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The influence of lightning conductor radii on the attachment of lightning flashes
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The influence of lightning conductor radii on the attachment of lightning flashes Vernon Cooray Department of Engineering Sciences, Uppsala University, Uppsala, Sweden
a r t i c l e
i n f o
Article history: Received 17 May 2016 Received in revised form 9 December 2016 Accepted 2 January 2017 Available online xxx Keywords: Lightning conductors Lightning attachment Blunt and sharp conductors Glow corona
a b s t r a c t The influence of the tip radius of lightning conductors on their lightning attractive distance as predicted by the self-consistent leader inception and propagation model (SLIM) is presented. The results show that in the absence of any glow corona from the tip of the conductor a smaller tip radius gives rise to a larger attractive radius than a larger radius. It is suggested that the reason for the experimental observations which show that blunt conductors are more efficient lightning receptors than sharp ones is the presence of glow corona at the tip of the sharp ones during the time of lightning strikes. Moreover, in a given background electric field, the probability of the inception of glow corona at the conductor tip increases with increasing conductor height and decreasing conductor radius. Thus, in a given electric field, as the conductor height increases its radius has to be increased to avoid the inception of glow corona at the tip. For this reason, the conductor radius that performs best as a lightning interceptor depends on the height of the conductor and the best performance shift from smaller radii to larger ones with increasing height of the conductor. © 2017 Elsevier B.V. All rights reserved.
1. Introduction In a classical experiment Moore et al. [1] demonstrated that moderately blunt conductors are more efficient than sharp or extremely blunt conductors. They speculated that the glow corona at the tip of the sharp conductors could be the reason for their inefficiency, in comparison to the blunt ones, in attracting lightning flashes. However, a question that will arise naturally when studying the results of this experiment is the following: What would be the effect of conductor radius on lightning attachment if glow corona is not present at the tip of the conductors? In order to investigate this we will derive and compare the height of attachment and the attractive radii of conductors of different tip radii pertinent to stepped leaders of lightning flashes. The model simulations are conducted using SLIM, a physics based model introduced by Becerra and Cooray [2,3, see also 4 for a detailed description]. Before proceeding further let us summarize the main features of SLIM. 2. The model SLIM The main steps that are included in the model are: (1) formation of a streamer discharge at the tip of a grounded object (first,
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second or third streamer bursts). (2) Transformation of the stem of the streamer into a thermalized leader channel (unstable leader inception). (3) Extension of the positive leader and its self-sustained propagation (stable leader inception). Let us consider these steps in details. The description given below is based on the work published by Becerra and Cooray [3,4]. Assume that the electric field at ground level as a function of time generated by the down coming stepped leader is known. How this is evaluated in the model is described in Section 3. The simulation consists of several main steps and let us consider them one by one.
(1) The first step is to extract the time or the height of the stepped leader when streamers are incepted from the grounded rod. Since the background electric field produced by the stepped leader is known (or given) the electric field at the tip of the grounded rod can be calculated, for example, by using charge simulation method. This field is used together with the avalanche to streamer transition criterion to investigate whether the electric field at the conductor tip is large enough to convert avalanches to streamers. In the analysis it is assumed that the electron avalanche will be converted to a streamer when the number of positive ions at the head of the avalanche exceeds about 108 [5]. The simulation continues using the time
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2
Voltage
Background potential
A Potential along streamers
Distance from the tip of the conductor. Fig. 1. Distance–Voltage diagram that illustrates how the charge associated with a streamer burst is obtained. The area between the two curves representing the background potential and the streamer potential is marked A.
varying electric field of the stepped leader until the streamer inception criterion is satisfied. (2) The moment the streamer inception criterion is satisfied a burst of streamers will be generated from the extremity of the object; in our case from the tip of the lightning rod. The next task is to calculate the charge in this streamer burst. The charge associated with these streamer bursts are calculated using a distance–voltage diagram with the origin at the tip of the grounded conductor as follows. The procedure is illustrated in Fig. 1. The streamer zone is assumed to maintain a constant potential gradient Estr . In the distance–voltage diagram this is represented by a straight line. On the same diagram the background potential produced by the thundercloud and the down-coming stepped leader at the current time is depicted. If the area between the two curves up to the point where they cross is A (see Fig. 1), the charge in the streamer zone is given by Qo ≈ KQ A
(1)
where KQ is a geometrical factor. Becerra and Cooray [3,4] estimated its value to be about 3.5 × 10−11 C/V m. In the analysis we assume Estr = 5 × 105 V/m [5]. This value of the electric field is valid for streamers propagating in normal atmospheric density. A slight change of this electric field around this value (10%) in the calculations to be presented does not change the conclusions to be reached. (3) The next task is to investigate whether this streamer burst is capable of generating a leader. This decision is based on the fact that in order to generate a leader a minimum of 1 C is required in the charge generated by the streamers [5]. If the charge in the streamer zone is less than this value then the procedure is repeated a small time interval later. Note that with increasing time the electric field generated by the stepped leader increases and, consequently, the charge in the streamer bursts increases. (4) Assume that at time t, the condition necessary for leader inception is satisfied. The next task is to estimate the length and the radius of this initial leader section. In doing this it is assumed that the amount of charge that is necessary to create a unit length of positive leader is ql . Becerra and Cooray [3] evaluated this parameter using the equations given by Gallimberti [5] and it was shown that it is a function of the leader speed. For low leader speeds (around 104 m/s) its value is about 65 C/m. In the analysis the value of ql is estimated using the relationship between this parameter and the leader speed as published by Becerra and Cooray [3]. Based on these considerations, the initial length of the leader section L1 is given by Qo /ql , where Qo is
