A Class of Analytical Functions to Study the Lightning Effects Associated With the Current Front | Lightning | Fourier Transform

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ETEP ETEP Vol. 12, No. 2, March/April 2002 141 A Class of Analytical Functions to Study the Lightning Effects Associated With the Current Front F. Heidler, J. Cvetic´ Abstract The lightning currents are measured on tall structures and by artificially initiated lightning. Additionally they are deduced from the remote electric or magnetic field radiated during the so-called return stroke phase, when the lightning current is flowing to earth. For this purpose return stroke models wer
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   ETEP    ETEP Vol. 12, No. 2, March/April 2002141  A Class of Analytical Functionsto Study the Lightning Effects AssociatedWith the Current Front  F. Heidler, J. Cvetic´ Abstract   The lightning currents are measured on tall structures and by artificially initiated lightning. Additionally they arededuced from the remote electric or magnetic field radiated during the so-called return stroke phase, when thelightning current is flowing to earth. For this purpose return stroke models were developed giving the dependencybetween the field data and the current along the ionized return stroke channel. The most frequently employed mod-els use the so-called channel-base currents, which are commonly expressed by analytical functions to simplify thecalculation procedure. A review is given about a class of analytical functions being very convenient in such cal-culations. The special attention is given to the type of current function most frequently used in lightning researchand standardization. With these functions the fine structure in the current front can be studied, which is of basicinterest with respect to the lightning protection.   1Introduction   The lightning currents are preferably studied on el-evated structures due to increasing probability of thestrike with the height [1]. In former lightning researchthey were measured with magnetic links [2] installedon various locations as power lines, masts, chimneysand high buildings (e.g. [1, 3, 4, 5, 6]). Because onlythe current peak proportional to the maximum magnet-ic field strength [2] is captured with this method, now-adays oscilloscopes are mainly employed for the re-cording of the current waveform. One of the first im-portant experiments was carried out on the EmpireState Building in New York City, USA [7]. Similar ex-periments were performed on a 60 m high mast inSouth Africa [8] and in Japan [9], where the currentswere measured in winter thunderstorms. In Russia evencaptive balloons connected with ground by a steel wire[10] were used.The majority of the currents, however, was meas-ured on tall telecommunication towers. The well-known experiments were carried out on two 40 m hightelecommunication towers in Italy [11], on a 248 mhigh telecommunication tower near St. Chrischona,Switzerland [12] and on a 160 m high telecommunica-tion tower located on the mountain Peissenberg nearMunich, Germany [13, 14]. The highest towers werethe 540 m high Ostankino tower in Moscow, Russia[15] and the 553 m high Toronto Canadian NationalTower, Canada [16]. The most important current data,however, stem from the experiments of Prof.  Berger    ,who had been recording the lightning currents duringabout 30 years on a telecommunication tower situatedon the mountain San Salvatore near Lugano, Switzer-land ([17, 18, 19]).Based on the current measurements and on the ob-servation of the lightning channel two different types of lightning to ground were identified, namely the cloud-to-ground lightning and the ground-to-cloud lightning(e.g. [7, 8, 18, 20, 21]). In the case of cloud-to-groundlightning the discharge process starts with a downwardgoing leader from the positive or negative charge cent-ers inside the thunder cloud [22]. In the striking point,however, the first current component is always associ-ated with an impulse current, which may be followedby a continuing current and individual impulse cur-rents, respectively. The cloud-to-ground lightning arecharacteristic for the flat country and small structuresup to a height of several tens of meters.On the other hand the ground-to-cloud lightningare typical for structures exceeding 60 m. Preferablythey start from the top, when the electric field is highenough to initiate an upward going leader. In the strik-ing point the leader is associated with a continuing cur-rent, which may be superimposed or succeeded by in-dividual impulse currents [14]. In comparison to thecloud-to-ground lightning the impulse currents of theground-to-cloud lightning are less severe, certainlywith respect to the impulse charge, the current peak andthe maximum current steepness [11]. Thus, for the