high abovethe tip of the semiellipsoid). For a location on axis,directly above the ellipsoid, Eq. (16) simplies to

z z . (17)0h c f

The hyperboloidal coordinate varies from 0 to 1; it is

dened as z

(18) h f

Calculation of the strength of the electric eld at variousdistances r above the tips center of curvature relativeto the eld strength at the tip indicates that

E ar (19) E 2r atip

for distances within 50 tip radii from sharp electrodes.In Fig. 11, a plot is shown of the calculated E r versus

r, measured in units of the tip radius of curvature a.

From this gure it can be seen that the strongly enhancedelds around the tip of a sharp electrode decrease rapidlywith distance; the eld strength at a distance of one tipradius from the tip surface has a strength that is one-third of that at the tip. On the other hand, since the eldstrengths scale inversely with the tip radius, they do notdecrease as rapidly above blunter electrodes. This be-havior leads to an interesting result: at distances in ex-cess of 6 mm or so, the enhanced eld in the absenceof ionization around a 10-mm-radius rod is much stron-ger than is the eld above a 0.1-mm-radius rod at the


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MAY 2000 603M O O R E E T A L .

FIG . 11. Plots showing the decrease of the distorted eld strengthwith distance above the tips of two, 6.4-m-high, prolate semi-ellipsoidal electrodes exposed to an axially directed ambient electriceld E 0 prior to the onset of discharges. The eld strength above thesharper electrode is given by E tip /[(2 r / a ) 1] out to distances inexcess of 50 radii.

FIG . 12. Plots of the strengths of the electric elds above the tipsof two blunt semiellipsoids relative to that above the tip of a similarbut sharp semiellipsoid with a 0.1-mm-radius tip, when all were ex-posed to the same, uniform electric eld oriented in the direction of the ellipsoids major axes, prior to the onset of discharges. The semi-major axis length of each semiellipsoid was 6.4 m.

FIG . 13. Plots of the relative strength of the electric eld just abovea sharp semiellipsoid with a 0.1-mm-radius tip compared with thoseabove the tips of two similarly exposed, blunt semiellipsoids whenall were exposed to the same, uniform electric eld oriented in thedirection of the ellipsoids major axes, prior to the onset of discharges.The semimajor axis length of each semiellipsoid was 6.4 m.

same height and exposed to the same ambient electriceld. Illustrations of this result are shown in Figs. 12and 13. This nding is important because, from the ear-lier discussion, the continued propagation of positivestreamers depends on the strength of the electric eld

ahead of the positive ions at the tip. It appears that thestronger elds above blunt rods outside the point dis-charge emission zone favor the propagation of anystreamer that forms.

These calculations for the variation of eld strengthabove the tip of an electrode subjected to a uniformelectric eld were used to estimate the thicknesses of the limited volumes in which electron avalanches occur,that is, Kips (1938) sensitive volumes. These esti-mates from the laboratory measurements are presentedin Table 2.

From Table 2 it can be seen that, at the onset of electron avalanches, the strong electric elds necessaryfor the initial liberation of electrons from negative ions

do not exist at distances greater than 0.25 mm from anyof these tips; Kips sensitive volumes are concentratedright around the electrode tips. We now turn to a studyof the point discharge processes.

6. An examination of the point discharge processes

The dependence of the point discharge currents (instill air) on the square of the applied electric elds, asindicated from the work by Warburg, Chapman, andothers, suggests that two different processes involving


12/17


TABLE 2. Calculations of the thicknesses of the sensitive volumes. Here, N 90 is dened as the distance, in tip-radius-of-curvature units,from the tip center of curvature to the location where the local E / p is 90 V cm 1 torr 1 (i.e., the location where electron avalanches maystart). Also, r 90 is dened as the actual distance from the tip where the local E / p is 90 V cm 1 torr 1 (i.e., the location where electronavalanches may start). This value is the calculated thickness of the region that Kip (1939) dened as the sensitive volume in which freeelectrons are produced around the tip.

