Analysis of the influencing factors of electric field distortion caused by the lightning rod

Abstract

The three-dimensional model of a lightning rod under a thunderstorm simulated by COMSOL Multiphysics software is established in this paper to study the influencing factors on electric field distortion including the position, height and shape of the lightning rod. In the model, the thunderstorm cloud is equivalent to a plate capacitor; and the building, the lightning rod and the ground are regarded as perfect conductors; whereas their surface potential is zero. The electric field distortion coefficient K is used to measure the influencing parameters on the lightning attachment to the lightning rod. The analysis of model leads to the following conclusions: influenced by the electric field distortion, a lightning rod at a higher height or at the edge of the building is more likely stuck by the lightning. Considering the effects of the upward connecting leaders in the attachment process, we can further infer that the upward leaders usually start from the tip with greater distortion electric field in the attachment process. However, when the tip of the rod is too sharp, it is often accompanied by the fast attenuation of the electric field, which is not beneficial for the continuous development of the upward connecting leaders. Therefore, in the process of lightning protection design, it is necessary to consider both the position and the height, as well as the shape of the lightning rod in a comprehensive manner.

Keywords

Lightning rod electric field distortion coefficient attachment process

1. Introduction

Since the 1950s, combined with the data of lightning observations and long-gap discharge experiment, the study on the attachment mechanism of lightning has a great advance. The lightning striking distance, which applied in the most commonly used electro-geometric model (EGM) [1], is obtained from the data of numerous long-gap discharge experiments. The rolling ball method derived from the EGM can easily determine the protection zone of the lightning rod under different protection circumstances. At the same time, researchers, such as Berger [2] and Eriksson [3], used camera to photograph lightning striking to the tower and found that the upward leader launched on the tower intercepted the downward one before the start of return stroke. Taking into account the influence of the connecting leader, researchers modified the EGM and developed the leader propagation model (LPM). Becerra and Cooray [4] further proposed the self-consistent leader inception and propagation model (SLIM). Yang et al. [5] used numerical method to simulate the two models and compared the differences between them For low-rise buildings, the impact of the connecting leader is relatively small, and protection zone of the EGM is in good agreement with that of the SLIM. However, as the height of building increases, the estimated protection zone of the EGM is smaller than that of the SLIM, and the error increases [4].

Moore et al. [6,7] have done a lot of pioneering work in the study of the geometric characteristics of the lightning rod. They compared and tested the attachment characteristics of lightning rods’ tips with different sharpness in natural flash. The results showed that blunt lightning rods may be more vulnerable to being striking in a certain condition. Sidik et al. [8] compared the development of corona discharge of lightning rod at the head of different tip by corona discharge experiments, and found that the corona area of the head of the blunt lightning rod has less shielding charge and lower breakdown voltage. It was also observed in the experiment that the origin of the corona was related to the coverage area of the tip.

In addition, many researchers have carried out a lot of researches on the influence of geometric features on the tip of the object using numerical simulation methods. Eriksson [3] discussed the influence of the ratio of height H to the radius R of the cylindrical lightning rod on the distortion coefficient of the electric field and worked out the logarithmic relationship between the two. Moore [9] approximated the tip of the lightning rod as a semi-ellipsoid, and discussed the relationship between the ratio of longer semi-axis of the ellipsoid c to the radius of curvature a at the apex and the distortion coefficient. Alessandro [10] used the finite element method to establish a three-dimensional model to calculate and analyze the electric field distortion caused by the geometric features of buildings and lightning rods. However, the relationship between the distortion coefficient K and the geometric features of the objects was not clearly given. Gen [11] found that as the height of the building increases, the vertical electric field at the tip of the lightning rod increases, and both increase approximately linearly. Wei [12] used EM simulation software to simulate the attachment process between the leader and the building, taking into account the building size, the potential of leader, the height of the lightning rod, and its location. After determining the attachment process and the lightning current, the coupling process of the lightning electromagnetic wave and the coaxial line [13] can be continued as well as the coupling of the lightning electromagnetic field and the distribution line [14]. That will be more conducive to lightning protection.

The COMSOL Multiphysics [5,6], which is based on the finite element method, is used here to simulate the electrostatic field at the tip of the lightning rod. The finite element method is based on the variational principle and the discretization method to obtain the approximate solution. The calculation using the differential form of Maxwell’s equations can be applied to the simulation of complex geometric structures or boundaries. Higher accuracy simulation results require more detailed meshing, which at the same time generates a huge amount of computation. The implementation of this algorithm is feasible due to the rapidly improved computer performance.

