The Lightning Electromagnetic Pulse (LEMP) Breakdown Current Reconstruction by the Leader

and Return Stroke TEM Multi-fractal Model Nan Wang

Abstract—This paper presents a uniform model for the LEMP

breakdown current normalized waveform reconstruction based on the lightning TEM multi-fractal model. Firstly, the length of the lightning discharging channel is estimated by the lightning engineering model and the coaxial transmission line (TL) model. Secondly, the lightning length model and the TEM multi-fractal model are applied to the lightning electromagnetic pulse (LEMP) breakdown current waveform reconstruction with the first and subsequent return strokes. Finally, it is found that the LEMP breakdown current waveform can be taken as an envelope of a sum of a series of superposed leaders or strokes sub-discharging breakdown currents with the overlapped life time.

Index Terms—Lightning protection, LEMP, return strikes, stepped leaders.

I. INTRODUCTION

HE lightning electromagnetic pulse (LEMP) has been known as a great hazard of the lightning electromagnetic

environment effect (E3) to the sensitive digital equipment. Indeed, not only the LEMP by return stroke, but also the LEMP originating in the preliminary breakdown in ground flashes and cloud flashes are responsible for the damages [1].

Many researches and attempts have been focused on the LEMP current for it is critical to the LEMP electromagnetic (EM) field calculation and the LEMP mechanism study. Several engineering return-stroke current models with the different specified channel-base current decay function and velocity assumptions have been proposed in the past years, such as the BG (Bruce and Golde,1941), the transmission line (TL), the MULS (Master, Uman, Lin and Standler), the traveling current source (TCS,Heidler,1985), the modified TL (MTL etc, Nucci MTLE and Rakov MTLL,1988), the DU (Deinodorfer and Uman), the MDU (Thottappilli), Cooray model (1993), the VDTC (Thottappilli and Uman,1994) and AT (Moini and Rakov, 2000) model, etc. The breakdown current in the inner hot corona core of such models are well-fitted, respectively [2][3].

According to the IEC-62305 and China GB50057-2004, the recommended LEMP current waveform of the first stroke and the subsequent stroke are 10/350μs and 0.25/100μs articulated by Heidler double-exponential function as follows [4]-[6]

( )( )

10

21

( ) exp1

n

n

tI ti ttτ

η ττ

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟

+ ⎝ ⎠⎣ ⎦ (1)

Author is with the Beijing Institute of Tracking and Telecommunications Technology, China PR. P.O.BOX 5131-6# (email:wangnan7869@163.com).

where I0 is the current peak value, τ1 is the rise time constant, τ2 is the decay time constant and η is the current correction factor, which are all listed in table I.

TABEL.I THE PARAMETERS OF THE FORMULA ABOVE WITH N=10 Parameters First stroke Second stroke

I0(kA) 200 50 η 0.93 0.993

τ1(μs) 19.0 0.454 τ2(μs) 485 143

The channel-based current in IEC-61024 is obtained by [2] 2

3 4

2

1 2

(0, ) 1 exp exp

1 exp exp

BD

C

I t ti t

I t tk k

χ τ τ

ξ

⎡ ⎤⎛ ⎞ ⎛ ⎞= − − −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞+ − − −⎢ ⎥⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎣ ⎦

(2)

where IBD and IC are the breakdown current and corona current, respectively. χ and ξ are the breakdown current and corona current modifying factor, respectively, which are given as follows

( ) ( ) 3 42 /4 3 4 3 3 42 / 2 / 2

τ τχ τ τ τ τ τ τ= + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ (3)

( ) ( ) 1 22 /2 1 2 1 1 22 / 2 / 2

k kk k k k k kξ = + +⎡ ⎤ ⎡ ⎤⎣ ⎦ ⎣ ⎦ (4)

The former part of the channel-base current i(0,t) formula is the breakdown current for BG, TL, MTL, MULS, TCS and DU models, whereas the latter part is the corona current part of the DU model [2].

The fitted parameters for DU and other models are suggested in table II and table III, respectively [2].

