*2.3. OPL Modeling*

The transversal view of the overhead power line is presented in Figure 1. In order to determine the magnetic flux density, both an analytical and a numerical approach can be used. In this paper the focus is on the magnetic field (the electric field is not monitored), so that the ground proximity of the line conductors plays a less important role since the Earth's magnetic permeability is close to μ0. Moreover, as stated before, the OPL works at 50 Hz (extra low frequency domain) which further simplifies the analysis. At such a low frequency, the wavelength is of the order of thousands of km, and the propagation of the electromagnetic field, which occurs in direction Oz (along the line) is characterized by the Poynting vector *S*.

$$\bar{S} = \bar{E} \times \bar{H} = \left(E\_\mathbf{x}\bar{i} + E\_\mathbf{y}\bar{j}\right) \times \left(H\_\mathbf{x}\bar{i} + H\_\mathbf{y}\bar{j}\right) = \left(E\_\mathbf{x}H\_\mathbf{y} - E\_\mathbf{y}H\_\mathbf{x}\right)\bar{k} \tag{5}$$

where <sup>−</sup> *i* , − *<sup>j</sup>* and <sup>−</sup> *k* are the unit vectors of the three axes Ox, Oy and Oz. In sinusoidal steady state, the vectors are complex. Comparing the dimensions of the line, which may be of the order of tens of kilometers, to the wavelength, we can assert that the line is in quasi-stationary magnetic regime. This means that the time variation of the electric field is negligible, and the equations for the determination of the magnetic field are the same as in stationary regime:

$$
\nabla \times \bar{H} = \bar{\bar{J}} : \bar{\mathbf{B}} = \mu \bar{H} ; \nabla \cdot \bar{\mathbf{B}} = 0 \tag{6}
$$

Thus, in an environment of constant permeability μ0, the magnetic flux density produced by a thin conductor can be determined using the Biot–Savart–Laplace formula (BSL):

$$\bar{\mathbf{B}}(P) = \frac{\mu\_0 \dot{\mathbf{r}}}{4\pi} \oint\_{\Gamma} \frac{\bar{dl} \times \bar{R}}{R^3} \tag{7}$$

which, for the long line conductor, becomes (Figure 2b) the following:

$$\bar{\mathbf{B}}\_{\mathbf{k}} = \frac{\mu\_0 \mathbf{L}\_{\mathbf{k}} \left[ \left( \boldsymbol{y} - \boldsymbol{y}\_{\mathbf{k}} \right) \bar{\boldsymbol{i}} + \left( \boldsymbol{x}\_{\mathbf{k}} - \boldsymbol{x} \right) \bar{\boldsymbol{j}} \right]}{2\pi \left[ \left( \boldsymbol{x} - \boldsymbol{x}\_{\mathbf{k}} \right)^2 + \left( \boldsymbol{y} - \boldsymbol{y}\_{\mathbf{k}} \right)^2 \right]} \tag{8}$$

In (8), *I*<sup>k</sup> is the complex rms value of the current in conductor k. In the case of the OPL line, there are six conductors which may or may not be current-carrying conductors (some currents may be zero).

The total magnetic field is obtained by superposition:

$$
\bar{\underline{\mathbf{B}}} = \sum\_{\mathbf{k}=1}^{6} \bar{\underline{\mathbf{B}}}\_{\mathbf{k}} \tag{9}
$$

**Figure 2.** Determination of the magnetic flux density with the BSL formula: (**a**) general case, (**b**) line conductors.

#### **3. Measurements**