the charge in the streamer burst that immediately precede the inception of the leader. The initial radius of the leader, aL1 (t) is assumed to be 10−3 m and the initial potential gradient of the leader section, EL1 (t) is assumed to be equal to the potential gradient of the streamer region i.e. 5.0 × 105 V/m. Now we proceed to the next time step, i.e. t = t + t (5) During the time interval t there will be a change in the background potential and we also have a small leader section of length L1 . Now the new charge in the streamer zone generated from the head of the new leader section is calculated as before but now including both the leader and its streamer zone in the distance–voltage diagram. The leader is represented by a line with a potential gradient EL1 (t). The charge generated in the current time step is obtained by subtracting from this the total charge obtained in the previous time step. Let the charge obtained thus be Q1 . This charge is used to evaluate the length of the new leader section L2 . Moreover, the flow of this charge through the leader channel changes the potential gradient and the radius of the older leader section L1 . The new potential gradient and the radius of L1 are given by EL1 (t + t) and aL1 (t + t). (6) Now let us consider the nth time step. There are n leader sections and they have their respective potential gradients and radii. The radius and the potential gradient of ith leader section are obtained from · a2Li (t + t) = · a2Li (t) + ELi (t + t) =
a2Li (t) a2Li (t + t)
−1 E (t) · ILi (t) · t · p0 Li
(2)
(3)
ELi (t)
In the above equation ELi (t) is the internal electric field, ILi (t) is the current of the leader section Li at time t, p0 is the standard atmospheric pressure and is the ratio between the specific heats at constant volume and constant pressure for air [5]. With these equations it is possible to calculate the time evolution of the internal electric field for each segment and the potential drop along the leader channel (at a given time) as follows: UL =
n
ELi (t) · Li
(4)
i=1
The steps described above can be used to simulate the inception and propagation of positive leaders. The calculation can be simplified if, instead of calculating the time evolution of leader potential gradient in each segment as above, one uses the expression derived by Rizk [6] for the potential of the tip of the leader channel which is given by
(i) Utip
=
(i) lL E∞
(i)
Estr − E∞ − Estr + xo E∞ ln − e E∞ E∞
l /x0 L
(5)
(i)
In the above equation lL is the total leader length at the current simulation step, E∞ is the final quasi-stationary leader gradient and x0 is a constant parameter given by the product v , where v is the ascending positive leader speed and is the leader time constant. (7) In the model the negative stepped leader is assumed to travel towards ground without being influenced by the connecting leaders issued from grounded structure unless final jump condition is established with one of them. In the original SLIM, the tip of the connecting leader is assumed to travel, at any given moment, towards the current location of the tip of the down coming stepped leader. In the simulations presented here this assumption is relaxed and the connecting leader is assumed
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to travel in the direction of the maximum background electric field at the current location of the tip of the connecting leader. (8) The condition for the final attachment is assumed to be fulfilled when the average potential gradient in the gap between the tips of the down-coming stepped leader and the upwardmoving connecting leader reaches a value 500 kV/m (i.e. final jump condition). 3. Charge distribution on the leader channel In a recent study, Cooray et al. [7] measured the charge brought to ground by the first 100 s of the return stroke. They found a strong correlation between the first return stroke current and this charge. Combining this information with the bi-directional leader model they have investigated how this charge was distributed along the stepped leader channel. In their analysis they have assumed that the stepped leader channel is vertical. According to their results the linear charge distribution (in C/m) on the stepped leader channel when its tip is at a height of zo above ground is given by
(z) = ao 1 − = (z − zo ) G(zo ) = 1 −
L − zo
z > zo zo L
J(zo ) = 0.3˛ + 0.7ˇ ˛=e ˇ=
−(zo −10)
zo 1− L
/75
G(zo )Ip +
Ip (a + b) 1 + c + d 2
J(zo )
(6) (7) (8) (9) (10) (11)
In the above equations z is the vertical height of the point of observation, L is the total length of the stepped leader channel in meters, (z) is the charge per unit length of a leader section located at height z, Ip is the return stroke peak current in kA, ao = 1.476 × 10−5 , a = 4.857 × 10−5 , b = 3.9097 × 10−6 , c = 0.522 and d = 3.73 × 10−3 . Note that the above equation is valid for zo ≥ 10 m. This charge distribution is utilized in calculating the electric field produced by the down coming stepped leader. 4. Results In the analysis the attractive radii of a conductor is obtained as follows. Starting with zero, the lateral distance to the stepped leader channel from conductor is gradually increased until the stepped leader is intercepted by the ground instead of the lightning conductor. This critical distance where the stepped leader is intercepted by the ground is taken to be the attractive radii of the lightning conductor pertinent to the prospective return stroke current associated with the stepped leader used in the simulation. The height of attachment is defined as the height of the stepped leader when the final jump condition is established between the connecting leader and the stepped leader. The height of attachment may vary depending on the lateral distance to the stepped leader from the conductor. In the simulations the conductor is assumed to be cylindrical in shape with a hemispherical tip. The tip radius is a variable in the simulation. In the simulation, for a given conductor height and lateral distance to the leader, the height at which the leader is intercepted by the connecting leader originating from the conductor is estimated. This parameter is obtained for three conductor radii and for four different conductor heights. The prospective return stroke current is kept constant at 30 kA in the simulation. The results of the simulation are shown in Fig. 2. Note that in the figures the largest lateral distance corresponding to a given radius is the attractive radius corresponding to that conductor.
