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Electric fields induced by a modified double exponential current waveform
Journal of Electrostatics 70 (2012) 152e156
Contents lists available at SciVerse ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Electric fields induced by a modified double exponential current waveform Scott L. Meredith a, *, Susan K. Earles a, Carlos E. Otero b a b
Department of Electrical and Computer Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA Department of Mathematics & Computer Science, University of Virginia’s College at Wise, Wise, VA 24293, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 May 2011 Accepted 11 December 2011 Available online 22 December 2011
This paper presents a novel approach for developing a closed form solution for a lightning generated electric field which uses a modified double exponential base-current waveform. In the literature, several models have employed the square pulse waveform to derive the corresponding electromagnetic fields. However, given the Department of Defense (DoD) has incorporated the double exponential current waveform as part of their “Electromagnetic Environmental Effects Requirements For Systems”, it is noteworthy to develop a solution for the electric field which utilizes this waveform. In order to facilitate the integration required for deriving the field, Taylor series expansion will be used for all variable dependent exponential terms within the current waveform. This methodology greatly simplifies the integration required to solve for the electric field in an approximated closed form. However, by utilizing Taylor series to approximate the variable dependent exponentials, the electric field’s accuracy becomes a function of the number of terms used. Consequently, a correction factor will need to be added in order to help reshape the electric field which resulted from the modified base-current waveform. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Lightning Double exponential Electric fields Indirect effect
1. Introduction Lightning can be regarded as a very complex and mysterious electrical discharge which occurs during the interaction between charges of the opposite polarity in and around thunderstorms. The exact physics behind the lightning phenomenon is shrouded by questions which have yet to be answered. However, a compelling effort has been waged to model and classify lightning’s effects into very distinct categories, each of which has a varying philosophy. Some of these include rigorous physics based approaches while others use empirical data procured during natural and rockettriggered lightning strike events. The latter empirically based approach is generally grouped into a class of “engineering” models. Engineering models can be separated into several sub-classes, two of which include: the transmission-line (TL), or current propagation models and the Traveling-Current-Source (TCS), or current generation models [1]. The transmission-line model can be viewed as providing a base-current waveform at the discharge channel which propagates upward without distortion or attenuation. This model can be further customized to include the modified transmission-line linear (MTLL) and modified transmission-line
* Corresponding author. Florida Institute of Technology, Department of Electrical and Computer Engineering, 150 W. University Blvd, Melbourne, Florida 32901, United States. E-mail address: smeredit@my.fit.edu (S.L. Meredith). 0304-3886/$ e see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2011.12.002
exponential (MTLE) models [1]. Although these models adopt the same ideology as the TL model of having no distortion, they do prescribe to a specified amount of attenuation. It is worth noting that one of the shortcomings with the TL model as described by Karwowski and Zeddam [2] is that it does not account for the vertical distribution of charge stored in the corona-sheath. This charge surrounds the current-carrying channel core and presumably contains the bulk of the channel charge [1]. Whereas the MTLE model can be used to account for the effects from this initial charge present within the corona-sheath [3], thus making it a more comprehensive alternative. However, if one assumes the current decay constant l (generally assumed to be 2000m) is very large, the current waveforms TL and MTLE are now equivalent. The TL model with a standard “pulse” current waveform has been used in literature to solve for the electromagnetic field expressions with two different techniques. These include the dipole technique, also referred to as the Lorentz condition, and the monopole technique, also referred to as the Continuity equation. Lorentz condition allows one to solve for the vector potential A, and scalar potential V, in terms of the current distribution alone. Whereas the Continuity equation, requires that the vector potential and scalar potential are solved in terms of the current and charge distributions respectively. Rubinstein and Uman [4] showed the two techniques are numerically equivalent while Safaeinili and Mina [5] demonstrated their analytically equivalency for a step function waveform traveling along a vertical antenna. These results
S.L. Meredith et al. / Journal of Electrostatics 70 (2012) 152e156
were extended by Thottappillil and Rakov [6] to include any arbitrary current and charge distribution. Their study found excellent agreement in the corresponding electric field curves, not limited to the transmission-line models, which rendered them almost indistinguishable when plotted. This paper will utilize the TL model’s dipole technique for evaluating the electric field expressions. This technique makes use of infinitesimal time-varying dipoles as the source of the electric and magnetic fields. The subsequent electric field waveforms, similar to pulse results obtained by TL model, are derived in a closed form using the classical double exponential function (DEXP) as the channel-base-current waveform. In doing so, Taylor series expansion is used to supplement the exponential terms required to carry out the integration used for providing a time based expression for the electric field components. Given the inherent disparity between the original and modified exponential terms warrants the introduction of a correction factor to offset the error that is inadvertently introduced.