Tip radius (mm) c / a k e E onset (kV m1

)

E / ptip

(V cm1

torr1

) N 90 r 90 (mm)0.010.06250.1250.250.50

14 0002 2601 130

565282

3 150634351197112

25.540.249.061.382.2

1 250395267187143

7.42.72.01.51.3

0.060.110.120.140.15

the strength of the ambient eld operate during theseemissions. This supposition is supported by an exami-nation of the charge burst emissions such as those shownin Figs. 3, 4, and 5, which suggest that the amount of positive ion space charge created in each burst is de-pendent on the excess strength of the electric eld above

a threshold and the time required to clear this spacecharge is a function of the ion motions under the inu-ence of the ambient electric eld and wind. Chapman(1967) recognized the dependence of the current emis-sions on the ion velocity and proposed relations similarto Eq. (20) to t his measurements:

I AF h( E 0 E threshold )[( KE 0 ) 2 2 ]1/2 , (20)

where AF is a proportionality coefcient, h is the heightof the electrode, is the permittivity of the air (8.85pF m 1 ), K is the positive ion mobility [nominally about1.4 10 4 m s 1 (V m 1 ) 1 for the conditions in Pilieslaboratory], and is the velocity of the wind past thetip of the electrode. Although Chapman calculated the

AF coefcient values for various electrodes, he presentedno analysis of their variations with electrode size. As aresult, his relation is not directly useful for an analysisof these data. There are also some problems in Eq. (20).Chapman assumed that the ion mobilities were the sameunder strong electric elds as they are in weak, neweather, electric elds, whereas Varneys (1953) mea-surements of the velocities of positively charged nitro-gen ions indicate that their mobilities under strong eldsmay be as little as one-third of the weak-eld values.Further, in his calculations, Chapman used the appliedelectric eld strengths directly and did not consider theenhanced eld strengths that acted on the ions at thetips of his electrodes. Nevertheless, Chapman provided

an explanation for Warburgs relation given in Eq. (1),and, from Chapmans approach, we can propose a sim-plied model of the charge transfer processes during theburst mode.

a. Pulse discharge regime relations

For this model, we suggest that, after a free electronis liberated above the tip of the electrode, electron av-alanches continue, leaving positive ions in their wakeuntil the electric eld caused by these ions so weakens

the local eld that the avalanches cease. The quantityof ion space charge created during such a burst can beestimated from the change in displacement charge onthe tip that occurs as the local eld decreases from thatat avalanche onset to that at cessation.

We now consider a vertical, cylindrical, metal light-

ning rod, formed into a hemisphere at its top, that isexposed to an ambient, uniform, vertical electric eldwith strength of E 0 . The displacement charge Qd in-duced by the electric eld on the hemispherical tip istaken to be

Q d a 2 k e E 0 (21)

where a is the radius of the hemisphere.We invoke Kips (1938) observations that point dis-

charges commence after the strength of the local electriceld at the tip exceeds a critical value E onset and ceasewhen the tip eld strength decreases below a turn-offvalue E cessation . Next, we assume that an upper limit onthe amount of new space charge in the air that surrounds

the tip following an electron avalanche episode is equiv-alent to the change Q d in the displacement charge onthe tip, which is approximated by

Q d a 2 ( E tip E cessation ). (22)

Here, E tip and E cessation cannot be measured directly,but the ambient eld strength, E 0 can be measured, andvalues for k e , the eld enhancement factor at the tip,can be calculated. The product k e E 0 then can be usedfor an estimate of E tip . Similarly, E cessation can be ap-proximated by use of the strength of the ambient ex-ternal electric eld E ex (the ambient eld strength whenpoint discharge activity extinguishes) times k e . (In thecurrent laboratory measurements, it was found that

E intercept was equal to E ex .) With these approximations,Eq. (22) can be rewritten as

Q d a 2 k e ( E 0 E ex ). (23)

In still air, the positive ion space charges will moveradially outward under the inuence of the local electriceld, which decreases with the distance r from the centerof curvature, and also as a result of their repulsion fromeach other. The duration of the interval between pulsesprobably is controlled by the clearing of ions fromaround the tip, because a period of many microseconds


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MAY 2000 605M O O R E E T A L .

TABLE 3. Calculations of the ion clearing timeelectric eld strength product.

Tip radius of curvature (mm) k e

Field strength atonset of steady cur-

rent (kV m 1)

I E ( E E )0 0 ex

[fA (V m 1) 2] E 0 t c (V s m 1)

a E t 0 c

[m 2 (V s) 1] t c a t E onset ( s)

0.01

0.06250.1250.250.50

3150

634351197112

25.5

40.249.061.382.2

1.134

1.3081.3621.3841.414

0.008

0.0530.110.250.55

0.0013

0.001180.001120.001010.00091

0.3

1.32.24.06.7

is required for the ion migrations whereas the electronavalanches occur quite rapidly; many of the measuredavalanches were completed within one microsecond orless.