3
One can see from these figures that, for a given conductor height, the conductor with a smallest radius will be able to intercept the stepped leader at the largest height. In other words the smaller the radius of the conductor the more efficient would it be in intercepting the stepped leader. Note also that the attractive radii of the conductor increase with decreasing conductor radius. The reason for this change in intercepting efficiency is the ability of the conductor with the smaller radius to generate a connecting leader earlier than a conductor with a larger radius. For example, the growth of the connecting leaders from conductors of different radii as the stepped leader approaches them is shown in Fig. 3. This figure corresponds to 30 kA current and 50 m tall conductor. On the x-axis of this figure one can read the height of the tip of the stepped leader at a given instant and on the y-axis one can read the length of the connecting leader at that instant. Observe how the conductor with the smallest radius gives rise to the longest connecting leader. This is due to the large electric field enhancement at the vicinity of the tip of the conductor with the smaller radius. Note however that the differences in the height of attachment from one conductor radius to another are not very large. However, this slight difference in the striking distance or attractive radius may play a significant role if different conductors are competing with each other during lightning attachment. It is important, however, to note here that a lightning conductor with a smaller radius will go into corona and develop glow corona in a lower background electric field than that is necessary to generate glow corona from a conductor with a similar height but with a larger radius. Let us consider this statement further. According to the Whitehead’s formula as given by Ref. [8] the electric field necessary at the tip of the conductor to initiate corona increases with decreasing radius. On the other hand the electric field at the tip of a grounded conductor placed in a background electric field is larger than the background electric field itself due to field enhancement caused by the accumulation of charges on the conductor. For a given conductor height this field enhancement increases with decreasing radius. For lightning conductors of practical heights (i.e. longer than about 1 m) this field enhancement more than compensates for the larger electric fields necessary for corona inception from conductors of smaller radii. Because of this as the background electric field increases a conductor with a smaller radius goes into corona first compared to a conductor with a same height but having a larger radius. Once glow corona is developed from the tip of the conductor it may impede the attachment process [9,10]. Thus in the presence of glow corona this natural advantage to attract lightning flashes disappears from the conductors with smaller radii. If several conductors with different radii are competing with each other, the conductor with the smallest tip radius that is not undergoing glow corona will win the competition. In order to investigate this point further the background electric fields necessary to generate corona from conductors of different radii and heights are calculated. In these calculations the corona inception is assumed to take place when the streamer inception criterion is established at the tip of the conductor immersed in the background electric field. The results of the exercise are shown in Fig. 4. Note that the background electric field necessary to generate corona from the tip of the conductors decreases with increasing conductor height and with decreasing conductor radius. The measurements indicate that the background electric field at ground level below thunderclouds may reach values on the order of 5 kV/m–10 kV/m [11,12]. The results presented in Fig. 4 shows that for a conductor of height 6 m the conductors with radii less than about 0.005–0075 m may go into corona in a 5 kV/m–10 kV/m background electric field. In the case of conductors of 20 m height the critical radii is about 0.02 m.
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104
140
Conductor height: 20 m
Conductor height: 50 m Current: 30 kA
.001 m
102
Height of attachment, m
Height of attachment, m
Current: 30 kA
100 .01 m
98 .1 m
130
.001 m
120
.01 m
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100
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94 56
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.1 m
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200
.001 m
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160
.1 m
120
100
80
80 80
100
120
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80
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120
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160
Lateral distance to leader, m
Lateral distance to leader, m
Fig. 2. The height of attachment of the stepped leader to the lightning conductor as a function of the lateral distance from the lightning conductor to the stepped leader. Note that different curves in each diagram correspond to different conductor radii and the corresponding values are shown in each diagram. The highest value of the lateral distance corresponding to a given curve is the attractive radius corresponding to that conductor.
Height of connecting leader, m
16
12
.001 m
8
.01 m
.1 m
4
0 0
200
400
600
Height of leader tip from ground, m Fig. 3. The growth of the connecting leader from conductors of different radii as the stepped leader approach towards them. The figure corresponds to the case with 30 kA current and 50 m tall conductor.
Let us now consider the experiment of Moore et al. [1]. The experiment was conducted near the 3288-m high summit of South Baldy Peak, New Mexico. In this experiment lightning conductors of different radii (between 0.0045 m to 0.0255 m) of height approximately 6 m were arranged symmetrically and the lightning strikes to various conductors were monitored for a period of seven years. The rods competed with each other to attract lightning flashes occurring in the vicinity. Over the seven-year study, none of the sharp rods were struck by lightning but 12 of the blunt rods were. All of the strikes were to blunt rods with radii ranging between 0.0063 m to 0.0127 m although most of the lighting flashes struck the 0.0095 m blunt rods. None of the adjacent blunt rods with radii 0.0045 m or 0.0255 m were hit. The authors concluded that moderately blunt rods are more likely to generate successful connecting leaders than sharp or extremely blunt rods. Observe that, in this experiment, the tips of the competing conductors were located at a height about 6 m from ground level. The results of the corona calculations presented earlier show that at 6 m height the conductors of radii less than about 0.005–0.0075 m could go into corona in a background electric field of 5–10 kV/m. This shows the possible establishment of glow corona at the tip of the conductors having radii less than about 0.0075 m reducing their ability to attract stepped leader in comparison to the con-
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Electric field, V/m
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40 30 20 10
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Radius: 0.02 m
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0
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20000
Electric field, V/m
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Electric field, V/m
Fig. 4. The background electric field (depicted on the x-axis) necessary to generate corona from conductors of different radii. On the y-axis the height of the conductor necessary to generate corona in a given background electric field is shown. The radius of the conductor corresponding to each diagram is given at the top of the diagram.