2. Literary background In 1929 Bewley [7] proposed using the double exponential to describe the wave shape a lightning current waveform may have. What he found were measurements obtained in both the field and lab closely matched wave shapes obtained by analysis. Since then, the research community has come to the consensus that the maximum dI/dt occurs near the current peak and not at t ¼ 0 as described by the double exponential waveform. A more accurate waveform can be represented by the Heidler function [8] which does not display a discontinuity in its time derivative t ¼ 0 [9]. Furthermore, the equation reproduces the observed concave rising portions of a typical current waveform [1] as opposed to the classical double exponential function which is distinguished by an unrealistic convex wavefront. But in light of the fact that the double exponential waveform plays an instrumental part in current U.S. military standards (MIL-STD), it becomes noteworthy to develop closed form solutions to the fields which result. Currently, the Department of Defense (DoD) utilizes the double exponential current waveform as part of their Electromagnetic Environmental Effects document [10] to describe the lightning current waveform. The parameters (I0, a and b) which make up the waveform are contingent upon the type and severity of the lightning strike. The double exponential base-current waveform I(t) described in [10] which replaces the standard pulse from the TL model is shown by,
IðtÞ ¼ I0 eat ebt
3. Propagation time delays As pointed out by Nevels [12], Lorenz proposed that the standard Neumann potentials in terms of instantaneous charge, current density, and position be modified to include the propagation of time from the source. This Retarded time t0 , is used as a means of describing the time delay that exists between a photon being emitted and when it is perceived by an observer at a distance R some R/c time later. The Retarded time or time delay is defined as,
t 0 ¼ t R=c
(2)
where t0 is the actually time of photon emission, t is the time perceived by an observer, and R/c is the time it took to travel. In principle, the lightning return stroke which is made up numerous adjoining current segments, imparts positive charges as it travels upward along the z-axis. In order to account for the reflections that occur between the lightning channel and surfaceground, one should treat this surface as a perfect conductor. In doing so, image theory can be used in the analysis. These images, or virtual sources, account for reflections which can then be added to the real source constituents to form a general equation. An illustration of this phenomenon is depicted in Fig. 1 which captures the lightning channel and its corresponding image. The magnitude R shown in Fig. 1 is describe by the following,
R ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ2 þr 2
(3)
where r is the observable distance, z0 is the height along the lightning channel at a given time t and z is the height at which the fields are evaluated. In order to account for the delay that exists between photon emissions given off by the upward traveling-current wavefront and when they are perceived at some observation point P, warrants the usage of the term jz0 j/v, where jz0 j is written to include all points along the z-axis both real and reflected. Knowing the time delay due to the upward propagating wavefront is in addition to the Retarded time (2) allows one to include both of these conditions when describing the time delay along the entire length of the lightning channel as experienced by an observer at distance R. Therefore, one can rewrite the double
(1)
where I0 is the current amplitude at the channel origin (z0 ¼ 0) and a, b are constants. For the purpose of this paper, the parameters for a severe stroke current waveform [10] are utilized. Thus, values for a and b are set to 11,354 s1 and 647,265 s1 respectively. Given this waveform’s usage in literature, a validation of the parameters used was warranted. Jia and Xiaoqing [11] used numerical trial and error to solve the double exponential parameters a and b. Upon substituting these values into the equation they found that the resulting waveform closely models the one which utilized the fixed parameters. However, the results presented in this paper go a step further by developing a closed form solution for the electric field induced by a modified double exponential current waveform. An approach which subsequently requires a correction factor be introduced to help reshape the electric field which results from the modified current used.
153
Fig. 1. Application of image theory used for the lightning channel.
154
S.L. Meredith et al. / Journal of Electrostatics 70 (2012) 152e156
exponential waveform from [10] in terms of both variables such that,
2.2
0 0 Iðz0 ;t R=cÞ ¼ I0 eaðtR=cjz j=vÞ ebðtR=cjz j=vÞ $uðt R=c
Given the complexity of the integration required to solve for the electric field, one must approximate the exponential terms using Taylor Series expansion. The expansion for the exponential terms can be approximated by,
ex y1 þ x þ
2!
þ
3!
þ.
(5)
In the case of the electric field, there are three double exponential waveforms that must be approximated. These include: the original waveform, the partially differentiated waveform and the integrated waveform, each of which will have been shifted in time by R/c þ jz0 j/v. This time shift represents the delay that exists between the lightning channel and observation point. From (4), one can separate the exponentials in terms of their constants and variables constituents. Doing so allows one to utilize Taylor series to account for the variable dependent terms while constants can be set aside and incorporated at a later time. Therefore from (4), one can distribute a, b and then regroup the exponential terms shown by (6),
h i 0 0 Iðz0 ;t R=cÞ ¼ I0 eat eaðR=cþjz j=vÞ ebt ebðR=cþjz j=vÞ $uðt R=c jz0 j=vÞ
ð6Þ
Given (6), one can make the following assignments,
R jz0 j xa ¼ a þ v c
(7)
and
xb ¼ b
1.8
Double Exponential Shifted by R/c + z'/v
(4)
4. Approximating the double exponential
x3
5
2
jz0 j=vÞ:
x2
x 10
1.6 1.4 1.2 1 0.8
0.4 0.2 0
0
0.5
1
1.5
(8)
where R is the distance between the lightning channel and observation point P, c is the speed of light in a vacuum and v is the speed of the front of the current wavefront which was taken to be c/3. For the purposes of this study, the variable dependent exponentials are approximated with the first two terms from Taylor series expansion as shown by (5). The effect from not carrying out the several additional terms described by Taylor series is marginalized by the introduction of the correction factor as illustrated in Fig. 2. In doing so, one can now leverage the first two terms from (5) where (7) and (8) equal the variable x, and then substitute into (6) as shown by,
R z0 R jz0 j ebt 1 þ b Iðz0 ; t R=cÞyI0 eat 1 þ a þ þ v c v c $uðt R=c jz0 j=vÞ:
ð9Þ
2.5
3
3.5
4
4.5
5
-5
x 10
Fig. 2. Modified and unmodified double exponential current waveforms time shifted by t ¼ R/c þ z0 /v.