In this model, the mean current I is dened by thechange Q d in displacement charge on the tip of therod caused by a charge-burst sequence divided by thetime t required for clearing the resulting positive ionsthat shield the tip from the ambient electric eld:

I a 2 k e ( E 0 E ex )/ t. (24)

Because the strength of the electric eld above thetip of the simulated elliptical electrode (in the absenceof any ions) scales in terms of the tip radius of curvature,

t can be dened as

Nat , (25)

where N is the number of tip radii a that the positiveions travel in the interval between point dischargebursts, and is the mean ion velocity, which heredepends on the strength of the local electric eld actingon the ions and on any wind moving past the tip. Thisrelationship leads to

Q d I ak ( E E ) . (26)e 0 ext N

To evaluate Eq. (26), and N, neither of which canbe measured directly, need to be known. Therefore, inthis analysis, the experimentally determined, ion-clear-ing times t are used in Eq. (24). We turn now to aconsideration of the glow regime currents.

b. Glow discharge regime relations

The laboratory experiments discussed in section 2d

showed that, as the applied eld became stronger, thefrequency of bursts increased and the time betweenbursts became so short that the bursts, in effect, becamecontinuous discharges. In the glow regime, the entiretip of the electrode appears covered with a luminositythat indicates that electronic excitation is occurring overthe entire area of the tip under the inuence of the localelectric elds; it suggests that photoionization processesare involved. Therefore the equivalent area for inductionof a near-hemispherical tip can be used with the cal-culated eld enhancement factor k e and the experimen-

tally determined slopes of the I / E versus E plots in Fig.7 to calculate relative ion-clearing times t c and to in-vestigate how they vary with tip radius. Inserting theslope I / E 0 ( E 0 E ex ) from the emissions plot for a givenelectrode into Eq. (24) gives

2 I a k e E , (27)0[ ] E ( E E ) t 0 0 ex cwhich can be used to obtain the product of t c and thestrength of the applied eld E 0 , provided that E 0 is equalto or greater than E onset / k e :

2 a k e E t . (28)0 c I [ ] E ( E E )0 0 ex

The E 0 t c values calculated for the ve electrodes usedin the laboratory measurements are shown in Table 3together with the values for t c calculated for E tip equalto E onset for each electrode. (For these calculations it wasassumed that k e did not change with the onset of ioni-zation; if the k e values had decreased with the creationof ions, the calculated eldclearing time productswould have decreased proportionately.) The ratio a / E 0 t cin Table 3 has mobility units but is not a measure of the true, positive ion mobility. It is of interest, never-theless, that this pseudomobility ratio decreases withincreasing tip radius. From the calculations of E 0 t c forrods at a given height, it appears that the calculatedrelative time t c required for ion clearing (s) varies withthe tip radius a (m) as

1.132940 a1.13t a , (29)c 4 E (3.4 10 ) E 0 0

which suggests that / N for the positive ions mi-

grating under the inuence of the electric elds aroundthese electrodes in still air is approximately equal to 3.4 10 4 [m s 1 (V m 1 ) 1 ] times E 0 . As a result, Eq.

(24) can be rewritten for the still-air, glow regime as

I a 0.87 k e ( E 0 E ex )(3.6 10 4 ) E 0 . (30)

One conclusion that can be drawn from these labo-ratory measurements is that, although ions are emittedin weaker electric elds from sharp-tipped electrodes,they are cleared away much more rapidly than they arearound blunter ones that have the same height and ex-


14/17


posure. The effect this nding would have on lightningprotection can now be considered; we turn now to anexamination of lightning return strokes.

7. The initiation of a return stroke

An interesting situation arises when the ambient elec-tric eld intensies so rapidly that the ion formation andmigration processes are not sufcient to limit thestrength of the eld around the tip. When the electricelds above the positive ions produced by earlier elec-tron avalanches have become so strong (probably great-er than an E / p of 90 V cm 1 torr 1 ) that new electronsare liberated at greater distances, the new avalancheswill create extended columns of positive ions fartherfrom the tip. As Kip (1939) pointed out, such a growthof the positive ion regions outward effectively extendsthe electrode tip in the direction of the eld, along thepositive ion trails. As a result, this process can pro-duce plasma streamers that continue to propagate as longas the eld strengths ahead of their tips are sufcientto produce adequate electron avalanches ahead of thestreamers.