ductor with the slightly higher radius that did not go into corona. This may explain the reason why conductors of radii around 0.01 m became better targets of lightning flashes than the conductors with smaller radii that went into corona. Since the ability to attract lightning flashes decreases with increasing radii, the conductor with the smallest radii that did not go into corona became a better attractor of stepped leaders than the conductors with larger radii or the conductors with smaller radii that went into corona. This could be the reason for the experimental observations of Moore et al. [1] where moderately blunt conductors were observed to be more efficient in attracting lightning flashes than sharp or extremely blunt conductors. The analysis presented in this paper also shows that, due to the effects of corona, there is no unique conductor radius that is better than other radii as lightning interceptors. The radius of the conductor that performs best as a lightning interceptor depends on the height of the conductors. For example, consider a conductor radius of 0.01 m. If the height of the conductor is less than about 5 m the background electric field necessary for corona generation is larger than about 20 kV/m. Since the background electric field generated by thunderclouds during thunderstorms is in the range 5–10 kV/m, no glow corona will set in at the tip of the conductor and the attachment to down coming stepped leaders could take place unhindered by glow corona. On the other hand, if the height of the conductor is increased to 15 m then the conductor tip will go into corona in a background electric field of 5–10 kV/m and the resulting glow corona could hinder the lightning attachment process. Thus, at small heights a conductor with a sharp radius may perform better in intercepting a lightning stepped leader while at larger heights this conductor may go into corona leaving a conduc-
tor with a larger radius that did not go into corona to perform better as a lightning interceptor. 5. Conclusions The main conclusion that can be extracted from the work presented here is that in the absence of glow corona sharp lightning conductors are more efficient in attracting lightning flashes than the conductors with blunt or larger tip radii. For example, a sharp conductor can intercept a lightning flash from a larger distance than does a blunt conductor. That is, the lightning attractive distance of a conductor increases with decreasing tip radius. However, this advantage disappears once the glow corona sets in at the tip of the conductor making the moderately blunt conductor a more preferable lightning attachment point than the sharp or extremely blunt conductor. Moreover, in a given background electric field, the probability of the inception of glow corona at the conductor tip increases with increasing conductor height and decreasing conductor radius. Thus, in a given electric field, as the conductor height increases its radius has to be increased to avoid the inception of glow corona at the tip. For this reason the conductor radius that performs best as a lightning interceptor depends on the height of the conductor and the best performance may shift from smaller radii to larger ones with increasing height of the conductors. References [1] C.B. Moore, G.D. Aulich, W. Rison, Measurements of lightning rod responses to nearby strikes, J. Geophys. Res. 27 (2000) 1487–1490. [2] M. Becerra, V. Cooray, A self-consistent upward leader propagation model, J. Phys. D: Appl. Phys. 39 (2006) 3708–3715. [3] M. Becerra, V. Cooray, Time dependent evaluation of the lightning upward connecting leader inception, J. Phys. D: Appl. Phys. 39 (2006) 4695–4702.
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[4] V. Cooray, M. Becerra, Attachment of lightning flashes to grounded structures, in: V. Cooray (Ed.), Lightning Protection, IET Publishers, UK, 2009 (Chapter 4). [5] I. Gallimberti, The mechanism of long spark formation, J. Phys. Coll. 40 (C7, Suppl. 7) (1972) 193–250. [6] F. Rizk, A model for switching impulse leader inception and breakdown of long air-gaps, IEEE Trans. Power Delivery 4 (1) (1989) 596–603. [7] V. Cooray, V. Rakov, N. Theethayi, The lightning striking distance—revisited, J. Electrostat. 65 (5–6) (2007) 296–306. [8] P.S. Maruvada, Corona Performance of High-Voltage Transmission Lines, Research Studies Press, Baldock, Hertfordshire, U.K, 2000, pp. 82–83. [9] E.M. Bazelyan, Yu P. Raizer, N.L. Aleksandrov, Corona initiated from grounded objects under thunderstorm conditions and its influence on lightning attachment, Plasma Sources Sci. Technol. 17 (2008) 024015.
[10] M. Becerra, Glow corona generation and streamer inception at the tip of grounded objects during thunderstorms: revisited, J. Phys. D: Appl. Phys. 46 (2013) 135205, http://dx.doi.org/10.1088/0022-3727/46/13/135205. [11] Serge Soula, Serge Chauzy, Multilevel measurement of the electric field underneath a thundercloud 2. Dynamical evolution of a ground space charge layer, J. Geophys. Res. 96 (D12) (1991) 22327–22336. [12] T.C. Marshall, M. Stolzenberg, Voltages inside and just above thunderstorms, J. Geophys. Res. 106 (D5) (2001) 4757–4768.
