A new approximate method for lightning-radiated extremely low-frequency (ELF) and very low-frequency (VLF) ground wave propagation over intermediate ranges is presented in this paper. In our approximate method, the original field attenuation function is divided into two factors in frequency domain representing the propagation effect of the ground conductivity and Earth’s curvature, and both of them have clearer formulations and can more easily be calculated rather than solving a complex differential equation related to Airy functions. The comparison results show that our new approximate method can predict the lightning-radiated field peak value over the intermediate range with a satisfactory accuracy within maximum errors of 0.0%, −3.3%, and −8.7% for the earth conductivity of 4 S/m, 0.01 S/m, and 0.001 S/m, respectively. We also find that Earth’s curvature has much more effect on the field propagation at the intermediate ranges than the finite ground conductivity, and the lightning-radiated ELF/VLF electric field peak value (V/m) at the intermediate ranges yields a propagation distance
Lightning-radiated very low-frequency (VLF) (3–30 kHz) and low-frequency (LF) (30–300 kHz) signals can propagate along a spherical earth with finite conductivity at an intermediate distance of hundreds to a couple of thousand kilometers, and the observed lightning-radiated electromagnetic field signal would be significantly attenuated and distorted due to the propagation effects (e.g., Wait [
The general problem of the radiation from an antenna and the propagation of ground wave over homogeneous earth has been studied since the 1910s (e.g., Watson [
In 2009, in order to study the propagation effect of lightning-radiated VLF/LF field along the spherical earth with finite conductivity, based on the formulas presented by Wait [
Therefore, in this paper, we will present an improved new approximate method for computing the propagation effect of lightning-produced extremely low-frequency (ELF) and very low-frequency (VLF) ground wave propagation over intermediate ranges. In our method, we will divide the ground wave attenuation function presented by Wait [
For a vertical electric dipole source located on the surface of a smooth spherical earth, the vertical electric field strength
For the electromagnetic field computation of the cloud-ground lightning, both the source and the receiver are assumed to be on the earth surface and the attenuation function
In order to obtain the roots
The ground conductivity and Earth’s curvature are the two factors affecting the propagation of electromagnetic wave over the earth surface; in our approximate method, the propagation attenuation function
The attenuation function
Attenuation function
The parameter
The error of this approximate expression (
Firstly, according to (
Comparison of the simulated amplitude of attenuation functions
In the following section, in order to validate the accuracy of our approximate method for predicting the lightning-radiated field in intermediate range, we adopt a commonly used current moment for lightning electromagnetic radiation at a significant distance (e.g., Cummer [
The current moment waveshape (a) in the time domain and (b) in the frequency domain used for computing the lightning VLF radiation at an intermediate distance.
Figure
Comparison of the vertical electric field calculated by our approximate formula in (
Figures
Similar to that of Figure
Similar to that of Figure
Relative errors of the peak value (
Peak value (V/m) | Error ( | |||
---|---|---|---|---|
Wait’s formula ( |
Our approximate formula ( | |||
4 | 200 | 1.070 | 1.070 | 0.0 |
500 | 0.349 | 0.349 | 0.0 | |
1000 | 0.111 | 0.111 | 0.0 | |
1500 | 0.046 | 0.046 | 0.0 | |
0.01 | 200 | 1.068 | 1.066 | −0.2 |
500 | 0.350 | 0.346 | −1.0 | |
1000 | 0.113 | 0.110 | −2.4 | |
1500 | 0.047 | 0.045 | −3.3 | |
0.001 | 200 | 1.030 | 1.024 | −0.6 |
500 | 0.330 | 0.321 | −2.6 | |
1000 | 0.108 | 0.101 | −6.2 | |
1500 | 0.046 | 0.042 | −8.7 |
Errors of the waveform rise time (
Waveform rise time ( |
Error ( | |||
---|---|---|---|---|
Wait’s formula ( |
Our approximate formula ( | |||
4 | 200 | 6.7 | 6.7 | 0.0 |
500 | 8.2 | 8.1 | −1.2 | |
1000 | 11.4 | 11.5 | 0.9 | |
1500 | 15.7 | 15.7 | 0.0 | |
0.01 | 200 | 7.7 | 7.8 | 1.3 |
500 | 9.7 | 9.7 | 0.0 | |
1000 | 13.7 | 13.7 | 0.0 | |
1500 | 18.6 | 18.3 | −1.6 | |
0.001 | 200 | 10.3 | 10.3 | 0.0 |
500 | 13.8 | 13.7 | −0.7 | |
1000 | 19.8 | 19.3 | −3.0 | |
1500 | 26.2 | 25.2 | −3.8 |
Also, it is shown that, for the planar earth, the field peak value decreases with the decrease of the ground conductivity, and the waveform rise times increase with the decrease of the ground conductivity, which is because the high frequencies are selectively attenuated by the finitely conducting ground, causing the amplitude of the electromagnetic fields to decrease and the rise time to increase (e.g., Cooray et al
In order to further explain the validity of our approximate method, we compare the attenuation functions in the frequency domain calculated using our approximate formula in (
The (a) amplitude and (b) phase angle of attenuation functions (
Figure
The lightning-radiated vertical electric field peak value as a function of propagation distance for (a) the planar earth and (b) the spherical earth. The lightning current source shown in Figure
Lightning discharges can radiate electromagnetic waves over a wide frequency range from a few Hz to many tens of MHz, but most of the electromagnetic energy is radiated in the ELF and VLF bands, and the higher frequency component is attenuated rapidly as the propagation distance increases [
The authors declare that they have no conflicts of interest.
This work was supported in part by the National Key Research and Development Program of China (2017YFC1501505) and the Research and Engineering Demonstration of Comprehensive Lightning Protection System for Distribution Network Mode (YNKJQQ00000274) and in part by the Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX17_0882).