TABEL.II THE PARAMETERS FOR DU MODEL [2] Parameters First stroke Second stroke

IBD(kA) 163.8 50.32 IC(kA) 212.3 53.70 τ3(μs) 3.859 0.0963 τ4(μs) 76.44 25.77 k1(μs) 80.20 26.02 k2(μs) 480.1 142.7

TABEL.III THE PARAMETERS FOR BG, TL, MTL, MULS, TCS MEDOL [2] Parameters BG,TCS,TL,MTL MULS

IBD(kA) 11 7.3524 τ3(μs) 0.0588 0.0426 τ4(μs) 5.2124 2.0324

T

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II. THEORY

A. The length of the stepped leader or return stroke

The leader channel consists of a thin core surrounded by a radially formed corona sheath [7]. This scenario is same to the return stroke, which is the basis of the TEM multi-fractal model. Set the radius of inner discharge core as b, and the radius of the outer corona sheath around as a. Given the length of the leaders or return stroke discharge core channel as h, then the capacitance and inductance by the unit length of the lightning corona sheath can be obtained by [8]

0 02 / ln hCa

πε ⎛ ⎞= ⎜ ⎟⎝ ⎠

, 00 ln

2hLb

μπ

= (5)

where μ0 and ε0 are the vacuum permeability and permittivity, respectively. The capacitance and inductance by the unit length coaxial transmission line are obtained by

0 2 / ln aCb

πε ⎛ ⎞= ⎜ ⎟⎝ ⎠

, 0 ln

2aLb

μπ

= (6)

where μ is the permeability, ε is the permittivity. According to (5) and (6), we can get

babh

r lnlnlnln

−−

=μ , ahba

r lnlnlnln

−−

=ε (7)

The TEM wave lightning characteristic impedance by the coaxial TL model is given by

02

0

1 1 lnLr r

aZb

με ε ε

= + (8)

And the coaxial TL characteristic impedance is given by

0 0

0 0

ln lnLL h hZC a b

με

= = (9)

Therefore, by (8) and (9), the length of the lightning can be obtained as follows

bahrr

lg1lg11lgεε

−⎟⎟⎠

⎞⎜⎜⎝

⎛+= (10)

B. The velocity of the lightning channel growth

The mean time of the lightning sub-discharge process is /t h v= (11)

The return stroke velocity is regarded as 3×108 m/s by TCS. And the return stroke velocity is equal to the stroke current wave velocity by TL, MTLE and MTLL. For MTL, a return-stroke velocity of 1.3×108 m/s has been assumed and the channel height has been fixed at 9 km. And the typical velocity of the stepped leader and the return stroke are 1.5×105 m/s and 1.5×108 m/s, respectively [2][3][9][10].

C. The coaxial TL lightning TEM multi-fractal model

When b<<1, the TEM mode characteristic impedance ZL

of lightning as follows [11]

( )ln( 0.5ln 1 2 )60 22 ln2 ln(1/ )L

r

DZ bD b

π κ

ε

⎛ ⎞− −⎜ ⎟= − +⎜ ⎟⎝ ⎠

(12)

where εr is the relative permittivity of the medium of the lightning corona sheath, D is the 2D-Hausdroff dimension of the lightning and the coefficient κ is given by[12]

( ) ( )( ) ( )5.00cossin25.02/

0

22 ≤≤−−= ∫ κθθθθθκπ

dRR TETM (13)

where RTM(θ) and RTE(θ) are the reflection coefficients for transverse electric and transverse magnetic waves, θ is the incident angle. Note κ is not an absorption coefficient since κ goes to 0.5 for perfectly absorbing walls.

When 0<b<<1, the radius of the lightning outer corona sheath and the inner main discharge channel yield[11]

( )2

120.5 ln 1 2D

Da bπ κ−⎡ ⎤= − −⎣ ⎦

(14)

( ) ( )3 1lg lg 0.5ln 1 2 0 22

b DD D

π κ⎛ ⎞ ⎡ ⎤= − − − < <⎜ ⎟ ⎣ ⎦−⎝ ⎠ (15)

And the maximum value of radius b is obtained by

( )max3(1 3) 1 3lg lg 0.5ln 1 2

22 3b π κ

⎛ ⎞+ + ⎡ ⎤= − + − −⎜ ⎟⎜ ⎟ ⎣ ⎦⎝ ⎠ (16)

where the Hausdorff dimension can be expressed by [11]

( )( ) ( )( )3

lg21ln5.0lg

2lg

21ln5.0lg1

2

−⎥⎥⎦

⎤

⎢⎢⎣

⎡ −−++

−−+=

bbD

κπκπ (17)

C. The normalized LEMP breakdown current In the channel bottom, the charge corresponding to the

corona current compared to that of breakdown current can be neglected [13]. By the charge conservation law, the charge transfer at the lightning striking point is given by [14]

0 2 / BDQ I I tτ η= = (18)

where the return stroke breakdown current wave traverses the leader-channel core and serves to bring it to ground potential. As a result, the leader charge stored in the corona sheath collapses into the channel core and is transferred to ground [15][16].