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The influence of lightning conductor radii on the attachment of lightning flashes Vernon Cooray Department of Engineering Sciences, Uppsala University, Uppsala, Sweden
a r t i c l e
i n f o
Article history: Received 17 May 2016 Received in revised form 9 December 2016 Accepted 2 January 2017 Available online xxx Keywords: Lightning conductors Lightning attachment Blunt and sharp conductors Glow corona
a b s t r a c t The influence of the tip radius of lightning conductors on their lightning attractive distance as predicted by the self-consistent leader inception and propagation model (SLIM) is presented. The results show that in the absence of any glow corona from the tip of the conductor a smaller tip radius gives rise to a larger attractive radius than a larger radius. It is suggested that the reason for the experimental observations which show that blunt conductors are more efficient lightning receptors than sharp ones is the presence of glow corona at the tip of the sharp ones during the time of lightning strikes. Moreover, in a given background electric field, the probability of the inception of glow corona at the conductor tip increases with increasing conductor height and decreasing conductor radius. Thus, in a given electric field, as the conductor height increases its radius has to be increased to avoid the inception of glow corona at the tip. For this reason, the conductor radius that performs best as a lightning interceptor depends on the height of the conductor and the best performance shift from smaller radii to larger ones with increasing height of the conductor. © 2017 Elsevier B.V. All rights reserved.
1. Introduction In a classical experiment Moore et al. [1] demonstrated that moderately blunt conductors are more efficient than sharp or extremely blunt conductors. They speculated that the glow corona at the tip of the sharp conductors could be the reason for their inefficiency, in comparison to the blunt ones, in attracting lightning flashes. However, a question that will arise naturally when studying the results of this experiment is the following: What would be the effect of conductor radius on lightning attachment if glow corona is not present at the tip of the conductors? In order to investigate this we will derive and compare the height of attachment and the attractive radii of conductors of different tip radii pertinent to stepped leaders of lightning flashes. The model simulations are conducted using SLIM, a physics based model introduced by Becerra and Cooray [2,3, see also 4 for a detailed description]. Before proceeding further let us summarize the main features of SLIM. 2. The model SLIM The main steps that are included in the model are: (1) formation of a streamer discharge at the tip of a grounded object (first,
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second or third streamer bursts). (2) Transformation of the stem of the streamer into a thermalized leader channel (unstable leader inception). (3) Extension of the positive leader and its self-sustained propagation (stable leader inception). Let us consider these steps in details. The description given below is based on the work published by Becerra and Cooray [3,4]. Assume that the electric field at ground level as a function of time generated by the down coming stepped leader is known. How this is evaluated in the model is described in Section 3. The simulation consists of several main steps and let us consider them one by one.
(1) The first step is to extract the time or the height of the stepped leader when streamers are incepted from the grounded rod. Since the background electric field produced by the stepped leader is known (or given) the electric field at the tip of the grounded rod can be calculated, for example, by using charge simulation method. This field is used together with the avalanche to streamer transition criterion to investigate whether the electric field at the conductor tip is large enough to convert avalanches to streamers. In the analysis it is assumed that the electron avalanche will be converted to a streamer when the number of positive ions at the head of the avalanche exceeds about 108 [5]. The simulation continues using the time
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2
Voltage
Background potential
A Potential along streamers
Distance from the tip of the conductor. Fig. 1. Distance–Voltage diagram that illustrates how the charge associated with a streamer burst is obtained. The area between the two curves representing the background potential and the streamer potential is marked A.
varying electric field of the stepped leader until the streamer inception criterion is satisfied. (2) The moment the streamer inception criterion is satisfied a burst of streamers will be generated from the extremity of the object; in our case from the tip of the lightning rod. The next task is to calculate the charge in this streamer burst. The charge associated with these streamer bursts are calculated using a distance–voltage diagram with the origin at the tip of the grounded conductor as follows. The procedure is illustrated in Fig. 1. The streamer zone is assumed to maintain a constant potential gradient Estr . In the distance–voltage diagram this is represented by a straight line. On the same diagram the background potential produced by the thundercloud and the down-coming stepped leader at the current time is depicted. If the area between the two curves up to the point where they cross is A (see Fig. 1), the charge in the streamer zone is given by Qo ≈ KQ A
(1)
where KQ is a geometrical factor. Becerra and Cooray [3,4] estimated its value to be about 3.5 × 10−11 C/V m. In the analysis we assume Estr = 5 × 105 V/m [5]. This value of the electric field is valid for streamers propagating in normal atmospheric density. A slight change of this electric field around this value (10%) in the calculations to be presented does not change the conclusions to be reached. (3) The next task is to investigate whether this streamer burst is capable of generating a leader. This decision is based on the fact that in order to generate a leader a minimum of 1 C is required in the charge generated by the streamers [5]. If the charge in the streamer zone is less than this value then the procedure is repeated a small time interval later. Note that with increasing time the electric field generated by the stepped leader increases and, consequently, the charge in the streamer bursts increases. (4) Assume that at time t, the condition necessary for leader inception is satisfied. The next task is to estimate the length and the radius of this initial leader section. In doing this it is assumed that the amount of charge that is necessary to create a unit length of positive leader is ql . Becerra and Cooray [3] evaluated this parameter using the equations given by Gallimberti [5] and it was shown that it is a function of the leader speed. For low leader speeds (around 104 m/s) its value is about 65 C/m. In the analysis the value of ql is estimated using the relationship between this parameter and the leader speed as published by Becerra and Cooray [3]. Based on these considerations, the initial length of the leader section L1 is given by Qo /ql , where Qo is
