# An Approach for Modelling Harnesses in the Extreme near Field for Low Frequencies
## 2 Near-Field Approximations Very Close to the Dipole Source
### Considerations for Observation Distances Comparable to the Cable's Length  
In order to investigate the applicability of Equations (1)-(3) or (4)-(6), when the observation distance is comparable to the source dimensions (harness length), we have implemented the following specific scenario: a radiating source consisting of a 2 m straight cable is fed with a 1 A current oscillating with a 9 kHz frequency. For a 2 m indicative observation distance, the second condition is not valid, so the cable cannot be considered as a single point source; however, after proper segmentation, the second condition holds true for each of the segments. If the number of source segments (Hertzian dipoles) is \(N\), the total field radiated from the cable assembly at the observation point P is given by:  
\[E_{cable}=\sum_{i=1}^{N}E_{i} \tag{9}\]  
The parameters of this segmentation are: (i) the number \(N\) of the segment dipoles that make up the whole cable, and (ii) the distance r from the center of the cable to the observation point P. In order to take into account the length \(L\) of the cable as well, we define and investigate the \(r/L\)_ratio_. Thus, the length itself is not a standalone variable anymore, and our study focuses on the relative to cable length distance. In order to showcase the effect of the parameter _ratio_ along with the segmentation number \(N\), we have investigated the following four indicative cases:  
1. 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. 2. 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 \(-L/2+L/2N+\vec{r}L/N\) (\(i=0,\,\ldots,\,N-1\)), and contributing to the total field with its segment field calculated from (1)-(3). 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. 4. 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 \(-L/2+L/2N+\vec{r}^{*}L/N\) (\(i=0,\,\ldots,\,N-1\)), and contributing to the total field with its segment field calculated from (4)-(6). We have studied the configuration of Figure 2 for several values of the elevation angle \(\theta\), but all results presented herein correspond to \(\theta=90^{\circ}\), for which segmentation of the source is needed for a larger distance, corresponding to _ratio_ = 4. Figure 2: Dipole segmentation for near-field approximation study. For the extreme near-field calculations of the complex values of the electric field component \(E_{x}\), \(E_{y}\), 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. As discussed earlier in Section 1, theoretically, an infinite number of infinitesimal sources are necessary for the accurate evaluation of the total electric field. However, practically, a sufficiently high number of segments can be used to approximate the total electric field without a significant loss (<1%) in the accuracy. In order to estimate the minimum number required of segments for an as accurate as possible (within acceptable limits) calculation of the total electric field, we have investigated the convergence point of the electric field for several values of \(N\) and _ratio_. Figure 3 reveals the impact of segmentation in the calculation of the electric field amplitude for \(N\) = 5, 50, and 50,000 segments.