# Laplace Transform for Finite Element Analysis of Electromagnetic Interferences in Underground Metallic Structures

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

**J**and

**A**along z, the problem can be formulated in the two-dimensional plane perpendicular to z. The governing equation is:

## 3. Simulation Results

#### 3.1. Time-Domain Current Response

- Dirac impulse $\delta \left(t\right)\to \mathcal{L}\left(\delta \right)=1$;
- Unit step $u\left(t\right)\to \mathcal{L}\left(u\right)=\frac{1}{s}$;
- Unit ramp $r\left(t\right)\to \mathcal{L}\left(r\right)=\frac{1}{{s}^{2}}$.

#### 3.1.1. Impulse Response

#### 3.1.2. Unit Step and Unit Ramp Response

#### 3.1.3. Influence of the Pipeline Magnetic Permeability

#### 3.2. Time-Domain Current Density and Magnetic Flux Density Field

#### 3.3. Pipeline Response to a Lightning Event Striking a Power Line Conductor

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Cross-section of the considered pipeline–power line corridor. The physical characteristics of the conductors are reported in Table 1.

**Figure 2.**Current response of Phase Conductor a (

**a**), the pipeline (

**b**), and the soil (

**c**), as a result of different voltage waveforms enforced in Phase Conductor a at $t=0$ s.

**Figure 3.**Current density distribution and magnetic field lines at $t\text{}=\text{}1\text{}\times \text{}{10}^{-3}$ s (

**a**) and $t=0.138$ s (

**b**) obtained from the impulse response in the Laplace domain.

**Figure 4.**Response of the phase conductor voltage to a lightning event from [29].

Conductor | x (m) | y (m) | Radius (m) | Conductivity (S m${}^{-1}$) |
---|---|---|---|---|

Phase a | 3.5 | 12 | 1.5 × 10^{−2} | 3.5 × 10^{7} |

Phase b | −3 | 14 | 1.5 × 10^{−2} | 3.5 × 10^{7} |

Phase c | 2.9 | 16 | 1.5 × 10^{−2} | 3.5 × 10^{7} |

pipeline | 2 | −1.1 | 5 × 10^{−1} | 5.5 × 10^{6} |

soil | - | - | 6 × 10^{2} | 0.01 |

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**MDPI and ACS Style**

Cristofolini, A.; Popoli, A.; Sandrolini, L.; Pierotti, G.; Simonazzi, M.
Laplace Transform for Finite Element Analysis of Electromagnetic Interferences in Underground Metallic Structures. *Appl. Sci.* **2022**, *12*, 872.
https://doi.org/10.3390/app12020872

**AMA Style**

Cristofolini A, Popoli A, Sandrolini L, Pierotti G, Simonazzi M.
Laplace Transform for Finite Element Analysis of Electromagnetic Interferences in Underground Metallic Structures. *Applied Sciences*. 2022; 12(2):872.
https://doi.org/10.3390/app12020872

**Chicago/Turabian Style**

Cristofolini, Andrea, Arturo Popoli, Leonardo Sandrolini, Giacomo Pierotti, and Mattia Simonazzi.
2022. "Laplace Transform for Finite Element Analysis of Electromagnetic Interferences in Underground Metallic Structures" *Applied Sciences* 12, no. 2: 872.
https://doi.org/10.3390/app12020872