ON DISCHARGES IN GAS INCLUSIONS IN SOLID INSULATION

V. S. Dmitrevsky

(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)

A number of works \([1, 2, 3]\) are devoted to the study of discharges in gas inclusions in a solid dielectric. It is stated in the literature that discharges in gas inclusions have a destructive effect on organic insulation and lead to its rapid breakdown. However, up to the present time the question of the nature of discharges when the sizes of pores in insulation change has been poorly covered in the literature. The questions of the influence of the sizes of gas pores on the destruction of dielectrics are insufficiently discussed.

For simplicity in considering the discharge mechanism, let us assume that the gas pore has the form of a cylinder whose axis coincides with the direction of the electric field. On the opposite walls of the pore, before a discharge occurs in it, specific energy will accumulate, determined by the formula

\[ W_{\text{sp}}=\frac{U^{2}C}{2S} =\frac{E_{\text{br}}^{2}d^{2}\xi S}{2\cdot 4\pi dS9\cdot 10^{11}} =\frac{E_{\text{br}}^{2}d\xi}{72\pi 10^{11}} \left[\frac{\text{W}\cdot\text{s}}{\text{cm}^{2}}\right]. \tag{1} \]

The specific surface charge accumulated on the opposite walls of the gas pore before a discharge occurs in it is found as

\[ q=\frac{UC}{S} =\frac{E_{\text{br}}d\xi S}{4\pi dS9\cdot 10^{11}} =\frac{E_{\text{br}}\xi}{36\pi\cdot 10^{11}} \left[\frac{\text{C}}{\text{cm}^{2}}\right]. \tag{2} \]

Through the dielectric connected in series with the gas pore, the current density under alternating voltage can be calculated before the moment of breakdown as

\[ j=\frac{U\omega C}{S} =E_{\text{br}}d\, \frac{\xi S2\pi f}{4\pi dS9\cdot 10^{11}} =\frac{E_{\text{br}}\xi f}{1.8\cdot 10^{12}} \left[\frac{\text{A}}{\text{cm}^{2}}\right]. \tag{3} \]

In these formulas:

\(U\) — voltage between the pore walls before the moment of discharge;
\(C\) — capacitance of the pore;
\(d\) — size of the pore in the field direction;
\(S\) — area of the pore;
\(\xi\) — dielectric permittivity of the gas filling the pore;
\(f\) — frequency of the alternating current;
\(E_{\text{br}}\) — electric strength of the gas filling the pore, at the corresponding distances \(d\) and gas pressure.

The results of calculations of the specific energy characteristics of a pore as a function of its thickness before discharge in the gas are given in Table 1.

In the calculations it was assumed that the pore contains air at a pressure of 760 mm Hg.

The energy characteristics, the specific surface charge, and the current density decrease monotonically with increasing thickness of the gas pore, while the specific energy has a minimum value in the range of thicknesses from 20 to 80 μm.

It was of interest to see what the conditions are for the development of electron avalanches in such a gas pore at field strengths equal to the breakdown values.

Table 1

Let us calculate the charge that can accumulate in the head of an avalanche at the end of a path segment equal to the thickness of the gas inclusion. According to Townsend’s equation,

\[ n = n_0 e^{\alpha d} \tag{4} \]

the coefficient of impact ionization by electrons, $\alpha$, depends on the field strength and the air pressure. Table 2 gives a calculation of the number of electrons arriving at the anode in a gas pore and the magnitude of the negative charge of one avalanche.

Table 2

During the motion of an electron avalanche, its head expands owing to electron diffusion. The radius of the electron head, determined from the diffusion conditions, is found as:

\[ r = \sqrt{2Dt}, \tag{5} \]

where $D$ is the diffusion coefficient;
$t$ is the time of motion of the electron avalanche.

An electron avalanche, upon reaching the anode, causes neutralization of the charge on the surface of the dielectric having area \(\pi r^2\). The charge present on the anode side of a pore of area \(\pi r^2\) can be found as

\[ Q_r=\pi r^2 q=2\pi Dtq. \tag{6} \]

The time for the electron avalanche to traverse the gas gap will be

\[ t=\frac{d}{\kappa E_{\mathrm{pr}}} \tag{7} \]

where \(\kappa\) is the mobility of the electrons.