pro-tection of buildings equipped with the sensitive electri-cal and electronic systems, the currents of cloud-to-ground lightning have primarily to be considered.Because the existing current data were mostly ac-cumulated on very high structures, they were chieflysrcinated from ground-to-cloud lightning. Only onrelatively small towers as on the two 40 m high tele-communication towers in Italy [11] and on the telecom-munication tower on the Monte San Salvatore (70 mincluding the lightning rod) [18, 21] a higher number of cloud-to-ground lightning could be measured. Essen-tially based on the measurements on the Monte SanSalvatore [23] the lightning parameters are fixed in var-ious standards as in the international standard IEC   ETEP    142ETEP Vol. 12, No. 2, March/April 2002   61312-1 [24]. The data base, however, is relativelypoor and limited to about 25 positive strokes, about 100negative first strokes and about 130 negative subse-quent strokes [23]. Furthermore the front of the fast ris-ing currents often could not be resolved sufficiently[18]. In addition it is distorted by current reflectionsdepending on the tower height [12, 13].Concerning the lightning protection the currentfront is one main subject of interest, because its high-frequency content is chiefly responsible for the coupling of over-voltages and disturbing currents intoelectrical circuits and electronic systems. Fig. 1   showsthe current rise proposed in [23] for negative cloud-to-ground lightning. The wave shape is concave,where the current steepness is continuously increas-ing up to the maximum current steepness (TANG) lo-cated at the 90%-value of the current peak. On theother hand, referring to comparable experiments in It-aly [11] such a concave wave shape was only foundfor the first negative strokes, while the currents of thenegative subsequent strokes were missing the initialslow rise portion. They were immediately startingwith a fast rise resulting in a convex waveform of thecurrent front.In a second method the lightning were artificiallytriggered by rockets pulling up a metal wire from theearth toward the thundercloud. The metal wire, how-ever, acts like a very high object, where the dischargeprocess is initiated by an upward going leader. Similarto the ground-to-cloud lightning the first current com-ponent is given by a continuing current, which may besuperimposed or succeeded by individual impulse cur-rents [25]. Thus the triggered lightning are much morelike the ground-to-cloud lightning.Trying to avoid this shortcoming in the newest ex-periments of the initiated lightning the so-called alti-tude triggered method is used [40]. The rocket pulls upa part of the metal wire a few hundreds of meters long(not connected to the ground). The rest of the wire ismade of a kevlar (insulator). The purpose of the wire isto stimulate the lightning discharge to occur and todirect it toward the ground. The lightning dischargebetween the lower end of the wire and the ground isconsidered to be more or less similar to the naturallightning.The disadvantages of the mentioned methods couldbe avoided, if the lightning currents are deduced fromthe remote electric or magnetic field: With this methodthe lightning to high structures can easily be excludedand the data of a high number of cloud-to-ground light-ning can be accumulated within a relatively short peri-od of time. For this purpose different return strokemodels were developed, providing the dependency be-tween the field data and the current along the returnstroke channel. From CIGRE Working Group 33.01[26] various return stroke models were tested and final-ly the following models were proposed for the calcula-tion purpose, namely the  Bruce-Golde   (BG) model[27], the transmission line model (TL) [28], the modi-fied transmission line model (MTL) [29], the travelingcurrent source model (TCS) [30] and the  Diendorfer-Uman   model (DU) [31].All of these models use the so-called channel-basecurrent, where the current in the striking point is a re-quired model parameter, while the electric and magnet-ic fields are an issue of the calculation process. The di-rect evaluation of the current from the field data istherefore impossible except of far distant lightning,where the use of far distant field approaches simplifiesthe calculation algorithm. Such approaches are avail-able for the DU-model [32], for the TCS-model [33]and for the TL-model [34]. In general, however, thecurrent is evaluated with an iterative process, where thecurrent waveform is varied as long as the calculatedand measured fields agree sufficiently. For this appli-cation analytical current functions are preferred allow-ing an easy variation of the current waveform.In conclusion, analytical current functions areneeded to simplify the evaluation of the current param-eters from the field data [35]. Besides they are also usedin standardization [24] and in simulation models, e.g.in computer codes to calculate the voltages and cur-rents coupled into cables and lines by lightning (e.g.