v 2 ½Iðz0 ; t R=cÞyI0 bebt aeat þ b ebt a2 eat vt R jz0 j $uðt R=c jz0 j=vÞ þ eat ebt þ v c R jz0 j at $dðt R=c jz0 j=vÞ þ ae bebt þ v c
ð11Þ
and integrated form,
" 0
½Iðz ; t R=cÞdtyI0
R þ c v
2
Time [seconds]
Zt
z0
Modified Waveform Modified Waveform with Correction Unmodified Waveform
0.6
R=cþz0 =v
R jz0 j þ v c
ebt
b
eat
$uðt R=c jz0 j=vÞ
a
þ
b a bt þ e eat ab ð12Þ
It should be noted that the partial differentiation and integration of the double exponential waveform (10) was exercised prior to utilizing Taylor series expansion. As the Fig. 2 shows, the modified waveform which utilized Taylor series expansion gives rise to a slightly larger magnitude and takes longer to decay. However, when the approximated waveform is multiplied by the seminal correction constituent 1.06exp1.25t, the resulting waveform illustrates how closely it models the original double exponential time shifted by R/c þ z0 /v. The derivation of this correction factor was an iterative process used to determine the appropriate term required to reshape the modified waveform to resemble the original. In doing so, one can apply these acquiescent similarities to approximate the electric field in a closed form.
Upon grouping the “like” terms, (9) can now be written as,
h R jz0 j Iðz0 ; t R=cÞyI0 eat ebt þ aeat bebt þ c v $uðt R=c jz0 j=vÞ with its partial differentiated form,
5. Evaluating the electric field
ð10Þ
In order to develop a closed form expression for the electric field induced by the double exponential waveform, we have adopted the electric field expression derived by [4] with the dipole as shown by,
S.L. Meredith et al. / Journal of Electrostatics 70 (2012) 152e156
" dz0 2ðz z0 Þ2 r 2 dEz ¼ 4pε0 R5 þ
2ðz z0 Þ2 r 2 cR4
Zt
I 1:06e1:25at Ez y 0 p 2 ε0
Iðz0 ; t R=cÞds
R=cþjz0 j=v
$Iðz0 ; t R=cÞ
r2
vIðz0 ; t R=cÞ
c2 R3
vt
ebt
(13)
b
" " I0 dz0 2ðz z0 Þ2 r 2 ebt eat b a bt þ þ e eat 5 a ab b 4pε0 R 0 R jz j 2ðz z0 Þ2 r 2 $uðt R=c jz0 j=vÞ þ þ v c cR4 h R jz0 j eat ebt þ aeat bebt $uðt R=c jz0 j=vÞ þ v c r2 h 2 2 3 bebt aeat þ b ebt a2 eat c R R jz0 j r2 h $uðt R=c jz0 j=vÞ 2 3 eat ebt þ v c c R R jz0 j at bt $dðt R=c jz0 j=vÞ ð14Þ þ ae be þ c v
bt
h ¼ J
a
þ
!
r 2 h2 þ r 2
3=2
r2
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 þ r 2
at
ðct JzÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðJct zÞ2 þr 2 1 J2 (16)
1 J2
6. Simulation of results The following illustrations utilized (15) in the time domain to describe the electric fields evaluated at given observable distances r. Fig. 3 depicts the electric fields for a current amplitude of 30000 A [4] whereas the fields shown in Fig. 4 use a current amplitude of 13 kA adopted from Heidler et al. [8]. Fig. 3 is a comparison between the electric fields derived by the modified double exponential with those from [4]. As this figure illustrates, magnitudes for both fields are quite similar at 100 m but
4
1100
5
1000
4.5
900
4
800
3.5 3 2.5 2
Modified DEXP Rubinstein and Uman [4]
700 600 500 400 300
1.5 Modified DEXP
1
200
Rubinstein and Uman [4]
100
0.5
0
0 0
1
2
3
4
Time [seconds]
5
6
7 x 10
-6
!
ba r 2 2h2 4 ffi þ
3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ r2 ab 2 2 3v h 3v h þ r
with the quantity J ¼ v/c being the ratio of the current propagation speed along the lightning channel to the speed of light and the variable z represents the position along the vertical axis.
Electric Field [V/m]
Electric Field [V/m]
x 10
eat
h3
where h, derived by Rubinstein and Uman [4], can be written as,
dEz y
Upon integrating (14) along the z0 axis from h to h, one can obtain a closed form solution for the electric field on the z ¼ 0 plane to include the correction factor as shown by the following,
!