the charge in the streamer burst that immediately precede the inception of the leader. The initial radius of the leader, aL1 (t) is assumed to be 10−3 m and the initial potential gradient of the leader section, EL1 (t) is assumed to be equal to the potential gradient of the streamer region i.e. 5.0 × 105 V/m. Now we proceed to the next time step, i.e. t = t + t (5) During the time interval t there will be a change in the background potential and we also have a small leader section of length L1 . Now the new charge in the streamer zone generated from the head of the new leader section is calculated as before but now including both the leader and its streamer zone in the distance–voltage diagram. The leader is represented by a line with a potential gradient EL1 (t). The charge generated in the current time step is obtained by subtracting from this the total charge obtained in the previous time step. Let the charge obtained thus be Q1 . This charge is used to evaluate the length of the new leader section L2 . Moreover, the flow of this charge through the leader channel changes the potential gradient and the radius of the older leader section L1 . The new potential gradient and the radius of L1 are given by EL1 (t + t) and aL1 (t + t). (6) Now let us consider the nth time step. There are n leader sections and they have their respective potential gradients and radii. The radius and the potential gradient of ith leader section are obtained from · a2Li (t + t) = · a2Li (t) + ELi (t + t) =
a2Li (t) a2Li (t + t)
−1 E (t) · ILi (t) · t · p0 Li
(2)
(3)
ELi (t)
In the above equation ELi (t) is the internal electric field, ILi (t) is the current of the leader section Li at time t, p0 is the standard atmospheric pressure and is the ratio between the specific heats at constant volume and constant pressure for air [5]. With these equations it is possible to calculate the time evolution of the internal electric field for each segment and the potential drop along the leader channel (at a given time) as follows: UL =
n
ELi (t) · Li
(4)
i=1
The steps described above can be used to simulate the inception and propagation of positive leaders. The calculation can be simplified if, instead of calculating the time evolution of leader potential gradient in each segment as above, one uses the expression derived by Rizk [6] for the potential of the tip of the leader channel which is given by
(i) Utip
=
(i) lL E∞
(i)
Estr − E∞ − Estr + xo E∞ ln − e E∞ E∞
l /x0 L
(5)
(i)
In the above equation lL is the total leader length at the current simulation step, E∞ is the final quasi-stationary leader gradient and x0 is a constant parameter given by the product v , where v is the ascending positive leader speed and is the leader time constant. (7) In the model the negative stepped leader is assumed to travel towards ground without being influenced by the connecting leaders issued from grounded structure unless final jump condition is established with one of them. In the original SLIM, the tip of the connecting leader is assumed to travel, at any given moment, towards the current location of the tip of the down coming stepped leader. In the simulations presented here this assumption is relaxed and the connecting leader is assumed
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to travel in the direction of the maximum background electric field at the current location of the tip of the connecting leader. (8) The condition for the final attachment is assumed to be fulfilled when the average potential gradient in the gap between the tips of the down-coming stepped leader and the upwardmoving connecting leader reaches a value 500 kV/m (i.e. final jump condition). 3. Charge distribution on the leader channel In a recent study, Cooray et al. [7] measured the charge brought to ground by the first 100 s of the return stroke. They found a strong correlation between the first return stroke current and this charge. Combining this information with the bi-directional leader model they have investigated how this charge was distributed along the stepped leader channel. In their analysis they have assumed that the stepped leader channel is vertical. According to their results the linear charge distribution (in C/m) on the stepped leader channel when its tip is at a height of zo above ground is given by
(z) = ao 1 − = (z − zo ) G(zo ) = 1 −
L − zo
z > zo zo L
J(zo ) = 0.3˛ + 0.7ˇ ˛=e ˇ=
−(zo −10)
zo 1− L
/75
G(zo )Ip +
Ip (a + b) 1 + c + d 2
J(zo )
(6) (7) (8) (9) (10) (11)
In the above equations z is the vertical height of the point of observation, L is the total length of the stepped leader channel in meters, (z) is the charge per unit length of a leader section located at height z, Ip is the return stroke peak current in kA, ao = 1.476 × 10−5 , a = 4.857 × 10−5 , b = 3.9097 × 10−6 , c = 0.522 and d = 3.73 × 10−3 . Note that the above equation is valid for zo ≥ 10 m. This charge distribution is utilized in calculating the electric field produced by the down coming stepped leader. 4. Results In the analysis the attractive radii of a conductor is obtained as follows. Starting with zero, the lateral distance to the stepped leader channel from conductor is gradually increased until the stepped leader is intercepted by the ground instead of the lightning conductor. This critical distance where the stepped leader is intercepted by the ground is taken to be the attractive radii of the lightning conductor pertinent to the prospective return stroke current associated with the stepped leader used in the simulation. The height of attachment is defined as the height of the stepped leader when the final jump condition is established between the connecting leader and the stepped leader. The height of attachment may vary depending on the lateral distance to the stepped leader from the conductor. In the simulations the conductor is assumed to be cylindrical in shape with a hemispherical tip. The tip radius is a variable in the simulation. In the simulation, for a given conductor height and lateral distance to the leader, the height at which the leader is intercepted by the connecting leader originating from the conductor is estimated. This parameter is obtained for three conductor radii and for four different conductor heights. The prospective return stroke current is kept constant at 30 kA in the simulation. The results of the simulation are shown in Fig. 2. Note that in the figures the largest lateral distance corresponding to a given radius is the attractive radius corresponding to that conductor.
