# An Approach for Modelling Harnesses in the Extreme near Field for Low Frequencies

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## Abstract

**:**

## 1. Introduction

## 2. Near-Field Approximations Very Close to the Dipole Source

#### 2.1. Near Field Representation of Harnesses in the Low-Frequency Domain

#### 2.2. Considerations for Observation Distances Comparable to the Cable’s Length

- Single Dipole Case: is when the electric field is evaluated from Equations (1)–(3), considering that the source (cable) is one dipole with length $L$ equal to the cable length.
- Segmented Cable Case: is when the electric field is evaluated from the superposition of the electric fields of N segment dipoles, each has a length equal to L/N laying consecutively on the cable path with its center at −L/2 + L/2N + i*L/N (i = 0, …, N − 1), and contributing to the total field with its segment field calculated from (1)–(3).
- Single Dipole Case with Near Field Approximation: is when the electric field is evaluated from Equations (4)–(6), considering that the source (cable) is one dipole with length L equal to the cable length.
- Segmented Cable Case with Near Field Approximation: is when the electric field is evaluated from the superposition of the electric fields of N segment dipoles, each having a length equal to L/N, laying consecutively on the cable path with its center at −L/2 + L/2N + i*L/N (i = 0, …, N − 1), and contributing to the total field with its segment field calculated from (4)–(6).

_{x}, E

_{y}, or E

_{z}, the imaginary part dominates over the real part, and its calculation convergence (in the Segmented Dipole Case) may be achieved for values of N higher than 20 segments when the ratio value is 0.8.

_{y}and E

_{z}electric field components.

## 3. Application of the Segmentation Technique in Complex Geometries

- Input (observation point coordinates)
- Input (cable segments start/endpoint coordinates)
- Input (current distribution)
- For i = 1 to number of segments
- Calculate mid-point coordinates for segments i,
- Calculate mid-point—observation point distance
- end
- set reference point coordinates equal to the coordinates of the mid-point with the minimum distance.
- Calculate ratio parameter.

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Comparison of the magnitude of the electric field (E

_{x}

^{2}+ E

_{y}

^{2}+ E

_{z}

^{2})

^{1/2}versus the ratio of the measurement distance to the cable length, for N = 0 (straight blue line), 5 (bulleted red line), 50 (x marks), and 50,000 (dashed line) segments.

**Figure 4.**Comparison of the real and imaginary parts of the electric field component E

_{x}versus the ratio of the measurement distance to the cable length, for the Single Dipole Case (x marks) and the Segmented Dipole Case for N = 50 segments (dashed lines).

**Figure 5.**Comparison of the real and imaginary parts of the electric field component E

_{x}versus the ratio of the measurement distance to the cable length, for the Single Dipole Case (circles), the Segmented Dipole Case for N = 50 segments (dashed lines), and the Single Dipole Case with Near Field Approximation (x marks).

**Figure 6.**Comparison of the real and imaginary parts of the electric field component E

_{x}versus the ratio of the measurement distance to the cable length, for the Segmented Cable Case (star marks) and the Segmented Cable Case with Near Field Approximation with N = 50 segments (dashed lines).

**Figure 7.**Comparison of the magnitude of the electric field (E

_{x}

^{2}+ E

_{y}

^{2}+ E

_{z}

^{2})

^{1/2}versus the ratio of the measurement distance to the cable length, for the Single Dipole Case (dashed line) and the Segmented Cable Case with N = 50 segments (solid line).

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**MDPI and ACS Style**

Baklezos, A.T.; Kapetanakis, T.N.; Vardiambasis, I.O.; Capsalis, C.N.; Nikolopoulos, C.D.
An Approach for Modelling Harnesses in the Extreme near Field for Low Frequencies. *Appl. Sci.* **2022**, *12*, 3202.
https://doi.org/10.3390/app12063202

**AMA Style**

Baklezos AT, Kapetanakis TN, Vardiambasis IO, Capsalis CN, Nikolopoulos CD.
An Approach for Modelling Harnesses in the Extreme near Field for Low Frequencies. *Applied Sciences*. 2022; 12(6):3202.
https://doi.org/10.3390/app12063202

**Chicago/Turabian Style**

Baklezos, Anargyros T., Theodoros N. Kapetanakis, Ioannis O. Vardiambasis, Christos N. Capsalis, and Christos D. Nikolopoulos.
2022. "An Approach for Modelling Harnesses in the Extreme near Field for Low Frequencies" *Applied Sciences* 12, no. 6: 3202.
https://doi.org/10.3390/app12063202