2. Basic theory

The distribution of an electrostatic field at any point in space can be obtained by the following equation in Maxwell’s equations: $\begin{eqnarray}\displaystyle {\nabla}\cdot D={\rho}. & & \displaystyle\end{eqnarray}$ (1) D is the electric displacement vector and 𝜌 is the free space charge density. Equation (1) shows that the electric field is an active field, and the distribution of the electrostatic field is only related to the distribution of space charge.

In an isotropic homogeneous medium, the electrical displacement vector D has the following constitutive relationship with the field strength E, and ϵ is the dielectric constant: $\begin{eqnarray}\displaystyle D={\varepsilon}E. & & \displaystyle\end{eqnarray}$ (2) In addition, the electric field E is the negative gradient of the potential: $\begin{eqnarray}\displaystyle E=-{\nabla}{\varphi}. & & \displaystyle\end{eqnarray}$ (3)

The Poisson equation describing the potential distribution in the medium can be obtained by combining the above equations ((1))–((3)): $\begin{eqnarray}\displaystyle {\nabla}^2{\varphi}=-{\rho}/{\varepsilon}. & & \displaystyle\end{eqnarray}$ (4) The induced charge only exists on the surface of the conductor. There is no free charge in the uniform air medium, i.e., 𝜌 = 0. Then, the electric field distribution in the environment can be obtained by the Laplace equation simplified from the Poisson equation: $\begin{eqnarray}\displaystyle {\nabla}^2{\varphi}=0. & & \displaystyle\end{eqnarray}$ (5) When a thunderstorm cloud with a large amount of charge appears above the lightning rod, the electric field around the lightning rod will increase and a certain charge will be induced on the surface of the lightning rod. Since the radius of the tip curvature of the lightning rod is generally small, the electric field lines near the tip are densely distributed, and therefore the electric field distortion becomes enhanced. When the field exceeds the air breakdown threshold, the lightning rod tip will begin to initiate the upward connection leader.

3. The establishment of model

The influence of parameters of the lightning rod on the distribution of the electric field in the lightning flash environment may be quite complicated. To simplify the problem for discussion, the following assumptions are made before the model is established:

(1) The complicated charge structure in a thunderstorm cloud can be represented by a typical tripolar charge structure. The upper part is the main positive charge region, the middle part is the main negative charge region, and there is a small amount of positive charge distribution at the cloud bottom. The horizontal scale of the main charge area in the cloud is much larger than its vertical scale. The height of the charge area is very high compared to the size of the objects on the ground. Therefore, the thunderstorm cloud can be seen as a charged plate in the calculation. This hypothesis implies that the distribution of the environmental electric field under the cloud is similar to that of the flat plate capacitor and is uniform.

(2) As the leader develops, the determination of the lightning strike point becomes extremely complicated due to numerous influencing factors. Because of the randomness of the discharge in the corona flow region, the electric field distribution is non-uniform, which makes the development direction of the leader appear random. At the same time, too much corona charge will weaken the electric field intensity in front of the streamer and inhibit the development of the leader in this direction. It can lead to the unstable development of the upward leader originating from the tip of the lightning rod, thereby reducing the possibility of the lightning striking the lightning rod. In this paper, the shielding effect of the corona charge is not taken into account, and the lightning attachment characteristics of the lightning rod is considered only from the viewpoint of electric field distortion. To visually describe the electric field distortion near the different lightning rods, the ratio of the electric field strength E at a certain point to the background electric field E ₀ is defined as the electric field distortion coefficient K at this point: $\begin{eqnarray}\displaystyle \textit{K}=E/E_0. & & \displaystyle\end{eqnarray}$ (6) Since the electric field distortion coefficient K is a normalized value, its size is only related to the geometry of the lightning rod and has nothing to do with the ambient electric field. Therefore, with K, a simpler uniform electric field distribution can be used to replace the complex electric field changes, and avoid the large amount of computation that may occur in numerical simulation. It is advisable to set the environmental electric field under thunderstorm cloud to 15 kV/m.

(3) Assuming that the ground is a flat conductor plane, the building and the lightning rod are metal conductors, and they have a good electrical connection with the ground. Then, it can be assumed that potential of the surface of the ground, buildings and lightning rods is zero. At the same time, taking into account the characteristics of the conductor under an external electric field, the potential inside the conductor is always zero with no induced charge, the induced charge only exists in the surface layer.

Fig. 1.

Spatial model.