3
One can see from these figures that, for a given conductor height, the conductor with a smallest radius will be able to intercept the stepped leader at the largest height. In other words the smaller the radius of the conductor the more efficient would it be in intercepting the stepped leader. Note also that the attractive radii of the conductor increase with decreasing conductor radius. The reason for this change in intercepting efficiency is the ability of the conductor with the smaller radius to generate a connecting leader earlier than a conductor with a larger radius. For example, the growth of the connecting leaders from conductors of different radii as the stepped leader approaches them is shown in Fig. 3. This figure corresponds to 30 kA current and 50 m tall conductor. On the x-axis of this figure one can read the height of the tip of the stepped leader at a given instant and on the y-axis one can read the length of the connecting leader at that instant. Observe how the conductor with the smallest radius gives rise to the longest connecting leader. This is due to the large electric field enhancement at the vicinity of the tip of the conductor with the smaller radius. Note however that the differences in the height of attachment from one conductor radius to another are not very large. However, this slight difference in the striking distance or attractive radius may play a significant role if different conductors are competing with each other during lightning attachment. It is important, however, to note here that a lightning conductor with a smaller radius will go into corona and develop glow corona in a lower background electric field than that is necessary to generate glow corona from a conductor with a similar height but with a larger radius. Let us consider this statement further. According to the Whitehead’s formula as given by Ref. [8] the electric field necessary at the tip of the conductor to initiate corona increases with decreasing radius. On the other hand the electric field at the tip of a grounded conductor placed in a background electric field is larger than the background electric field itself due to field enhancement caused by the accumulation of charges on the conductor. For a given conductor height this field enhancement increases with decreasing radius. For lightning conductors of practical heights (i.e. longer than about 1 m) this field enhancement more than compensates for the larger electric fields necessary for corona inception from conductors of smaller radii. Because of this as the background electric field increases a conductor with a smaller radius goes into corona first compared to a conductor with a same height but having a larger radius. Once glow corona is developed from the tip of the conductor it may impede the attachment process [9,10]. Thus in the presence of glow corona this natural advantage to attract lightning flashes disappears from the conductors with smaller radii. If several conductors with different radii are competing with each other, the conductor with the smallest tip radius that is not undergoing glow corona will win the competition. In order to investigate this point further the background electric fields necessary to generate corona from conductors of different radii and heights are calculated. In these calculations the corona inception is assumed to take place when the streamer inception criterion is established at the tip of the conductor immersed in the background electric field. The results of the exercise are shown in Fig. 4. Note that the background electric field necessary to generate corona from the tip of the conductors decreases with increasing conductor height and with decreasing conductor radius. The measurements indicate that the background electric field at ground level below thunderclouds may reach values on the order of 5 kV/m–10 kV/m [11,12]. The results presented in Fig. 4 shows that for a conductor of height 6 m the conductors with radii less than about 0.005–0075 m may go into corona in a 5 kV/m–10 kV/m background electric field. In the case of conductors of 20 m height the critical radii is about 0.02 m.
Please cite this article in press as: V. Cooray, The influence of lightning conductor radii on the attachment of lightning flashes, Electr. Power Syst. Res. (2017), http://dx.doi.org/10.1016/j.epsr.2017.01.002
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104
140
Conductor height: 20 m
Conductor height: 50 m Current: 30 kA
.001 m
102
Height of attachment, m
Height of attachment, m
Current: 30 kA
100 .01 m
98 .1 m
130
.001 m
120
.01 m
110 .1 m
100
96
90
94 56
60
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68
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80
240
Conductor height: 75 m Current: 30 kA
160 .001 m
140 .01 m
120
.1 m
Height of attachment, m
Height of attachment, m
100
110
120
Lateral distance to leader, m
Lateral distance to leader, m
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90
Conductor height: 100 m Current: 30 kA
200
.001 m
.01 m
160
.1 m
120
100
80
80 80
100
120
140
80
100
120
140
160
Lateral distance to leader, m
Lateral distance to leader, m
Fig. 2. The height of attachment of the stepped leader to the lightning conductor as a function of the lateral distance from the lightning conductor to the stepped leader. Note that different curves in each diagram correspond to different conductor radii and the corresponding values are shown in each diagram. The highest value of the lateral distance corresponding to a given curve is the attractive radius corresponding to that conductor.
Height of connecting leader, m
16
12
.001 m
8
.01 m
.1 m
4
0 0
200
400
600
Height of leader tip from ground, m Fig. 3. The growth of the connecting leader from conductors of different radii as the stepped leader approach towards them. The figure corresponds to the case with 30 kA current and 50 m tall conductor.
Let us now consider the experiment of Moore et al. [1]. The experiment was conducted near the 3288-m high summit of South Baldy Peak, New Mexico. In this experiment lightning conductors of different radii (between 0.0045 m to 0.0255 m) of height approximately 6 m were arranged symmetrically and the lightning strikes to various conductors were monitored for a period of seven years. The rods competed with each other to attract lightning flashes occurring in the vicinity. Over the seven-year study, none of the sharp rods were struck by lightning but 12 of the blunt rods were. All of the strikes were to blunt rods with radii ranging between 0.0063 m to 0.0127 m although most of the lighting flashes struck the 0.0095 m blunt rods. None of the adjacent blunt rods with radii 0.0045 m or 0.0255 m were hit. The authors concluded that moderately blunt rods are more likely to generate successful connecting leaders than sharp or extremely blunt rods. Observe that, in this experiment, the tips of the competing conductors were located at a height about 6 m from ground level. The results of the corona calculations presented earlier show that at 6 m height the conductors of radii less than about 0.005–0.0075 m could go into corona in a background electric field of 5–10 kV/m. This shows the possible establishment of glow corona at the tip of the conductors having radii less than about 0.0075 m reducing their ability to attract stepped leader in comparison to the con-
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8
4
Height necessary for corona, m
12
30
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10
0
0 0
4000
8000
12000
16000
4000
8000
120 Height necessary for corona, m
40
20
0
16000
20000
4000
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12000
Electric field, V/m
16000
20000
40 30 20 10
4000
8000
12000
16000
20000
16000
20000
Electric field, V/m 500
Radius: 0.02 m
80
40
Radius: 0.1 m
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0 0
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Radius: 0.01 m
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Radius: 0.0075 m
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Height necessary for corona, m
16
5
0
4000
8000
12000
16000
20000
Electric field, V/m
0
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8000
12000
Electric field, V/m
Fig. 4. The background electric field (depicted on the x-axis) necessary to generate corona from conductors of different radii. On the y-axis the height of the conductor necessary to generate corona in a given background electric field is shown. The radius of the conductor corresponding to each diagram is given at the top of the diagram.
