# An Efficient Method for Calculating the Lightning Electromagnetic Field Over Perfectly Conducting Ground

^{*}

## Abstract

**:**

## 1. Introduction

_{r},

_{ideal}and H

_{φ},

_{ideal}) accounted for most of the computation time. Therefore, increasing the efficiency of the first step is more conducive to improving the overall computational efficiency. The relevant studies about this tend to be neglected. The formulae for evaluating E

_{r},

_{ideal}and H

_{φ},

_{ideal}have been established based on the dipole method, and they are composed by integrals with respect to the lightning channel. The common method to evaluate these integrals are mainly based on the numerical integration, by means of a discretization of the lightning channel. However, only a sufficiently small discretization step is essential to get an accurate result, which leads to a relatively large number of calculations and results in a lengthy computation time. Besides, the programming is relatively complicated because the propagation of the lightning current along the channel must be considered.

## 2. Method for Calculating the Lightning Electromagnetic Field over Perfectly Conducting Ground

#### 2.1. Review of the Lightning Return Stroke Model

#### 2.2. Fundamental Formulation of Lightning Electromagnetic Field over Perfectly Conducting Ground

_{ideal}(φ, r, z, t)—ideal horizontal electric field at observation point P(r, z), H

_{ideal}(φ, r, z, t)—ideal tangential magnetic field at observation point P(r, z), h—the height of the lightning channel, ε

_{0}—vacuum permittivity, r—horizontal distance from the lightning channel to point P(r, z), R

_{z}′ and R

_{−z}′—distance from the dipole to point P(r, z), z and z′—the z-coordinate of point P(r, z) and the dipole, respectively, and c—speed of light in vacuum.

#### 2.3. Proposed Method for Calculating the Lightning Electromagnetic Field over Perfectly Conducting Ground

_{base}(t-z′/v) distribution at all points along the channel direction, the system can be transformed into a time-invariant system. Then, we can transform the time-invariant system into the frequency domain based on the Fourier transform. Finally, the analytical formulae for most of the integral in Equation (3) can be obtained by means of some derivations.

_{above}and E

_{under}have the same form, so only the derivation for E

_{above}is provided. The above-ground lightning channel-generated horizontal electric field can be divided into three parts, i.e., E

_{A}, E

_{B}, and E

_{C}, as shown in Equation (5):

_{A}can be rewritten as:

_{A}

_{2}can be described as Equation (7) based on $\int \frac{(z-{z}^{\prime})}{{R}_{{z}^{\prime}}^{3}}}d{z}^{\prime}=\frac{1}{{R}_{{z}^{\prime}}$ and $\int {u}^{\prime}vdz=uv-{\displaystyle \int u{v}^{\prime}dz}$:

_{A}

_{22}, according to $\int \frac{1}{{R}_{{z}^{\prime}}{\left(z-{z}^{\prime}\right)}^{2}}}d{z}^{\prime}=\frac{{R}_{{z}^{\prime}}}{{r}^{2}\left(z-{z}^{\prime}\right)$ and $\int {u}^{\prime}vdz=uv-{\displaystyle \int u{v}^{\prime}dz}$, it will be:

_{A}

_{3}with E

_{B}and using $\int \frac{(z-{z}^{\prime})}{{R}_{{z}^{\prime}}^{4}}}dz{}^{\prime}=\frac{1}{2{R}_{{z}^{\prime}}^{2}$, we can get:

_{B2}+ E

_{A24}= 0 and E

_{B3}+ E

_{C}= 0, Equation (5) can be rewritten as:

_{h}—the distance between the highest point of the lightning channel and point P(r, z), and R

_{0}—the distance between the lightning strike point and point P(r, z).

_{under}can also be achieved, only by replacing z–z′ and R

_{z′}by z + z′ and R–

_{z′}, respectively. Finally, the horizontal electric field E

_{ideal}can be obtained by adding them together.

_{integral}. Moreover, the numerical calculation of E

_{integral}is very simple, because it only contains the integral of the current along the channel and its derivatives. Generally, comparing Equations (12)–(14) with Equation (3), it can be seen that the proposed method equates the integral operation with a simple arithmetic operation and a simple integral. The arithmetic operation involves fewer steps, requires less data, and produces more accurate results. The integral is much simpler than that of the conventional method. Thus, doing this makes the proposed method more efficient and easily programmed.

## 3. Results

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**The time-domain evolutions of the horizontal electric field at the observation point under different dz′; (

**a**) calculated by the conventional method; (

**b**) calculated by the proposed method.

**Figure 4.**The two parts of the horizontal electric field at the observation point: (

**a**) the arithmetic parts calculated by the conventional method; (

**b**) the arithmetic parts calculated by the proposed method; (

**c**) the integral part calculated by the proposed method.

**Figure 5.**The horizontal electric field curves at the observation point calculated by the proposed method and the conventional method under different dz′.

**Figure 6.**The horizontal electric field curves at the different observation points calculated by the proposed method and the conventional method under different dz′. (

**a**) the horizontal electric field at r = 250 m calculated by two methods; (

**b**) the horizontal electric field at r = 500 m calculated by two methods.

i_{01} | i_{02} | τ_{11} | τ_{12} | τ_{2}_{1} | τ_{22} | n_{1} | n_{2} |
---|---|---|---|---|---|---|---|

10.7 × 10^{3} A | 6.5 × 10^{3} A | 0.25 × 10^{−6} s | 2.5 × 10^{−6} s | 2 × 10^{−6} s | 230 × 10^{−6} s | 2 | 2 |

t | v | r | z | c | α | dt | ε_{0} |
---|---|---|---|---|---|---|---|

30 μs | 1.3 × 10^{8} m/s | 50/250/500 m | 10 m | 3 × 10^{8} m/s | 1700 m | 10 ns | 8.854 × 10^{−12} |

dz′ | Conventional Method | Proposed Method |
---|---|---|

0.05 m | 5.776 s | - |

2 m | 0.121 s | 0.143 s |

dz′ | Conventional Method | Proposed Method | ||
---|---|---|---|---|

- | r = 250 m | r = 500 m | r = 250 m | r = 500 m |

0.05 m | 4.688 s | 4.673 s | - | - |

2 m | 0.203 s | 0.163 s | 0.150 s | 0.142 s |

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**MDPI and ACS Style**

Liu, X.; Ge, T.
An Efficient Method for Calculating the Lightning Electromagnetic Field Over Perfectly Conducting Ground. *Appl. Sci.* **2020**, *10*, 4263.
https://doi.org/10.3390/app10124263

**AMA Style**

Liu X, Ge T.
An Efficient Method for Calculating the Lightning Electromagnetic Field Over Perfectly Conducting Ground. *Applied Sciences*. 2020; 10(12):4263.
https://doi.org/10.3390/app10124263

**Chicago/Turabian Style**

Liu, Xin, and Tianping Ge.
2020. "An Efficient Method for Calculating the Lightning Electromagnetic Field Over Perfectly Conducting Ground" *Applied Sciences* 10, no. 12: 4263.
https://doi.org/10.3390/app10124263