WebNIR

Purpose

This tool allows you to determine the levels of electric field dispersed by one or more bundles (or systems) of indefinite rectilinear conductors, all parallel to each other. A model of this type is commonly used to represent power lines, as also provided for by the CEI 211-4 guide Guide to the methods of calculating electric and magnetic fields generated by power lines and stations, at least for the part relating to two-dimensional models.

The tool is therefore useful to evaluate the exposure to the electric field of those who, for work or other reasons, must remain near power lines, in cases where a two-dimensional model allows you to obtain an accurate result. This occurs, in particular, in the presence of overhead lines, around which the conductors can be considered parallel to each other and to the ground surface, at least for a sufficiently long stretch with respect to the distances between the conductors themselves.

General setting

The application allows you to consider up to 10 bundles of conductors, each of which represents, in the most typical case, a three-phase power line; however, the general approach adopted makes the application very flexible, able to adapt to even very different situations.

Each conductor system is described through the starting data entered in a specific numbered sub-tab, located in the Data Entry tab. These data consist of the position of the conductors in space (specified by locating the point of intersection of the conductor with the calculation plane) and the values of the intensity and phase of the current on them.

The processing produces, in the Results tab, a summary with the peak value and the distance from the origin of the point where this is reached, and a table containing the distances from the origin at which a series of significant values ​​are obtained (1, 2 and 5 V/m).

A graph is also displayed, showing the trend of the intensity of the electric field along a segment positioned on the calculation plane and parallel to the ground (see the following figure, click to enlarge).

Calculation details

Geometry of conductors

Conductors are considered to be cylindrical in shape, with a constant diameter and small compared to their mutual distances. Unlike the calculation procedure used in standard 211-4, in which at a certain point the one-dimensional conductors are considered, in this case the radius of the same is considered in all the passages.

Bundled conductors

In this case it is assumed that:

• the distance between subconductors of a phase is small compared to the distance between conductors at different potential;
• all the subconductors of a phase are equal to each other;
• in a normal section of the bundle the centers of the subconductors of a phase lie on a circumference.

In the case of n conductors of radius r, arranged around a circumference of radius R, where s is the distance between adjacent subconductors, the equivalent radius ρ is used for the calculation:

Approximations

The presence of support pylons or pylons, buildings, vegetation and any other object in the area of interest is ignored.

The ground, flat and free of irregularities, is considered to be perfectly conductive, and is taken into account using the theory of image charges.

Calculation setup

In addition to the phase conductors, the guard cables are also considered (in light blue in the figure), placed at earth potential.

Data N_C conductors, indicated with:

• V₁, V₂, ... V_{2NC le respective voltages (including images)}
• λ₁, λ₂, ... λ_{2NC the charge densities;}
• P₁, P₂, ... P_{2NC the positions of the conductors;}

the electric field at a point Q is given by:

The voltage of the l-th conductor with respect to ground can be expressed as:

where:

• h_k is the altitude above the ground of the k-th conductor;
• d_kl is the distance from the center of the k-th conductor to the surface of the l-th conductor;
• d_kk is the radius of the l-th conductor.

To determine the charge densities, it is necessary to solve a linear system of N_C equations in N_C unknowns (the densities λ_k) of this type.

In matrix form, this system can be expressed, indicating with [V(t)] and [λ(t)] the vectors of the voltages and the charge densities, as:

and the linear charge densities are obtained by inverting the matrix of the parameters:

having placed:

In particular, the i-th component of [λ(t)] is obtained, indicating with c_ik the element of [C], from:

where V_k and φ_k indicate respectively the voltage and phase of the k-th conductor, and the last equality is possible by virtue of the fact that the frequency of the signal is the same for all conductors.

Using phasors:

You get:

Once the charge densities have been obtained, they are used in the initial formula to determine the effective electric field. Remembering that the component parallel to the conductor is zero:

Limitations

The basic condition of the model, i.e. the validity of a two-dimensional representation of the problem, cannot be considered satisfied in close proximity to supports or intersections between two or more lines and, in general, in electrical stations.