In combination with the above assumptions, the finite element method of the COMSOL in electrostatic module is used to simulate the electric field distortion of the lightning rod tip at different height, radius of curvature of the tip, and placement position. In the COMSOL, a three-dimensional calculation area of 200 m × 200 m × 200 m above ground is established, as Fig. 1. The free tetrahedral mesh is used to divide the calculation area. The maximum unit size is 5 m and the minimum unit size is 0.3 m. The potential on the upper surface of the calculation area is set as 3000 kV (15 kV/m × 200 m). The potential of the lower surface of the calculation area, the building above it and the lightning rod on the building are all set as 0. The material of the building and the lightning rod is selected as copper, and the remaining computation space is filled with air medium.

4. Simulation results and analysis

4.1. The effect of lightning rod height

To study the electric field distortion of individual lightning rods at different heights, the height of the lightning rod tip ranges from 5 m to 50 m, and each interval is 5 m. The lightning rod in the simulation is replaced with a cylinder with a hemisphere at the top, the radius of the top semicircle is set as 1 m. And the simulation results are shown in Fig. 2.

Fig. 2.

Electric field distribution of lightning rod tip at different heights.

From the distribution of the potential contours, it is easy to see that as the height of the lightning rod increases, the potential contours near its tip become denser. The electric field strength E is a negative gradient of the electric potential V, so the denser the electric potential contour is, the greater the field strength and the more severe the electric field distortion will be.

As is shown in the Fig. 3, the relationship between the maximum electric field distortion coefficient of the lightning rod tip and the height of the lightning rod tip is shown, and the dash line is the fitting result. The maximum distortion coefficient at the tip is approximately proportional to the height, and the maximum electric field distortion coefficient increases in a linear manner with height. At H = 50 m, the maximum distortion coefficient deviates from the fitted line, which may be due to insufficient selection of computational space. In the model, it is directly assumed that the upper surface potential on the calculation area is constant, but this assumption requires that the height of the upper surface is at least more than 5 times the size of the calculated object as it is shown in Fig. 3. Otherwise, the potential of the upper surface under the influence of the ground object is actually indeterminate. Substitution of a constant potential plane results in a small calculated electric field distortion.

Fig. 3.

Lightning rod tip height and maximum electric field distortion coefficient.

4.2. The influence of the radius of curvature of the lightning rod tip

Considering the influence of the curvature radius of the lightning rod tip on the electric field distribution, the height of the lightning rod tip is fixed at 30 m in the simulation, and the tip of the lightning rod is equivalent to a spheroid.

As shown in Fig. 4, it is the projection of the spheroid in the xz-axis plane. C is the length of the upper long semi-axis on the z-axis, the upper short-minor axis a on the x-axis is equal to the upper short-axis b on the y-axis. The radius of curvature of the tip of the lightning rod is equal to the radius of curvature of the projected ellipse at the apex A of the z-axis, and the radius of curvature of point A is r = a ²∕c.

Fig. 4.

Projection of an ellipsoid.

Let a = b = 1 m, c is set as 20 m, 10 m, 5 m, 1 m and 0, respectively, corresponding to the radius of curvature of the tip 1/20 m, 1/10 m, 1/5 m, 1 m, ∞. The lightning rod is located at the center of the top surface of a 20 m × 20 m × 20 m cube building.

As is shown in Fig. 5, the smaller the radius of curvature of the tip of the lightning rod is, that is, the sharper the tip is, the stronger the effect of the electric field distortion caused by it will be. The peak of each curve has been marked in the figure. It should be noted that the peak value shown by curve c = 0 is not the maximum distortion coefficient at the tip of the lightning rod. c = 0 means that the lightning rod is a cylindrical rod with a flat top surface, the true maximum electric field distortion appears at the edge of the top surface.

Fig. 5.

Vertical distribution of the maximum electric field distortion coefficient of the tip of the lightning rod with different radius of curvature.

As shown in Fig. 6, at the center axis, the smaller the radius of the tip curvature is, the greater the maximum electric field distortion coefficient at the corresponding tip becomes. However, in the process that field point is far away from the center axis horizontally, the distortion of the electric field caused by the tip with a smaller radius of curvature will decrease more quickly. After exceeding a certain distance, the distance is about 0.4 m in the figure. The degree of the electric field distortion caused by the “blunt” tip will exceed that of the ”sharper” tip. So, compare to the shape rod, lightning strike probability will be increased when the tip of the lightning rod is blunt appropriately. Which is consistent with the results of Moore et al. [6,7]. In addition, as mentioned above, for the flat cylindrical lightning rod on the top surface, the electric field distortion value is the largest at the top edge and smallest in the center region.

Fig. 6.

Variation of electric field distortion along the horizontal direction at the tip of different radius of curvature.