[36]). As mentioned above one important source of in-terference is the high-frequency content of the light-ning current. Concerning this requirement in the furthertext a review is given about a class of analytical currentfunctions, which especially allows to model the finestructure during the current rise. 2 Basic features of the lightning currentfunctions   2.1 Requirement of the analytical representationof the lightning current   In the calculation with the return stroke models thecurrent as well as the current steepness and the chargeare needed at each instant of time and in each point of the ionized return stroke channel [34]. For the consid-ered return stroke models using channel-base currentthis requirement includes that the current i   (   t    ), the cur-rent steepness d   i    /d   t    and the charge Q   =  ∫    i   · d   t    are need-ed in the striking point. If the field derivatives are in-vestigated, the second time-derivation of the current,d   2   i    /d   t    2   , is required, too. Therefore, a current functionshould be able to be differentiated at least twice without TANG  I  90  I  100 i kA  I  TRIG 0 20 40 µ s t  Fig. 1. Typical current rise of a negative cloud-to-groundlightning adopted from [23]   ETEP    ETEP Vol. 12, No. 2, March/April 2002143   any discontinuity. Especially, the first time-derivationis not allowed to have a discontinuity at the instant of time t    = 0 and (d   i    /d   t    )   t = 0   must be equal to zero. Thiscondition is not fulfilled in case of the double exponen-tial current function, given bywith the following correction coefficient of the currentpeak  i   max   :withDue to the discontinuity of the first time derivativeat t    = 0, several modifications of eq. (1) are proposed toreduce the slope of the current at this instant of time[37, 38]. The condition (d   i    /d   t    )   t = 0   = 0, however, is notfulfilled by any of these functions.The electric field of lightning can be separated intoa near distant field component, an intermediate distantfield component and a far distant field component. Thenear distant field component is determined by thecharge, the intermediate distant field component by thecurrent and the far distant field component by the cur-rent derivative [34]. Concerning lightning protectionthe maxima of these parameters are taken into account,namely the maximum current derivative (d   i    /d   t    )   max   , thecurrent peak  i   max   and the total charge Q   =  ∫    i   max   · d   t    [39]. To investigate the influences of these parametersindependently a current function should allow theirseparate variation.From the experimental results it can be concluded,that an analytical current function should additionallyallow to vary the location of the maximum currentsteepness in a wide range. Even the location of (d   i    /d   t    )   max   at the 90%-current value should be possibleas proposed in [23] ( Fig. 1).   2.2Deduction of the current function In the following it is assumed that the current is start-ing to flow at t    = 0. Before this time the current is con-sidered to be zero: i   = 0, for t    ≤   0. To enable a separatevariation of the total charge, the maximum current steep-ness and the current peak  i   max   a rise function  x   (   t    ) and adecay function  y   (   t    ) are defined as follows [33, 41]:   i   = i   max   ·  x   (   t    ) ·  y   (   t    )(3)The concept considers, that the rise function  x   (   t    )determines only the current rise and the decay function    y   (   t    ) only the current decay. A decoupling between thetwo functions is achieved, if during the current rise thedecay function becomes  y   (   t    ) ≈   1 and if during the cur-rent decay the rise function becomes  x   (   t    ) ≈   1. Becausedue to the lightning protection the fine structure in thedecaying current is of minor interest, generally an ex-ponential decay is taken into account. For the decayfunction one obtains:The exponential decay function fulfills the condi-tion of the current rise:  y   (   t    ) ≈   1. The fine structure in thecurrent rise is modeled by the following class of func-tions [33]:During the current decay the function  x   (   t    ) is ap-proximately equal to 1, if the exponent n   is chosen highenough. The term (   t     /    T    )   n   determines the change fromthe current rise to the current decay at the instant of time t    ≈   T    . With the correction factor for the currentpeak, η   , the expression of the current function finallyfollows to:Due to (   t     /    T    )   n   = 0 for t    = 0 only the ratio  f    (   t    )/    g   (   t    )determines the beginning of the current rise. A lot of various functions as trigonometric or polynomial func-tions can be applied successfully to  f    (   t    ) and g   (   t    ). With arelatively high exponent n   a very good decouplingbetween the current rise and the current decay can beachieved. On the other hand a lower exponent may beadvantageous to make this change smoother.   