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! h2 þ r 2 h 3h þ 2ln e e þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r h2 þ r 2 ! aeat bebt 3r 2 2 2 2 þ 3 þ ln þ r ln h r c2 h2 þ r 2 ! r aeat bebt h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bebt aeat 3 cv c c2 h2 þ r 2 2 h r r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 ebt a2 eat þ tan1 r c2 v h2 þ r 2 c v 2 ð15Þ b ebt a2 eat
where the lower integration boundary was changed from [4] given the current wavefront begins propagating along the channel at time t ¼ R/c þ z0 /v. The first portion on the right hand side of (13) is known as the static term and is dependant upon the change of charge along the lightning channel. The second portion, or induction term, decays relatively slowly and is dependent upon the channel current. The third portion, the radiation or far-field term, decays most slowly. Given (13), substitute (10), (11) and (12) in place of the current pulses for the second, third and first terms of (13) respectively.
5.5
155
"
1.5
2
2.5
3
3.5
4
4.5
Time [seconds]
5
5.5
6 6.5 -5 x 10
Fig. 3. Comparison of the electric fields from [4] to the modified double exponential current waveform when r ¼ 100 m (left) and r ¼ 5 km (right).
156
S.L. Meredith et al. / Journal of Electrostatics 70 (2012) 152e156 450 r = 5000m r = 7000m r = 9000m
400
20
Electric Field [V/m]
Electric Field [V/m]
350
300 250 200
150 100
15
10
5
50 0 2
2.5
3
Time [seconds]
3.5
4 x 10
0 3.3
3.32 3.34 3.36 3.38
-5
3.4
3.42 3.44 3.46 3.48
Time [seconds]
3.5
x 10
-4
Fig. 4. Electric fields induced by the modified double exponential waveform for a current amplitude I ¼ 13 kA when r ¼ 5000 km, 7000 km and 9000 km (left) and r ¼ 100 km (right).
become less comparable at 5000 m. However, in both cases the modified double exponential decays much faster as one would expect. Fig. 4 depicts the modified electric fields evaluated at given increments of r. The fields illustrated by this figure tend to be much greater than those reported in literature [1,6]. This is primarily due to the dominate electrostatic field component not being as influenced by the radial distance from the lightning channel as reported by other TL models.
7. Conclusion A closed form solution for the electric field induced by the double exponential current waveform has been presented. The approach presented is unique in that it utilized Taylor series to approximate the variable dependent exponential terms within the current waveform. This greatly simplified the mathematics which helped facilitate the integration required to solve for the solution in closed form. However, this approximation caused the modified current waveform to grow in both magnitude and decay time. With this in mind, it is logical to presume that the resulting field would be subject to this same anomaly. Realizing a departure existed between the modified and unmodified waveforms, warranted the use of a correction factor to recapture its original form. Doing so ensured the electric fields which followed, would tend to more closely model those from the original unmodified current waveform.
References [1] V.A. Rakov, M.A. Uman, Review and evaluation of lightning return stroke models including some aspects of their application, IEEE Trans. Electromagn. Capability 40 (4) (November 1998) 403e426. [2] A. Karwowski, A. Zeddam, Transient currents on lightning protection systems due to the indirect effect, IEE Proc.eSci. Meas. Technol. 142 (3) (May 1995) 213e222. [3] F. Rachidi, V.A. Rakov, C.A. Nucci, J.L. Bermudez, Effect of vertically extended strike object on the distribution of current along the lightning channel, J. Geophys. Res. 107 (2002). doi:10.1029/2002JD0021119 pp. 16e1e16e6, no. D23, 4699. [4] M. Rubinstein, M.A. Uman, Methods for calculating the electromagnetic fields from a known source distribution: application to lightning, IEEE Trans. Electromagn. Compatibility 31 (2) (May 1989) 183e189. [5] A. Safaeinili, M. Mina, On the analytical equivalence of electromagnetic fields solutions from a known source distribution, IEEE Trans. Electromagn. Compatibility 33 (1) (February 1991) 69e71. [6] R. Thottappillil, V.A. Rakov, On different approaches to calculating lightning electric fields, J. Geophys. Res. 106 (2001) 14191e14205. [7] L.V. Bewley, Traveling waves due to lightning, AIEE Trans. 49 (1929) 1050e1064. [8] F. Heidler, J.M. Cveti c, B.V. Stani c, Calculation of lightning current parameters, IEEE Trans. Power Deliv. 14 (2) (April 1999) 399e404. [9] A. Andreotti, S. Falco, L. Verolino, Some integrals involving heidler’s lightning return stroke current expression, Electrical Eng. 87 (April 2004) 121e128. doi:10.1007/s00202-004-0240-8. [10] Department of Defense Interface Standard, Electromagnetic Environmental Effects Requirements for Systems, MIL-STD-464, March 1997. [11] W. Jia, Z. Xiaoqing "Double-exponential Expression of Lightning Current Waveforms", CEEM 2006/Dalian 3A1-09, pp 320e323. [12] R. Nevels, C. Shin, Lorenz, Lorentz, and the Gauge, IEEE Antenn. Propagat. Mag. 43 (2001) 70e72.
