ESTIMATION OF THE INCEPTION TIME OF PARTIAL BREAKDOWN CHANNELS IN SOLID DIELECTRICS IN A NONUNIFORM ELECTRIC FIELD

V. S. DMITREVSKII, M. S. ZEMEROV

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

The process of breakdown development in solid dielectrics in a nonuniform electric field may be divided into stages: the inception of partial breakdown channels, the development of breakdown channels, and the final stage. Accordingly, the time to breakdown is expressed as

\[ \tau_{\mathrm{br}}=\tau_{z}+\tau_{r}+\tau_{k}, \tag{1} \]

where \(\tau_{z}\) is the time from the beginning of voltage application to the appearance of the first irreversible destruction (microchannel) in the solid dielectric;

\(\tau_{r}\) is the time of growth of the partial breakdown channels from the moment of microchannel inception until contact with the opposite electrode;

\(\tau_{k}\) is the time from the moment the electrodes are connected by the channel until short-circuiting.

In the present work \(\tau_{z}\) is investigated; it may be estimated on the basis of the thermofluctuation theory of polymer destruction [1, 2].

Under the action of an electric field and mechanical stress, \(\tau_{z}\) is expressed by the formula

\[ \tau_{z}=\tau_{0}\exp\left\{ \frac{D}{2KT} \left[ \sqrt{1-2x} - x\ln\left( \frac{1}{x}+\frac{1}{x}\sqrt{1-2x}-1 \right) \right] \right\}, \tag{2} \]

where

\[ x=\sqrt{ \left( \frac{A\cdot e^{-bT}\cdot \beta\cdot f\cdot E}{D} \right)^{2} +(\gamma\sigma)^{2} }; \]

\[ \frac{1}{\tau_{0}} \]
is the natural frequency of atomic vibrations;

\(D\) is the bond dissociation energy, approximately equal to the rupture energy;

\(K\) is Boltzmann’s constant;

\(T\) is the absolute temperature;

\(A,\gamma\) are structure-sensitive coefficients;

\(b\) is a coefficient allowing for the change in the density and modulus of elasticity of the dielectric with temperature;

\(\beta\) is a coefficient allowing for inhomogeneity in the dielectric;

\(\sigma\) is the mechanical stress;

\(f\cdot E\) is the maximum electric-field strength.

The study of \(\tau_{z}\) was carried out on specimens of organic glass (PMMA), polystyrene (PS), and a cold-curing epoxy compound.

(EC) (100 parts ED-5 + 9 parts PEPA) in a “needle–plane” electrode system at room temperature. Needles having a tip radius of \(10 \pm 1\ \mu\) were pressed into PMMA and PS at a temperature above the glass-transition temperature. In EC the needles were embedded in the usual way. The specimens underwent the following curing regime: \(20^\circ\mathrm{C}\)—24 h, \(80^\circ\mathrm{C}\)—12 h, with subsequent lowering of the temperature at a rate of \(10\ \mathrm{deg/h}\).

The distances between the electrodes in PMMA, PS, and EC were, respectively, 4 mm, 4.2 mm, and 3 mm. The plane electrode was a semiconducting layer made from a mixture of epoxy compound and graphite. The specimens were tested in batches of 20 at an alternating (50 Hz) voltage of 27–36 kV. Transformer oil was used as the insulating medium.

At specified time intervals the specimens were examined under a microscope and \(\tau_3\) was recorded.

The maximum electric-field strength near the needle was determined from formula (3)

\[ fE=\frac{2\cdot U}{r\ln\left(1+\frac{4d}{r}\right)}, \tag{3} \]

where \(u\) is the applied voltage;
\(r\) is the radius of curvature of the needle;
\(d\) is the distance between the electrodes.

From the experimental values \(\tau_3=f(E)\), the coefficients \(D\) and \(A\cdot e^{-bT}\cdot\beta\) entering into equation (2) can be calculated. Table 1 gives the values of these parameters, calculated from the results of our tests and from the experimental data given in [7, 8]. In the calculations it was assumed that the internal mechanical stresses near the needle are zero \((\sigma=0)\).