When the intensifying eld is caused by an initiating,negative stepped leader descending from a thunder-cloud, the resulting positive streamer arising from agrounded electrode can intensify and propagate as apositive leader to meet the approaching stepped leader,with the energy required for the ionization coming fromthe ambient electric eld. When the two leaders meet,electrons from successively higher portions of the neg-ative leader will pass down the conducting path towardthe earth, heating the channel and creating the upward-

going luminosity called the return stroke. A consistenttreatment of the processes involved has been given re-cently by Bondiou and Gallimberti (1994).

The formation of a positive leader depends on thedevelopment of a strong electric eld that acts on thepositive ions around the tip of an electrode; it thereforeis worth estimating the rates of eld intensication re-quired to exceed the eld-limiting ion-transfer ratesfrom the air terminals. For this exercise, the time rateof increase in the displacement charge on the terminaltip is given by

dQ d / dt d ( a 2 k e E 0 )/ dt a 2 k e dE 0 / dt. (31)

Stronger electric elds acting on the ions will produce

new electron avalanches ahead of them, thereby creatingmore positive ions farther from the tip. Therefore, fromEqs. (24) and (31), the condition for initiating the prop-agation of a positive streamer in still air may be thatthe eld must intensify faster than it is limited by theion current such that

E E 0 exdE / dt . (32)0 t

Since E 0 intensies greatly during the approach of anegative stepped leader, it eventually dominates the ( E 0

E ex ) term. In this regime, the critical rate of eldintensication for streamer launching, in effect, be-comes

E 0(dE / dt ) , (33)0 critical t

which can be rewritten as

1 dE 10 ; (34) E dt t 0

this equation indicates that, for the initiation of a stream-er, the fractional rate of electric eld intensication mustexceed the rate 1/( t ) at which ions are cleared.

8. Electric eld intensications caused by negativeleaders

At the current time, no measures of the eld inten-sication rates that occur during the initiation of a light-ning strike exist (although direct determinations of theseare possible), but Gallimberti (1979) earlier reportedthat eld intensications in excess of 10 11 V m 1 s 1were required for the formation of long sparks in hislaboratory.

From Coulombs law, one can obtain an estimate of the eld intensication that would be produced by apoint charge Q at a height H descending vertically to-ward a at earth at a velocity Q :

Q Q|dE / dt | . (35)

3 H

Similarly, the strength of the electric eld E 0 on the

earth directly beneath the point charge Q isQ

E . (36)0 22 H

Accordingly, when an elevated point charge descendsabove a conducting plane and induces corona dischargesfrom an exposed tip just above the plane, the conditionthat the fractional rate of ambient electric eld inten-sication exceed the rate at which ions are cleared fromabove the tip is

Q 2 1 dE Q 1Q Q0 . (37)3 2 E dt H 2 H H t 0

Under this criterion for the initiation of a streamerfrom a given lightning rod, the eld-intensication char-acteristic time dened by the ratio H /(2 Q ) must be lessthan the ion-clearing time. From Eq. (37), the fractionalrate of electric eld intensication at the earth below astepped leader at a height of 30 m (the height used inthe 1997 NFPA 780 Lightning Protection Standard asthe nominal minimum distance over which the nalbreakdown of the initial stroke occurs) and descendingat the rate of 3 10 5 m s 1 (Uman 1969; Berger 1977)is about 20 000 per second. It is expected that in this


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MAY 2000 607M O O R E E T A L .

situation successful, upwardly propagating, positive re-turn-stroke leaders would not be launched to meet anegative stepped leader from any lightning rod under-neath for which the ion-clearing time was much lessthan (1/20 000) s (i.e., much less than 50 s). Whenthe relative ion-clearing timetip radius relation givenin Eq. (29) is combined with Eq. (37), the condition forthe initiation of a leader (at the conditions during themeasurements: a pressure of about 0.85 atm and a tem-perature of 290 K) may be that

1.13a H t . (38)

4(3.4 10 ) E 2 0 Q

The preceding exercise can provide an explanation asto why none of the sharpened rods in the test were struck by lightning but several blunter rods in their vicinityhave been struck repeatedly. The calculated ion-clearingtimes for the blunt rods (several milliseconds for tipdiameters greater than 10 mm at the 6.4-m heights) aremuch longer than the 16- s intervals measured for thesharp, UL-approved Franklin rods. Further, the electricelds at heights greater than 1 cm above these bluntrods at the onset of the ionization would become stron-ger than those over the sharp rods because of the blunt-rod geometry and because the locally strong elds atthe tips of the sharp rods were limited by the emissionand migration of the positive ions. As a result, if theelds intensify so rapidly after the onset of ionizationthat electrons are liberated from the air ahead of theresidue of the initially formed positive ions before theyhave time to migrate, new electron avalanches can formfarther from the tip, creating new regions of positivelycharged ions at ever greater distances. Under rapidlystrengthening electric elds, this process can continueuntil it results in the formation of a streamer of positiveions propagating in the direction of the electric eld andbecoming an upward-going, return-stroke leader.