Contents lists available at SciVerse ScienceDirect
Journal of Electrostatics journal homepage: www.elsevier.com/locate/elstat
Electric fields induced by a modified double exponential current waveform Scott L. Meredith a, *, Susan K. Earles a, Carlos E. Otero b a b
Department of Electrical and Computer Engineering, Florida Institute of Technology, Melbourne, FL 32901, USA Department of Mathematics & Computer Science, University of Virginia’s College at Wise, Wise, VA 24293, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 26 May 2011 Accepted 11 December 2011 Available online 22 December 2011
This paper presents a novel approach for developing a closed form solution for a lightning generated electric field which uses a modified double exponential base-current waveform. In the literature, several models have employed the square pulse waveform to derive the corresponding electromagnetic fields. However, given the Department of Defense (DoD) has incorporated the double exponential current waveform as part of their “Electromagnetic Environmental Effects Requirements For Systems”, it is noteworthy to develop a solution for the electric field which utilizes this waveform. In order to facilitate the integration required for deriving the field, Taylor series expansion will be used for all variable dependent exponential terms within the current waveform. This methodology greatly simplifies the integration required to solve for the electric field in an approximated closed form. However, by utilizing Taylor series to approximate the variable dependent exponentials, the electric field’s accuracy becomes a function of the number of terms used. Consequently, a correction factor will need to be added in order to help reshape the electric field which resulted from the modified base-current waveform. Ó 2011 Elsevier B.V. All rights reserved.
Keywords: Lightning Double exponential Electric fields Indirect effect
1. Introduction Lightning can be regarded as a very complex and mysterious electrical discharge which occurs during the interaction between charges of the opposite polarity in and around thunderstorms. The exact physics behind the lightning phenomenon is shrouded by questions which have yet to be answered. However, a compelling effort has been waged to model and classify lightning’s effects into very distinct categories, each of which has a varying philosophy. Some of these include rigorous physics based approaches while others use empirical data procured during natural and rockettriggered lightning strike events. The latter empirically based approach is generally grouped into a class of “engineering” models. Engineering models can be separated into several sub-classes, two of which include: the transmission-line (TL), or current propagation models and the Traveling-Current-Source (TCS), or current generation models [1]. The transmission-line model can be viewed as providing a base-current waveform at the discharge channel which propagates upward without distortion or attenuation. This model can be further customized to include the modified transmission-line linear (MTLL) and modified transmission-line
* Corresponding author. Florida Institute of Technology, Department of Electrical and Computer Engineering, 150 W. University Blvd, Melbourne, Florida 32901, United States. E-mail address: smeredit@my.fit.edu (S.L. Meredith). 0304-3886/$ e see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.elstat.2011.12.002
exponential (MTLE) models [1]. Although these models adopt the same ideology as the TL model of having no distortion, they do prescribe to a specified amount of attenuation. It is worth noting that one of the shortcomings with the TL model as described by Karwowski and Zeddam [2] is that it does not account for the vertical distribution of charge stored in the corona-sheath. This charge surrounds the current-carrying channel core and presumably contains the bulk of the channel charge [1]. Whereas the MTLE model can be used to account for the effects from this initial charge present within the corona-sheath [3], thus making it a more comprehensive alternative. However, if one assumes the current decay constant l (generally assumed to be 2000m) is very large, the current waveforms TL and MTLE are now equivalent. The TL model with a standard “pulse” current waveform has been used in literature to solve for the electromagnetic field expressions with two different techniques. These include the dipole technique, also referred to as the Lorentz condition, and the monopole technique, also referred to as the Continuity equation. Lorentz condition allows one to solve for the vector potential A, and scalar potential V, in terms of the current distribution alone. Whereas the Continuity equation, requires that the vector potential and scalar potential are solved in terms of the current and charge distributions respectively. Rubinstein and Uman [4] showed the two techniques are numerically equivalent while Safaeinili and Mina [5] demonstrated their analytically equivalency for a step function waveform traveling along a vertical antenna. These results
S.L. Meredith et al. / Journal of Electrostatics 70 (2012) 152e156
were extended by Thottappillil and Rakov [6] to include any arbitrary current and charge distribution. Their study found excellent agreement in the corresponding electric field curves, not limited to the transmission-line models, which rendered them almost indistinguishable when plotted. This paper will utilize the TL model’s dipole technique for evaluating the electric field expressions. This technique makes use of infinitesimal time-varying dipoles as the source of the electric and magnetic fields. The subsequent electric field waveforms, similar to pulse results obtained by TL model, are derived in a closed form using the classical double exponential function (DEXP) as the channel-base-current waveform. In doing so, Taylor series expansion is used to supplement the exponential terms required to carry out the integration used for providing a time based expression for the electric field components. Given the inherent disparity between the original and modified exponential terms warrants the introduction of a correction factor to offset the error that is inadvertently introduced.
