# An Approach for Modelling Harnesses in the Extreme near Field for Low Frequencies
## 2 Near-Field Approximations Very Close to the Dipole Source  
### Near Field Representation of Harnesses in the Low-Frequency Domain  
In the low-frequency domain, the electric length of a typical harness with straight path geometry (up to some meters) is a minuscule fraction of the wavelength, thus its radiation can be approximated by an infinitesimal dipole for most intents and purposes. It should be noted, that for frequencies below 200 kHz, and wavelengths larger than 1.5 km, any current (CM or DM) in a route of several meters can be assumed constant. In general, the electric field of a Hertzian dipole is calculated by applying the following equations [6]:  
\[E_{r}=\eta\frac{I_{l}cos\theta}{2\pi r^{2}}\bigg{[}1+\frac{1}{j\mathrm{kr}} \bigg{]}e^{-j\mathrm{kr}} \tag{1}\]  
\[E_{\theta}=j\eta\frac{kI_{l}sin\theta}{4\pi r}\bigg{[}1+\frac{1}{j\mathrm{kr}} -\frac{1}{(kr)^{2}}\bigg{]}e^{-j\mathrm{kr}} \tag{2}\]  
\[E_{\varphi}=0 \tag{3}\]  
In real cases, harness routing is not limited to single straight lines, but it follows complex geometries. In order to account for such cables, we propose the division of the complex geometry into the minimum number of necessary straight paths, and the representation of each of them as a Hertzian dipole centered in the corresponding path. An indicative geometry showcasing the proposed rationale is depicted in Figure 1. The length of each segment dipole is equal to one of the corresponding straight paths (e.g., segments OA, AB, BC, CD). Figure 1: Application of the near-field dipole segmentation technique in a complex cable geometry. For cleanliness purposes, in the case of spacecraft sensors and other sensitive payloads, their location is considered as the observation point P, at which the emissions can be calculated as the superposition of the electric field of each segment dipole. Obviously, in the low-frequency domain, the near-field approximation is well justified, so we use Equations (4)-(6) (\(kr\ll 1,\;kr\ll\lambda,\;r\ll 2\pi\)) [6] in order to calculate the fields, instead of the general Equations (1)-(3). \[E_{r}\cong-j\eta\frac{I_{o}le^{-jkr}}{2\pi kr^{3}}cos\theta \tag{4}\]  
\[E_{\theta}\cong-j\eta\frac{I_{o}le^{-jkr}}{4\pi kr^{3}}sin\theta \tag{5}\]  
\[E_{\varphi}=0 \tag{6}\]  
In order to carry out the superposition in Cartesian coordinates, the electric field components that are given by Equations (1)-(3) or (4)-(6) in spherical coordinates need to be converted according to the following transformations [6]:  
\[ \begin{split} E_{x}&=E_{r}sin\theta cos\varphi+E_{ \theta}cos\theta cos\varphi-E_{\varphi}sin\varphi\\ E_{y}&=E_{r}sin\theta sin\varphi+E_{\theta}cos \theta sin\varphi+E_{\varphi}cos\varphi\\ E_{z}&=E_{r}cos\theta-E_{\theta}sin\theta\end{split}  \tag{7}\]  
The above electric field component expressions are valid at every point, except from the source itself [6]. The main condition for this formulation to hold is that the source length has to be electrically small in comparison to the wavelength, which is valid for the harness case under study. However, another condition that has to be also fulfilled is that the source has to be short enough in order for the distance between a random source point and an arbitrary observation point P to be identical to the distance between the source center and P. In the case of a sensor (observation point P) located near the spacecraft (and to the cable also), their distance is comparable to cable's length, resulting in that the second condition does not hold true and extra steps are necessary to accurately calculate the harness emissions. Equations (1)-(3) and (4)-(6) are valid when (i) the wire is very short (with length \(l\ll\lambda\)) and (ii) very thin (with diameter \(\alpha\ll\lambda\)), while (iii) the distance \(r\) between any point on the source and the observation point can be considered constant [6]. This last condition for \(r\) does not hold true when the observation point is very close to the source, which is the case under study.