From all of the earlier studies, it appears that theessential requirement for the initiation of a streamer isthat the electric elds ahead of the ions intensify fasterthan the ions can move to counteract the intensication;the essential requirement for the propagation of a leaderis that sufciently strong electric elds for the liberationof free electrons continue to exist ahead of its tip. Aftera leader has become well developed and an appreciablyconductive plasma channel connects it to the electrode

tip, however, the charge carried on the tip of the leadercan create much of the electric eld necessary for itscontinued propagation. When this condition occurs (asit does during the initiation of lightning by the rapidinjection of wires into subcloud regions with strongelectric elds), after the conductive channel has bridgedatmospheric potential differences of several megavolts(Willett et al. 1999), a leader can propagate into regionswhere the strengths of the ambient elds are weakerthan those found to be necessary initially by Phelps(1974) and by Allen and Ghaffar (1995).

9. Discussion of improved lightning rodcongurations

a. Requirements for strike protection against normal-polarity lightning

Benjamin Franklins signicant contribution to light-ning protection was his suggestion for erecting groundedmetal rods to provide preferential paths to the earth forlightning. In retrospect, his method for providing thisprotection has been made less effective than it could beby his urging that the tips of lightning rods be sharpened;we think that his rods would provide better protectionif they were not so effective in limiting the strength of the local electric elds. From the foregoing analysis, therequirements for the preferential formation of an up-ward-going, positive, return-stroke leader from the ex-posed object that is to receive lightning can now belisted. These requirements are that

1) the object be an exposed conductor of electricity thatis connected to the earth;

2) very strong, upwardly directed, electric elds de-velop in the air above the objects tip, with strengthssufcient to liberate electrons from negative ions inthe air, creating electron avalanches and columns of positive ions that accumulate more rapidly than theyare carried away by ion migration and repulsion pro-cesses; and

3) the electric eld in the air just above the accumu-lation of positive ions intensify to strengths greaterthan Loebs 90 V cm 1 torr 1 (6.8 MV m 1 at sealevel) threshold necessary for the liberation of elec-trons from negative ions in air. The resulting positiveleader streamer that develops can propagate to meeta downcoming, negative stepped leader if thestrength of the electric eld ahead of the positive tipcontinues to be above the level required for electronavalanche formation.

To meet requirement 2, it is desirable that the upperportion of a preferred strike object be exposed to theatmospheric eld in such a manner that the eld strengthat the top is increased greatly over the ambient valueeven before the approach of an initiating lightningstepped leader. Most of the lightning strikes to the com-peting, nominally 6.4-m-high lightning rods were tothose with hemispherical tip diameters that ranged be-tween 12.7 and 19 mm, although blunter rods with 25.4-

mm-diameter, hemispherical tips received two strikesduring this study. The tip heighttotip radius of cur-vature ratios c:a for the rods that were struck by light-ning ranged between about 500 and 1000. From theseratios, the useful eld enhancement factors k e were cal-culated, and ranged between about 180 and 300. Thebest estimates at present are that the optimum tip heighttotip radius of curvature ratio is about 680:1 and thatthe eld strength enhancement at the top should be about230-fold greater than the undisturbed, ambient eld. Aplot of the recommended ranges of eld enhancement


16/17


FIG . 14. Plots of the enhancement of the electric elds at the tipsof vertical, conducting, prolate semiellipsoids, prior to the onset of discharges, as functions of the electrode tip height and diameter. Thevertical line on the plot intersects the lines of constant eld enhance-ment to show the range of tip diameters for the 6.4-m-high rods thatwere struck during this experiment.

for various tip heights and diameters is shown in Fig.14. As an example illustrating the use of this chart, theintersections of the vertical line for the 6.4-m tip heightwith the contours of equal eld enhancement dene therange of tip diameters for the rods that were struck bylightning during the experiments.