2. Literary background In 1929 Bewley [7] proposed using the double exponential to describe the wave shape a lightning current waveform may have. What he found were measurements obtained in both the field and lab closely matched wave shapes obtained by analysis. Since then, the research community has come to the consensus that the maximum dI/dt occurs near the current peak and not at t ¼ 0 as described by the double exponential waveform. A more accurate waveform can be represented by the Heidler function [8] which does not display a discontinuity in its time derivative t ¼ 0 [9]. Furthermore, the equation reproduces the observed concave rising portions of a typical current waveform [1] as opposed to the classical double exponential function which is distinguished by an unrealistic convex wavefront. But in light of the fact that the double exponential waveform plays an instrumental part in current U.S. military standards (MIL-STD), it becomes noteworthy to develop closed form solutions to the fields which result. Currently, the Department of Defense (DoD) utilizes the double exponential current waveform as part of their Electromagnetic Environmental Effects document [10] to describe the lightning current waveform. The parameters (I0, a and b) which make up the waveform are contingent upon the type and severity of the lightning strike. The double exponential base-current waveform I(t) described in [10] which replaces the standard pulse from the TL model is shown by,
IðtÞ ¼ I0 eat ebt
3. Propagation time delays As pointed out by Nevels [12], Lorenz proposed that the standard Neumann potentials in terms of instantaneous charge, current density, and position be modified to include the propagation of time from the source. This Retarded time t0 , is used as a means of describing the time delay that exists between a photon being emitted and when it is perceived by an observer at a distance R some R/c time later. The Retarded time or time delay is defined as,
t 0 ¼ t R=c
(2)
where t0 is the actually time of photon emission, t is the time perceived by an observer, and R/c is the time it took to travel. In principle, the lightning return stroke which is made up numerous adjoining current segments, imparts positive charges as it travels upward along the z-axis. In order to account for the reflections that occur between the lightning channel and surfaceground, one should treat this surface as a perfect conductor. In doing so, image theory can be used in the analysis. These images, or virtual sources, account for reflections which can then be added to the real source constituents to form a general equation. An illustration of this phenomenon is depicted in Fig. 1 which captures the lightning channel and its corresponding image. The magnitude R shown in Fig. 1 is describe by the following,
R ¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðz z0 Þ2 þr 2
(3)
where r is the observable distance, z0 is the height along the lightning channel at a given time t and z is the height at which the fields are evaluated. In order to account for the delay that exists between photon emissions given off by the upward traveling-current wavefront and when they are perceived at some observation point P, warrants the usage of the term jz0 j/v, where jz0 j is written to include all points along the z-axis both real and reflected. Knowing the time delay due to the upward propagating wavefront is in addition to the Retarded time (2) allows one to include both of these conditions when describing the time delay along the entire length of the lightning channel as experienced by an observer at distance R. Therefore, one can rewrite the double
(1)
where I0 is the current amplitude at the channel origin (z0 ¼ 0) and a, b are constants. For the purpose of this paper, the parameters for a severe stroke current waveform [10] are utilized. Thus, values for a and b are set to 11,354 s1 and 647,265 s1 respectively. Given this waveform’s usage in literature, a validation of the parameters used was warranted. Jia and Xiaoqing [11] used numerical trial and error to solve the double exponential parameters a and b. Upon substituting these values into the equation they found that the resulting waveform closely models the one which utilized the fixed parameters. However, the results presented in this paper go a step further by developing a closed form solution for the electric field induced by a modified double exponential current waveform. An approach which subsequently requires a correction factor be introduced to help reshape the electric field which results from the modified current used.
153
Fig. 1. Application of image theory used for the lightning channel.
154
S.L. Meredith et al. / Journal of Electrostatics 70 (2012) 152e156
exponential waveform from [10] in terms of both variables such that,
2.2
0 0 Iðz0 ;t R=cÞ ¼ I0 eaðtR=cjz j=vÞ ebðtR=cjz j=vÞ $uðt R=c
Given the complexity of the integration required to solve for the electric field, one must approximate the exponential terms using Taylor Series expansion. The expansion for the exponential terms can be approximated by,
ex y1 þ x þ
2!
þ
3!
þ.
(5)
In the case of the electric field, there are three double exponential waveforms that must be approximated. These include: the original waveform, the partially differentiated waveform and the integrated waveform, each of which will have been shifted in time by R/c þ jz0 j/v. This time shift represents the delay that exists between the lightning channel and observation point. From (4), one can separate the exponentials in terms of their constants and variables constituents. Doing so allows one to utilize Taylor series to account for the variable dependent terms while constants can be set aside and incorporated at a later time. Therefore from (4), one can distribute a, b and then regroup the exponential terms shown by (6),
h i 0 0 Iðz0 ;t R=cÞ ¼ I0 eat eaðR=cþjz j=vÞ ebt ebðR=cþjz j=vÞ $uðt R=c jz0 j=vÞ
ð6Þ
Given (6), one can make the following assignments,
R jz0 j xa ¼ a þ v c
(7)
and
xb ¼ b
1.8
Double Exponential Shifted by R/c + z'/v
(4)
4. Approximating the double exponential
x3
5
2
jz0 j=vÞ:
x2
x 10
1.6 1.4 1.2 1 0.8
0.4 0.2 0
0
0.5
1
1.5
(8)
where R is the distance between the lightning channel and observation point P, c is the speed of light in a vacuum and v is the speed of the front of the current wavefront which was taken to be c/3. For the purposes of this study, the variable dependent exponentials are approximated with the first two terms from Taylor series expansion as shown by (5). The effect from not carrying out the several additional terms described by Taylor series is marginalized by the introduction of the correction factor as illustrated in Fig. 2. In doing so, one can now leverage the first two terms from (5) where (7) and (8) equal the variable x, and then substitute into (6) as shown by,
R z0 R jz0 j ebt 1 þ b Iðz0 ; t R=cÞyI0 eat 1 þ a þ þ v c v c $uðt R=c jz0 j=vÞ:
ð9Þ
2.5
3
3.5
4
4.5
5
-5
x 10
Fig. 2. Modified and unmodified double exponential current waveforms time shifted by t ¼ R/c þ z0 /v.