Then formula (6) will have the form

\[ Q_r=\frac{2\pi Ddq}{\kappa E_{\mathrm{pr}}}. \tag{6a} \]

The ratio of the diffusion coefficient to the mobility of the electrons is determined by the gas pressure, temperature, and a number of other factors. For our case one may use the expression

\[ \frac{D}{\kappa}=\frac{\kappa T_e}{q_0}, \tag{8} \]

where \(\kappa\) is Boltzmann’s constant;
\(T_e\) is the temperature of the electrons in the avalanche;
\(q_0\) is the electron charge.

Then

\[ Q_r=\frac{2\pi\kappa}{q_0}\,\frac{dT_e}{E_{\mathrm{pr}}}\,q \tag{6b} \]

We shall denote by \(Q\) the charge carried to the anode by one electron avalanche (Table 2). Let

\[ An_0=\frac{Q_r}{Q}, \tag{9} \]

The quantity \(An_0\) indicates the number of electron avalanches that must reach the anode in order to neutralize the charge over an area \(\pi r^2\). Using expressions (4) and (6b), we find

\[ An_0=\frac{2\pi\kappa}{q_0^2}\,\frac{d}{E_{\mathrm{pr}}e^{\alpha d}}\,T_eq. \tag{10} \]

For discharges in air, the electron temperature is \(2000\text{--}4000^\circ\mathrm{K}\) [4].

Having specified a definite temperature, it is not difficult to calculate the quantity from formula (10). Table 3 gives the results of calculations of \(An_0\) for various thicknesses of the gas pore.

It follows from Table 3 that, as the pore thickness increases, the number of avalanches required to neutralize the charge on the pore wall decreases.

Considering Tables 2 and 3, it should be noted that, as the thickness of the gas inclusion increases, the charge in the electron avalanche increases, and consequently so does the positive charge in the volume of the pore.

Analyzing Tables 2 and 3, one may assume that in pores of small size the discharge represents a series of small avalanches following one another, the number of which increases as the field strength in the inclusion increases.

As the thickness of the gas inclusion increases, the number of charges neutralized by an avalanche on the pore wall increases. As a result—

the neutralization of the surface charge at the point where the electron avalanche strikes, there will be a sharp decrease in the potential at this point, and a significant tangential component of the field strength will appear, leading to the occurrence of a surface discharge.

Table 3

A surface discharge in a pore 0.3 mm thick was observed by Mason [1]. Owing to surface discharges, the energy released in the gas pore increases sharply, which will lead to destruction of the dielectric. A significant magnitude of the tangential component of the field strength will appear under the condition of complete neutralization of the surface charge at the point where the electron avalanche strikes.

Thus, for small pore thicknesses, individual electron avalanches will be observed in the pores, occurring in rapid succession and more or less distributed over the area of the pore wall. In this case the destruction of the dielectric material will be very slight. For large gas-inclusion sizes, the character of the discharge takes the form of a spark, with a rather considerable release of energy and rapid destruction of the material constituting the pore wall.

A discharge in small gas inclusions can exist for a very long time without causing noticeable wear of the insulation. The criterion for transition to a spark discharge in a gas pore may be the parameter $An_0$. If $An_0 = 1$, then the discharge becomes spark-like and the destruction of the insulation will proceed very intensively.

REFERENCES

1. Mason. The deterioration and breakdown of dielectrics resulting from internal discharges. Proc. IEE, part 1, Vol. 98, No. 109, 1951.

2. A. V. Dmitriev. The nature of a gas discharge in a cavity surrounded by a dielectric or bounded by the surfaces of a dielectric and a conductor. In: Ionization Aging, Corona Resistance, and Test Methods for High-Voltage Insulation, TsINTI Publishing House, 1963.

3. Ch. M. Dzhuvarly, G. V. Vechkhaizer, Yu. V. Gorik, P. V. Leonov. Volt-ampere characteristics of an electric discharge in a gas cavity bounded by a dielectric. Izv. AN Azerb. SSR. Series of Physical-Technical and Mathematical Sciences, 1966, No. 2, 139–146.

4. J. Meek, J. Craggs. Electrical Breakdown in Gases. Foreign Literature Publishing House, 1960.