Table 1

It is seen from Table 1 that \(D_{\text{exp}}\) agrees well with the rupture energy of the C—C bond \((4.4–5.5\cdot10^{19}\ \text{erg}\cdot\text{eV}^{-1}\ [4])\), which is the principal bond in polymer chains.

As the tested volume \(v\) of the dielectric decreases, the probability of the appearance in this volume of inhomogeneities leading to an increase in the maximum electric-field strength decreases, i.e., as \(v\to0\) the coefficient \(\beta\to1\). In the specimens tested, the inception of the channel

always occurred near the needle in a volume of polymer limited to several cubic microns; therefore one may expect that the coefficient \(\beta\) is close to unity.

The quantity \(Ae^{-bT}\) was estimated theoretically from the relation [2]

\[ Ae^{-bT}= \frac{a\cdot e\cdot M\cdot \sqrt{\varepsilon\cdot G_{\mathrm{m}}}} {3.22\cdot N_A\cdot \rho\cdot n\left(1-\dfrac{2m}{m_{\mathrm{c}}}\right)}, \tag{4} \]

where \(a\) is the coefficient of disorder of the molecular chains;
\(M\) is the molecular weight of the monomer unit of the polymer;
\(\varepsilon\) is the dielectric constant;
\(G_{\mathrm{m}}\) is the elastic modulus;
\(N_A\) is Avogadro’s number;
\(\rho\) is the density;
\(m\) is the molecular weight of the end of a polymer chain capable of slippage;
\(m_{\mathrm{c}}\) is the molecular weight of the polymer;
\(n\) is the number of C—C bonds in the main chain of the monomer unit.

The dynamic elastic modulus reflects more accurately the elastic properties of intramolecular bonds than the static modulus; therefore, in calculating the coefficient \(Ae^{-bT}_{\mathrm{theor}}\), the dynamic elastic modulus at a temperature of \(293^\circ\mathrm{K}\) was chosen. The ratio \(\dfrac{2m}{m_{\mathrm{c}}}\) was taken to be zero. The quantities needed for the calculations and reflecting the properties of the polymers are given in Table 2, according to the literature [5, 6].

Table 2

The coefficient \(a\) may vary within the range from 1 to 3. In Table 1, \(Ae^{-bT}_{\mathrm{theor}}\) is calculated for \(a=1\). The value \(Ae^{-bT}\beta_{\mathrm{exp}}\) is also given there. Assuming that \(\beta=1\), the coefficient of disorder of the molecular chains of the polymers was determined as the ratio

\[ \frac{Ae^{-bT}\cdot \beta_{\mathrm{exp}}}{Ae^{-bT}_{\mathrm{theor}}}. \]

The coefficients \(a\) of the polymers determined in this way lie within the limits predicted by the theory. It is evident from the table that PE and EC possess the most disordered structure.

The experiments and calculations performed show that the time of inception of microchannels under prolonged application of voltage in a nonuniform field is well described by the thermofluctuation theory of material breakdown.

References

1. A. A. Vorob'ev, V. S. Dmitrevskii. Proceedings of the All-Union Conference “Physics of Dielectrics and Prospects for Its Development,” vol. 2, Leningrad, 1973.

2. V. S. Dmitrevskii. Calculation of Electrical Insulation. Tomsk, TPI, 1971.

3. N. N. Mirolyubov et al. Methods for Calculating Electric Fields. Moscow, “Vysshaya shkola,” 1963.

4. Concise Handbook of Physicochemical Quantities. Ed. by K. P. Mishchenko and A. A. Ravdel'. 6th ed. Leningrad, “Khimiya,” 1972.

5. Proceedings of the Rostov Institute of Railway Transport. Issue 79, 1971.

6. K. I. Chernyak. Nonmetallic Materials in Ship Electrical and Radio-Technical Equipment. “Shipbuilding” handbook. Leningrad, 1970.

7. M. Nawata, H. Kawamura and Jeda. Voltage and Temperature Dependences of Treeing Breakdown in Organic Solid Insulators. Electr. Eng. Jap., vol. 91, No. 4, 1971.

8. W. Mosch, J. Pilling, B. Tschacher, K. Eckholz. Kanaleinsatzzeit und Durchschlagzeit zur Beurteilung des Langzeitverhaltens von Feststoffisolierungen. Elektrie, 11, 1972.