v 2 ½Iðz0 ; t R=cÞyI0 bebt aeat þ b ebt a2 eat vt R jz0 j $uðt R=c jz0 j=vÞ þ eat ebt þ v c R jz0 j at $dðt R=c jz0 j=vÞ þ ae bebt þ v c
ð11Þ
and integrated form,
" 0
½Iðz ; t R=cÞdtyI0
R þ c v
2
Time [seconds]
Zt
z0
Modified Waveform Modified Waveform with Correction Unmodified Waveform
0.6
R=cþz0 =v
R jz0 j þ v c
ebt
b
eat
$uðt R=c jz0 j=vÞ
a
þ
b a bt þ e eat ab ð12Þ
It should be noted that the partial differentiation and integration of the double exponential waveform (10) was exercised prior to utilizing Taylor series expansion. As the Fig. 2 shows, the modified waveform which utilized Taylor series expansion gives rise to a slightly larger magnitude and takes longer to decay. However, when the approximated waveform is multiplied by the seminal correction constituent 1.06exp1.25t, the resulting waveform illustrates how closely it models the original double exponential time shifted by R/c þ z0 /v. The derivation of this correction factor was an iterative process used to determine the appropriate term required to reshape the modified waveform to resemble the original. In doing so, one can apply these acquiescent similarities to approximate the electric field in a closed form.
Upon grouping the “like” terms, (9) can now be written as,
h R jz0 j Iðz0 ; t R=cÞyI0 eat ebt þ aeat bebt þ c v $uðt R=c jz0 j=vÞ with its partial differentiated form,
5. Evaluating the electric field
ð10Þ
In order to develop a closed form expression for the electric field induced by the double exponential waveform, we have adopted the electric field expression derived by [4] with the dipole as shown by,
S.L. Meredith et al. / Journal of Electrostatics 70 (2012) 152e156
" dz0 2ðz z0 Þ2 r 2 dEz ¼ 4pε0 R5 þ
2ðz z0 Þ2 r 2 cR4
Zt
I 1:06e1:25at Ez y 0 p 2 ε0
Iðz0 ; t R=cÞds
R=cþjz0 j=v
$Iðz0 ; t R=cÞ
r2
vIðz0 ; t R=cÞ
c2 R3
vt
ebt
(13)
b
" " I0 dz0 2ðz z0 Þ2 r 2 ebt eat b a bt þ þ e eat 5 a ab b 4pε0 R 0 R jz j 2ðz z0 Þ2 r 2 $uðt R=c jz0 j=vÞ þ þ v c cR4 h R jz0 j eat ebt þ aeat bebt $uðt R=c jz0 j=vÞ þ v c r2 h 2 2 3 bebt aeat þ b ebt a2 eat c R R jz0 j r2 h $uðt R=c jz0 j=vÞ 2 3 eat ebt þ v c c R R jz0 j at bt $dðt R=c jz0 j=vÞ ð14Þ þ ae be þ c v
bt
h ¼ J
a
þ
!
r 2 h2 þ r 2
3=2
r2
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 þ r 2
at
ðct JzÞ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðJct zÞ2 þr 2 1 J2 (16)
1 J2
6. Simulation of results The following illustrations utilized (15) in the time domain to describe the electric fields evaluated at given observable distances r. Fig. 3 depicts the electric fields for a current amplitude of 30000 A [4] whereas the fields shown in Fig. 4 use a current amplitude of 13 kA adopted from Heidler et al. [8]. Fig. 3 is a comparison between the electric fields derived by the modified double exponential with those from [4]. As this figure illustrates, magnitudes for both fields are quite similar at 100 m but
4
1100
5
1000
4.5
900
4
800
3.5 3 2.5 2
Modified DEXP Rubinstein and Uman [4]
700 600 500 400 300
1.5 Modified DEXP
1
200
Rubinstein and Uman [4]
100
0.5
0
0 0
1
2
3
4
Time [seconds]
5
6
7 x 10
-6
!
ba r 2 2h2 4 ffi þ
3=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ r2 ab 2 2 3v h 3v h þ r
with the quantity J ¼ v/c being the ratio of the current propagation speed along the lightning channel to the speed of light and the variable z represents the position along the vertical axis.
Electric Field [V/m]
Electric Field [V/m]
x 10
eat
h3
where h, derived by Rubinstein and Uman [4], can be written as,
dEz y
Upon integrating (14) along the z0 axis from h to h, one can obtain a closed form solution for the electric field on the z ¼ 0 plane to include the correction factor as shown by the following,
!