2.3 Application of analytical unit step function   From eq. (6) one obtains a special type of functionassuming g   (   t    ) = 1. Eq. (6) becomes:Because the first term of eq. (7) determines the cur-rent rise only, the decay function is approximated by    y   = 1 for this term. Hence it follows [33]:with the functions ii t t  = −     − −           max exp exp , ( ) η τ τ  1 2 1 ητ τ  = −     − −      exp exp , ( ) max max t t  1 2 2a t  max ln . ( ) =−      τ τ τ τ τ τ  1 21 212 2b  y t t  ( ) = −     exp . ( ) τ  4  x t  f t t T g t t T  nn ( ) = ( ) +     ( ) +     . ( )5 i t if t t T g t t T  y t  nn ( ) = ( ) +     ( ) +    ⋅ ( ) max . ( ) η 6 i t i f t t T  y t t T t T  y t  nnn ( ) = ( ) +    ⋅ ( ) +    +    ⋅ ( )  max . ( ) η 1 17 i t i A t f t B t y t  ( ) = ( ) ⋅ ( ) + ( ) ⋅ ( ) [ ] max , ( ) η 8a  A t t T  B t t T t T  nnn ( ) =+     ( )     +     11 18; . ( )b  ETEP  144ETEP Vol. 12, No. 2, March/April 2002 For a high value of the exponent n the functions  A ( t  ) and  B ( t  ) result in  A   → 1 and  B   → 0, for t  < T  ; for t  > T  they become  A   → 0 and  B   → 1. Therefore, theycan be interpreted as analytical formulations of unitstep functions switching off   f  ( t  ) and switching on  y ( t  )at the instant of time t  = T  . A lot of various functionscan successfully be applied to  f  ( t  ) and  y ( t  ). For asmooth change from the current front to the currentdecay, however, both functions should have approxi-mately the same value at the switching time T  :  f  ( t  = T  ) ≈    y ( t  = T  ). Fig. 2 shows an example, where the current rise isdominated by the function  f  ( t  ), which is identical to thesquare of the sine function. The maximum currentsteepness is approximately located at the 55%-currentlevel [33]. The front duration results in T  1 = 10 µs andthe time to half a value in T  2 = 350 µs. Considering thecurrent peak  i max = 100 kA this 10/350 µs-currentwaveform is related to the first stroke of the standardIEC 61312-1, protection level III-IV [24]. 2.4 Application of power functions With regard to Fig. 1 it should be possible to varythe location of (d i  /d t  ) max between the 0%- and the 90%-current levels. For this purpose in [33, 42] the use of power functions is proposed in combination with theexponential decay function of eq. (4). With the expo-nents k  i , m i < n this type of function is given by:To avoid the discontinuity at the instant of time t  =0 the coefficients k  i , m i should be greater than 1. Light-ning currents with a concave current rise (Fig. 1) areachieved with the following simplification considering k  < n [33, 42]: Fig. 3 shows two examples both representing a10/350 µs-current waveform analogously to Fig. 2. Theinitial slow rise of the concave current front is associ-ated with a relatively low value of  k  . For the current of Fig. 3a the maximum current steepness is located at the70%-current level. If the exponent n increases, themaximum current steepness also increases and isplaced closer to the current peak. For the consideredexponent n = 60 (Fig. 3b) the maximum current steep-ness is located at the 90%-current level as proposed in[23] (Fig. 1). 2.5Lightning currents with two different riseportions In [43] it is reported, that the front of the lightningradiated fields is sometimes not increasing continuous-ly, but in two distinct rise portions. One obtains such afield waveform considering a current showing also tworise portions [43]. This kind of current is achieved withthe following adaptation of eq. (9): 50 i kA0 20 40 µ s t  1000100 %90 %10 % T  1 Fig. 2. Lightning current based on eq. (8) with the currentpeak  i max = 100 kA and the coefficients η = 0.981, n = 10,and T  = 13 µs. The functions are chosen to  y = exp(  –t   /  τ  ),with τ  = 485 µs and  f  ( t  ) = sin 2 ( ω  t  ), with ω  = 1.1 · 10 5 (1/s) i t iat T at T t T b bt T bt T t T t  k k nm m n ( ) =    +    + +    +    +    + +    −     max exp . ( ) ητ  1 20 1 2 1 21 2 9 LL i t iat T t T t T t  k nn ( ) =    +    +    −     max exp . ( ) η τ  110 50 i kA0 20 40 µ s t  1000 a) 50 i kA0 20 40 µ s t  1000 b) Fig. 3. Lightning currents based on eq. (10) with the currentpeak  i max = 100 kA. The coefficients are given by:a) η = 1.04, a = 1, k  = 2.2, n = 10, T  = 13 µs, τ  = 485 µs;b) η = 0.983, a = 1, k  = 2.5, n = 60, T  = 14 µs, τ  = 485 µs i t i X t c Y t t  ( ) = ( ) + − ( ) ( ) [ ] −     max exp , ( ) η τ  1 11a
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