Figure 4 presents the color map of the electric field in the critical zone of air for 500V applied between the copper wire and the silver conducting paint. This voltage corresponds to half the peak voltage of the PDIV measured on twisted pairs. Indeed, when twisted pairs are supposed to be made of two identical enameled wires covered by an equipotential conducting layer, the voltage applied between the too copper conductors is split in two equal parts. The highest field values, where PDs may appear, are concentrated near the end of the silver conducting paint. A detailed analysis of the FE simulation results can be made using Paschen theory, for predicting the PDIV of the twisted pair made by an enameled wire covered by a conducting paint.

Figure 4 Geometry and mesh of the polymer layer and the ambient air in the critical zone where PDs may appear

Let us remind that, more than a century ago, Paschen studied the ignition of the ionization of a pressurized gas, under an electric field [11]. At the same period, Townsend worked on gas ionization processes, where free electrons are accelerated by electric fields [12]. Similar studies were performed for typical non-uniform fields due to specific electrode shapes used for high-voltage applications [13]. These approaches are based on the acceleration of free electrons, in the mean free path between gas molecules at a given pressure. When the free path is too short (high pressures), the electron gives up its energy in a series of non-ionizing collisions. When the mean free path is too long (low pressures), the electron reaches the anode before colliding with a gas molecule. In both cases, there is no gas ionization. The electronic avalanche occurs when the accelerated free electrons have enough energy for ionizing gas molecules when they collide. Then, a complex chain reaction leads breakdown, if the electric field remains high enough [14].

The Paschen’s empiric law defining the breakdown voltage in a gas is still a reference. This law was built on the results of many experiences made with two plane metallic electrodes in a pressurized gas. Therefore, the electric fields of Paschen’s experiences are uniform; free electrons accelerate along straight lines toward the anode. In practical cases, the electric field is never uniform. Figure 5 shows the field lines in the critical zone of air for our problem. The field lines are curved; the field in not uniform.

The interesting part of each field line starts from the polymer surface and finishes on the silver conducting paint. Unlike Paschen’s experiments, the electron inertia may have an influence: it is not obvious that the free electrons follow the field lines. The consequences of this major difference must be carefully studied before applying Paschen’s law. Two basic questions must be considered:

–

Do free electrons follow a filed line as they do in the Paschen’s experiences?

–

If the first answer is “yes”, a second question arises: does the variations of the field magnitude along a field line influence the Paschen’s threshold?

Figure 5 Field lines in the critical zone of air

For answering the first question, the ionization energy of molecules must be considered (13.6 eV for oxygen and 14.5 eV for nitrogen). It is supposed that the voltage is equal to the PDIV. For this voltage corresponding to the very beginning of the electronic avalanche. This hypothesis can be formulated in other words considering a microscopic point of view. It is supposed that the electric field accelerates a free electron until it acquires a kinetic energy equal to the gas molecule ionization one. Then a collision with a molecule absorbs this energy for creating a new free electron, which is also accelerated by the field. The random aspects of the problem are also neglected.

With this simplifying hypothesis, it is possible to estimate the trajectory of successive free electrons by solving the mechanical equation of a solid charged object placed in the Field force $(\overrightarrow{F}=q\overrightarrow{E})$, defined in the critical zone (the electron mass is *m* = 9.109 10^{−31}*kg* and its charge *q* = −1.602 10^{−19}*C*). The hypotheses suppose that the gas pressure is adapted to the voltage imposed on the twisted pair: the ionizing collisions occur exactly when free electrons have enough kinetic energy.

Figure 6 shows simulation results of the mechanical problem, that consider the mass of moving electrons. A free electron at zero speed is supposed to appear at the origin of the filed line 15 of figure 5 Two voltage levels are considered: 500V, which correspond to the PDIV in air at atmospheric pressure, and half of this value that corresponds to a lower pressure. The time step used for solving the mechanical equation is 0.01ps and the geometry of the critical zone is a square mesh of 0.5*μm*.

Figure 6 Trajectories of free electrons for two voltages and the field line 15 copied from Fig. 5

Figure 7 Kinetic energy of free electrons between collisions for two voltages and the field line 15

The small crosses plotted on figure 6 correspond to the ionizing collisions with a *N*_{2} or a *O*_{2} molecule considering an average energy of 14*eV*. The electron trajectories cumulated from the starting point on the polymer layer toward the end point on the silver paint. The successive electron trajectories are not exactly superimposed to the field lines; the difference is higher for lower fields because the Coulomb forces are lower for the same inertial ones. For instance, at 250*V* and for a cumulated electron trajectory that starts at the beginning of the field line 15, the trajectory length is 41.5μm (1.5*μm* longer than the field length). This cumulated trajectory is the same that the field line for 500V (the difference is lower than the mesh grid). Figure 7 shows that, for lower fields, free electrons need more time, and consequently more space, for acquiring the ionization energy of 14*eV*.

This simplified approach shows that, for the considered geometry, the field line curvatures can be neglected at ambient pressure, but not for very low pressures.

The energy acquired by an accelerated free electron is a key value for answering the second question relative to the influence of the variations of the field magnitude along a field line. For an elementary electron displacement $\overrightarrow{d}l$ this energy is;
$$dw=\overrightarrow{F}\cdot \overrightarrow{d}l=q\overrightarrow{E}\cdot \overrightarrow{d}l$$(1)
When the free electrons follow the field lines, $\overrightarrow{d}l$ becomes an element of the field line, which is collinear to the electric field $\overrightarrow{E}$. The dot product of (1) disappears, the elementary energy is expresses as a function do the field magnitude at any point of the field line.
$$dw=q|\overrightarrow{E}||\overrightarrow{d}l|$$(2)

Consequently, the energy acquired by an electron moving on a field line from point A to point B is equal to:
$$W=\underset{A}{\overset{B}{\int}}|\overrightarrow{F}||\overrightarrow{d}l|=q\underset{A}{\overset{B}{\int}}|\overrightarrow{E}||\overrightarrow{d}l|=q({V}_{B}-{V}_{A})$$(3)
This energy does not depend on the variation of the magnitude of the electric field between points A and B of the field line, but only on the potential difference between the considered points.

The simplified approach presented above, made for a voltage corresponding to the electronic avalanche threshold, shows that the collisions with the molecules of the gas limit the free electron speed. Consequently, the inertia forces related to the curvature of the trajectory, remain low relatively to the Coulomb ones, at atmospheric pressure. The Paschen’s law can be used, substituting the field line length to the distance between the plane electrodes of Paschen’s experiences. This equivalence is not valid for very low pressures that remains over the Paschen’s minimum. For lower PDIV, Coulomb forces are also lower, but the maximum free electron speed remains the same. Consequently, the forces due to electron inertia are no longer negligible, comparing to the Coulomb forces. The trajectories of free electrons is no longer superimposed to the field lines. Equation (3) is no longer valid. In this case, more detailed simulations must be performed for estimating the kinetic energy of free electrons in the field force.

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