ductor with the slightly higher radius that did not go into corona. This may explain the reason why conductors of radii around 0.01 m became better targets of lightning flashes than the conductors with smaller radii that went into corona. Since the ability to attract lightning flashes decreases with increasing radii, the conductor with the smallest radii that did not go into corona became a better attractor of stepped leaders than the conductors with larger radii or the conductors with smaller radii that went into corona. This could be the reason for the experimental observations of Moore et al. [1] where moderately blunt conductors were observed to be more efficient in attracting lightning flashes than sharp or extremely blunt conductors. The analysis presented in this paper also shows that, due to the effects of corona, there is no unique conductor radius that is better than other radii as lightning interceptors. The radius of the conductor that performs best as a lightning interceptor depends on the height of the conductors. For example, consider a conductor radius of 0.01 m. If the height of the conductor is less than about 5 m the background electric field necessary for corona generation is larger than about 20 kV/m. Since the background electric field generated by thunderclouds during thunderstorms is in the range 5–10 kV/m, no glow corona will set in at the tip of the conductor and the attachment to down coming stepped leaders could take place unhindered by glow corona. On the other hand, if the height of the conductor is increased to 15 m then the conductor tip will go into corona in a background electric field of 5–10 kV/m and the resulting glow corona could hinder the lightning attachment process. Thus, at small heights a conductor with a sharp radius may perform better in intercepting a lightning stepped leader while at larger heights this conductor may go into corona leaving a conduc-
tor with a larger radius that did not go into corona to perform better as a lightning interceptor. 5. Conclusions The main conclusion that can be extracted from the work presented here is that in the absence of glow corona sharp lightning conductors are more efficient in attracting lightning flashes than the conductors with blunt or larger tip radii. For example, a sharp conductor can intercept a lightning flash from a larger distance than does a blunt conductor. That is, the lightning attractive distance of a conductor increases with decreasing tip radius. However, this advantage disappears once the glow corona sets in at the tip of the conductor making the moderately blunt conductor a more preferable lightning attachment point than the sharp or extremely blunt conductor. Moreover, in a given background electric field, the probability of the inception of glow corona at the conductor tip increases with increasing conductor height and decreasing conductor radius. Thus, in a given electric field, as the conductor height increases its radius has to be increased to avoid the inception of glow corona at the tip. For this reason the conductor radius that performs best as a lightning interceptor depends on the height of the conductor and the best performance may shift from smaller radii to larger ones with increasing height of the conductors. References [1] C.B. Moore, G.D. Aulich, W. Rison, Measurements of lightning rod responses to nearby strikes, J. Geophys. Res. 27 (2000) 1487–1490. [2] M. Becerra, V. Cooray, A self-consistent upward leader propagation model, J. Phys. D: Appl. Phys. 39 (2006) 3708–3715. [3] M. Becerra, V. Cooray, Time dependent evaluation of the lightning upward connecting leader inception, J. Phys. D: Appl. Phys. 39 (2006) 4695–4702.
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[4] V. Cooray, M. Becerra, Attachment of lightning flashes to grounded structures, in: V. Cooray (Ed.), Lightning Protection, IET Publishers, UK, 2009 (Chapter 4). [5] I. Gallimberti, The mechanism of long spark formation, J. Phys. Coll. 40 (C7, Suppl. 7) (1972) 193–250. [6] F. Rizk, A model for switching impulse leader inception and breakdown of long air-gaps, IEEE Trans. Power Delivery 4 (1) (1989) 596–603. [7] V. Cooray, V. Rakov, N. Theethayi, The lightning striking distance—revisited, J. Electrostat. 65 (5–6) (2007) 296–306. [8] P.S. Maruvada, Corona Performance of High-Voltage Transmission Lines, Research Studies Press, Baldock, Hertfordshire, U.K, 2000, pp. 82–83. [9] E.M. Bazelyan, Yu P. Raizer, N.L. Aleksandrov, Corona initiated from grounded objects under thunderstorm conditions and its influence on lightning attachment, Plasma Sources Sci. Technol. 17 (2008) 024015.
[10] M. Becerra, Glow corona generation and streamer inception at the tip of grounded objects during thunderstorms: revisited, J. Phys. D: Appl. Phys. 46 (2013) 135205, http://dx.doi.org/10.1088/0022-3727/46/13/135205. [11] Serge Soula, Serge Chauzy, Multilevel measurement of the electric field underneath a thundercloud 2. Dynamical evolution of a ground space charge layer, J. Geophys. Res. 96 (D12) (1991) 22327–22336. [12] T.C. Marshall, M. Stolzenberg, Voltages inside and just above thunderstorms, J. Geophys. Res. 106 (D5) (2001) 4757–4768.
Please cite this article in press as: V. Cooray, The influence of lightning conductor radii on the attachment of lightning flashes, Electr. Power Syst. Res. (2017), http://dx.doi.org/10.1016/j.epsr.2017.01.002