Since the breakdown current duration is much shorter than that of the corona current, τ1<<τ2 [14], the total energy of the lightning is obtained by

20 2 /BD Lenergy UQ I Z t Pτ η= = = (18)

which can be taken as a constant in the breakdown current duration. The normalized breakdown current is obtained by

( )( )min

max

LBD

BD L

Z tII Z t

= (19)

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III. THE LEMP BREAKDOWN CURRENT WAVEFORM REPRODUCTION AND MECHANISM DISCUSSIONS

The stepped leaders and return strokes quantities of the LEMP breakdown current reconstruction are listed below. TABEL.IV THE PARAMETERS FOR THE LEMP CURRENT RECONSTRUCTION

Quantity Stepped leaders

First return stroke

Subsequent return stroke

Κ 0.2 0.46 0.40 v(m/s) 1.5×105 5×107 1.0×108 b(mm) 0.5~48 0.5~2.5 0.5~6 εr 5~36 4~25 2.5~20

A. The stepped leaders

The length of the stepped leaders is shown in Fig.1.

Fig. 1 Length of stepped leader and radius of inner discharge channel

The lengths of the stepped leaders are consistent with the observed value from 10m to 200m by natural or triggered lightning stepped leaders [7][10].

B. The first stroke

The length of the first stroke is shown in Fig.2.

Fig. 2 Length of stepped leader and radius of inner discharge channel

It is noticeable that the length of the lightning is always longer than the height of the flashes. And it is reasonable to conclude that the larger the Hausdorff fractal dimensions of the lightning sub-processes, the larger the gaps among them. The normalized breakdown current of the first stroke is compared and shown in Fig.3.

Fig. 3 Normalized Current and Time of first stroke

By this figure, it is found that the first stroke breakdown current waveform is well fitted along the DU breakdown current, which can be taken as an envelope of a series of superposed sub-discharge breakdown currents with the listed permittivity and overlapped life time.

C. The Subsequent stroke

The length of the subsequent stroke with the inner thermal corona discharge channel variation is shown in Fig.4.

Fig. 4 Length of subsequent stroke and radius of inner discharge channel

The normalized subsequent stroke breakdown current waveform is compared and shown as Fig.5. The breakdown current waveform is well along with the BG/TL/MTL/TCS breakdown current when the permittivity is not less than 5,

249 978-1-4673-0029-2/12/$26.00 ©2011 IEEE

since the subsequent stroke velocity is set to be constant in the reconstruction, which should slow down with the height growth [16][17]. The breakdown current reconstruction is well fitted along with the DU model breakdown current when the permittivity is not more than 5.

Fig. 5 Normalized Current and Time of second stroke dart leaders

The corresponding maximum length of the subsequent stroke inner channel is about 20km of permittivity 5 by fig4. Consequently, there is a possible and typical scenario when the subsequent stroke height below about 20km, the neutralization of the breakdown current is not the majority until above 20km.

D. The κ discussion

The value variation of κ in the model has a great impact on the length of the leader or return stroke. Thus, the LEMP rise time and the sub-discharges life time is sensitive. Thus, κ can be taken as a significant characteristic quantity on the various lightning processes. The κ of the stepped leaders, the first stroke and subsequent stroke are 0.2, 0.46 and 0.40 by fig IV, respectively. The physical background of 2κ is the absorbing coefficient of a giant EM manifold enclosure consists of all the flashes and its EM field radiation wave-fronts. The lightning EM manifold likes a chamber with an oscillating charge-discharge generator. The κ of the dart leader just before the subsequent stroke may be relatively small than stepped leaders, since the length of the dart leaders are almost shorter. Therefore, the value figure of κ is a zigzag and the bigger κ means a quasi anechoic chamber for the return stroke and the relative smaller κ means a quasi reverberation chamber for the stepped leaders with bigger Hausdorff dimension and lower voltages.

IV. CONCLUSIONS

The LEMP breakdown current waveform can be regarded as an envelope of a series of the superposed flashes currents with the overlapped various sub-discharges life time. The

DU is a better model for first stroke breakdown current and the BG/TL/MTL/TCS models are better for the subsequent stroke breakdown current by the TEM multi-fractal model.

ACKNOWLEDGMENT

Thank my family in Wuhan city of central China for their support on my EMC research in Beijing.

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