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! h2 þ r 2 h 3h þ 2ln e e þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r r h2 þ r 2 ! aeat bebt 3r 2 2 2 2 þ 3 þ ln þ r ln h r c2 h2 þ r 2 ! r aeat bebt h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bebt aeat 3 cv c c2 h2 þ r 2 2 h r r pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 b2 ebt a2 eat þ tan1 r c2 v h2 þ r 2 c v 2 ð15Þ b ebt a2 eat
where the lower integration boundary was changed from [4] given the current wavefront begins propagating along the channel at time t ¼ R/c þ z0 /v. The first portion on the right hand side of (13) is known as the static term and is dependant upon the change of charge along the lightning channel. The second portion, or induction term, decays relatively slowly and is dependent upon the channel current. The third portion, the radiation or far-field term, decays most slowly. Given (13), substitute (10), (11) and (12) in place of the current pulses for the second, third and first terms of (13) respectively.
5.5
155
"
1.5
2
2.5
3
3.5
4
4.5
Time [seconds]
5
5.5
6 6.5 -5 x 10
Fig. 3. Comparison of the electric fields from [4] to the modified double exponential current waveform when r ¼ 100 m (left) and r ¼ 5 km (right).
156
S.L. Meredith et al. / Journal of Electrostatics 70 (2012) 152e156 450 r = 5000m r = 7000m r = 9000m
400
20
Electric Field [V/m]
Electric Field [V/m]
350
300 250 200
150 100
15
10
5
50 0 2
2.5
3
Time [seconds]
3.5
4 x 10
0 3.3
3.32 3.34 3.36 3.38
-5
3.4
3.42 3.44 3.46 3.48
Time [seconds]
3.5
x 10
-4
Fig. 4. Electric fields induced by the modified double exponential waveform for a current amplitude I ¼ 13 kA when r ¼ 5000 km, 7000 km and 9000 km (left) and r ¼ 100 km (right).
become less comparable at 5000 m. However, in both cases the modified double exponential decays much faster as one would expect. Fig. 4 depicts the modified electric fields evaluated at given increments of r. The fields illustrated by this figure tend to be much greater than those reported in literature [1,6]. This is primarily due to the dominate electrostatic field component not being as influenced by the radial distance from the lightning channel as reported by other TL models.
7. Conclusion A closed form solution for the electric field induced by the double exponential current waveform has been presented. The approach presented is unique in that it utilized Taylor series to approximate the variable dependent exponential terms within the current waveform. This greatly simplified the mathematics which helped facilitate the integration required to solve for the solution in closed form. However, this approximation caused the modified current waveform to grow in both magnitude and decay time. With this in mind, it is logical to presume that the resulting field would be subject to this same anomaly. Realizing a departure existed between the modified and unmodified waveforms, warranted the use of a correction factor to recapture its original form. Doing so ensured the electric fields which followed, would tend to more closely model those from the original unmodified current waveform.
References [1] V.A. Rakov, M.A. Uman, Review and evaluation of lightning return stroke models including some aspects of their application, IEEE Trans. Electromagn. Capability 40 (4) (November 1998) 403e426. [2] A. Karwowski, A. Zeddam, Transient currents on lightning protection systems due to the indirect effect, IEE Proc.eSci. Meas. Technol. 142 (3) (May 1995) 213e222. [3] F. Rachidi, V.A. Rakov, C.A. Nucci, J.L. Bermudez, Effect of vertically extended strike object on the distribution of current along the lightning channel, J. Geophys. Res. 107 (2002). doi:10.1029/2002JD0021119 pp. 16e1e16e6, no. D23, 4699. [4] M. Rubinstein, M.A. Uman, Methods for calculating the electromagnetic fields from a known source distribution: application to lightning, IEEE Trans. Electromagn. Compatibility 31 (2) (May 1989) 183e189. [5] A. Safaeinili, M. Mina, On the analytical equivalence of electromagnetic fields solutions from a known source distribution, IEEE Trans. Electromagn. Compatibility 33 (1) (February 1991) 69e71. [6] R. Thottappillil, V.A. Rakov, On different approaches to calculating lightning electric fields, J. Geophys. Res. 106 (2001) 14191e14205. [7] L.V. Bewley, Traveling waves due to lightning, AIEE Trans. 49 (1929) 1050e1064. [8] F. Heidler, J.M. Cveti c, B.V. Stani c, Calculation of lightning current parameters, IEEE Trans. Power Deliv. 14 (2) (April 1999) 399e404. [9] A. Andreotti, S. Falco, L. Verolino, Some integrals involving heidler’s lightning return stroke current expression, Electrical Eng. 87 (April 2004) 121e128. doi:10.1007/s00202-004-0240-8. [10] Department of Defense Interface Standard, Electromagnetic Environmental Effects Requirements for Systems, MIL-STD-464, March 1997. [11] W. Jia, Z. Xiaoqing "Double-exponential Expression of Lightning Current Waveforms", CEEM 2006/Dalian 3A1-09, pp 320e323. [12] R. Nevels, C. Shin, Lorenz, Lorentz, and the Gauge, IEEE Antenn. Propagat. Mag. 43 (2001) 70e72.