The preceding analysis suggests that the rate of eldintensication required for the launching of an upward-going positive leader is attained earlier and at lowervalues over blunter rods during the approach of a neg-ative leader than it is over the currently used, sharp-tipped Franklin rods. Note also that, after a streamer islaunched, the eld strengths it would encounter at dis-tances greater than 6 mm or so above a blunt rod aremuch stronger than those over a similarly exposed sharprod. Therefore, during the approach of a negative leader,requirement 2 is met better with moderately blunt light-ning rods than it is with any of the sharpened devicesand wires that have been devised for lightning protec-tion. On the other hand, very blunt rods (i.e., those withsmall height-totip size ratios) are not useful becausepoint discharges from the rod tips are necessary to ini-tiate positive-polarity return strokes, and these discharg-es do not occur readily from objects over which theambient electric elds are not appreciably enhanced. Inour view, either too much or too little electric eld en-hancement can decrease the lightning-protecting utilityof a rod.

b. Protection against the relatively infrequent, positive discharges

We turn now to a brief consideration of protectionagainst positive lightning. It is clear that negative leaderdischarges arising from an elevated lighting rod in re-sponse to approaching positive leaders would behavedifferently than would the positive discharges respond-ing to normal, negative lightning. Under the inuenceof the strong eld created by an approaching positiveleader, electrons from the tip region would migratequickly outward and, on collision with neutral air mol-ecules, would free other electrons, creating relativelyslow but inwardly moving positive ions in their wake.Because the electric elds acting on these fast-movingelectrons must continue to be strong for them to continuethe ionizations, it appears that the geometry of a bluntrod (with its stronger elds at distances of several mil-limeters in comparison with those over very sharp rods)will favor it to become the preferential receptor of pos-itive strikes.

Another consideration is that because the laboratorymeasurements indicate that the ions formed around neg-ative electrodes move faster than the positive ions cre-ated around positive electrodes of the same size andexposure, we expect that sharpened rods will protectthemselves even better against positive polarity strikesthan they do for normal, negative lightning. The ionsaround the negative electrodes clear in about two-thirdsthe time required for the normal-polarity ones, whichsuggests that 50% greater rates of eld intensicationare required for return-stroke emissions during positivestrikes than are required for the normal, negative ones.

As a result, moderately blunt rods may be expected tobe the favored receptors for positive strikes as well asfor the normal, negative discharges. Nevertheless, be-cause no measurements that show how lightning rodsinteract with positive lightning yet exist, a study of theirresponses is needed.

10. Conclusions

We conclude, from this analysis and from the resultsof the lightning strike competition, that moderately bluntFranklin rods with tip heighttotip radius of curvatureratios of about 680:1 (i.e., with electric eld enhance-ments of about 230 fold at their tips) are more likelyto furnish return strokes and therefore to provide betterprotection against lightning than can either very bluntones or the traditional, sharp rods.

Acknowledgments. Many of the ideas discussed hereabout the nature of positive discharges, the effects of ion collisions with air molecules, and the conversion of positive space charges into plasma streamers were cov-ered earlier in lucid and instructive papers, on the con-duction and breakdown processes of gases, by A. F. Kip


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MAY 2000 609M O O R E E T A L .

(1938, 1939), by A. von Hippel (1959), and by I. Gal-limberti (1979).

We thank Charles H. Ackerman (East Coast LightningEquipment Co.), Harold G. Van Sickle (AC LightningSecurity Co.), Patricia L. Robbins (Robbins Lightning,Inc.), Thomas J. Cottle Jr. (Capital Lightning ProtectionCo.), Robert E. Cripe Jr. (Independent Protection Co.,Inc.), Dennis Dillon (Bonded Lightning Protection Sys-tems Co.), Dennis A. Jenkins (Power Technologies Co.),G. Maxwell (Maxwell Lightning Protection of FloridaCo.), Will Priestley (Priestley Lightning Protection Co.),Tony A. Riley (Advanced Lightning Technology Co.),and D. J. Stepka (ERICO) for their generous supportthat made this study possible. We also are deeply in-debted to William P. Winn and to Sandra Kieft for theirmany contributions to this study and to Marx Brook forhis loan of the digitizer. We thank the reviewers for theirinsights and suggestions.

REFERENCES

Allen, N. L., and A. Ghaffar, 1995: The conditions required for thepropagation of a cathode-directed positive streamer in air. J.Phys. D, 28, 331337.

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