4.3. Influence of the position of lightning rod on tip of the building

In the model, the building is set as a 20 m × 20 m × 20 m cube with a 5 m lightning rod on top. The tip of the lightning rod is a hemisphere and its radius is 1 m. The distortion values of surrounding electric field are compared when the lightning rod is located at A, B, C and D. As is shown in the Fig. 7, A, B, C and D are on the surface of the building, with their coordinates. A is at the center of the surface, B and C are closer to the corner of the building, C is in the corner of the upper surface and D is near the middle point of the edge of the upper surface of the building. Line segment EF represents the edge of the building.

Fig. 7.

Selection of the location of the lightning rod on top of the cube building.

The influence on the distortion of electric field near edge of the building EF when the lightning rod is set at A, B, C and D is shown in the Fig. 8. The solid line shows the distortion of electric field when the building has no lightning rod on it. The projection of the building in the xz-axis plane is shown in the lower part of the figure.

Fig. 8.

Influence of different positions of the lightning rod on the electric field near ab edge of the building.

It can be easily found out that when there is no lightning rod, the distortion of the electric field in the corner of the edge of the building EF is relatively larger than that in the middle. As the lightning rod moves closer to the corner, the distortion in the corner becomes even stronger. When the lightning rod is at A and D, the distortion of electric field is only a little bit stronger, while at C and B, the distortion of electric field becomes remarkably stronger. The closer the lightning rod is to the corner, the stronger the distortion of electric field will be. At the same time, changes of the distortion of electric field in horizontal distance become even stronger and the curve becomes more precipitous. But when the lightning rod is a few meters away from the corner of the building, the electric field of all curves will attenuate to almost the same level with environmental electric field. Limited by the capacity of personal computer, the maximum size of the grid in the simulation is 5 m and a more detailed dissection will lead to too much computation task to accomplish. Therefore, there may exist certain error in the simulation result of the electric field distribution in the middle of the edge.

As is shown in the Fig. 9, the maximum coefficient of distortion of electric field near the tip of lightning rod changes together with the position of the rod. The distortion becomes much stronger when the rod is on the edge and in the corner of the upper surface of the building and the maximum distortion is observed when the rod is in the corner. When the lightning rod is put in the corner of the upper surface of the building, the electric field distortion on the tip of the lightning rod will be enhanced. The closer the lightning rod is to the corner, the effect of such enhancement will be observed. And in the surrounding of the point with the strongest electric field distortion, the gradient of electric field has a remarkable change.

Fig. 9.

Vertical changes of distortion of electric field on the central axis of lightning rod in different positions.

5. Summary

In this paper, the electric field distortion coefficient K of different positions and different geometric structures of the lightning rod at the top of the building is analyzed. From the above analysis of the model, the following conclusions can be drawn:

(1) The higher the height of the lightning rod is, the stronger the electric field distortion of its tip will be, and the maximum electric field distortion coefficient is approximately linearly related to the height.

(2) The smaller the radius of curvature of the tip of the lightning rod is, the stronger the distortion of the electric field at the tip will be. But at the same time, the sharper tip will also cause a rapid attenuation of electric field distortion at the horizontal distance. After a certain distance, the electric field distortion caused by the lightning rod with a larger tip curvature radius will be larger than the electric field distortion caused by the lightning rod with a smaller tip curvature radius.

(3) For the position of the lightning rod, a tip more closer to the edge or the corner of the building will get larger maximum coefficient of electric field distortion. At the same time, the electric field distortion in the corner of the building is also enhanced.

Therefore, in order to protect against lightning better, the selection of the position and the geometry structure of a lightning rod should meet the requirements of the lightning interception ability of it in a given environment. If it is required to have a strong capability of leader interception, the lightning rod with a large radius of curvature of the tip, a suitable height and mounting position should be chosen. Under a certain ambient electric field, the tip of the lightning rod can initiate an upward leader more easily. The attenuation of the slowing electric field distortion at tip enables the originating upward leader to further develop. Sometimes we do not want the lightning rod to be a frequent leader intercept device, for protecting the flammable and explosive goods warehouse, because the frequent lightning attachment to the lightning rod may cause the probability of accidents to increase. At this time, we may consider sharper lightning rod tips. Although the tip of the lightning rod will frequently originate the upward leader, due to the rapid electric field attenuation, the upward leader can only continue to develop and connect with it when the downward leader is close enough.

Footnotes

Acknowledgements

This work was supported by the National Key R&D Program (2017YFC1501501) and Jiangsu Overseas Research & Training Program for University Prominent Young & Middle-aged Teachers and Presidents (2018).

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