On the Measurement of Breakdown Energy in Solid Dielectrics

A. A. Vorob'ev and A. F. Kalganov

Introduction

Breakdown of a dielectric occurs when the energy necessary for this is released in it. This energy is taken from the electric field created by the voltage applied to the dielectric. According to modern concepts, the loss of energy during breakdown is associated with the process of acceleration of electrons in the dielectric. Energy is expended on ionization of the particles of the dielectric structure, on excitation of their vibrations, and on radiation.

Thus, in the process of breakdown of air, for one act of ionization under normal conditions the energy expended is \(W=\dfrac{E}{\alpha}=2630\) eV, where \(E=30\) kV/cm is the breakdown field strength for air, and \(\alpha=11.4\) is the number of ionizations in air per 1 cm of path at such a field strength [1]. The greater part of the energy goes into heating the gas and radiation, since only about 15 eV is expended directly on ionization [2].

As B. M. Gokhberg found, breakdown of gases occurs as a result of reaching a definite value of the coefficient of volume ionization \(\alpha\), approximately the same for different gases [3, 4]. From the relation \(E=W\cdot\alpha\) it is seen that, for a constant value of \(\alpha\), the breakdown field strength is determined by the magnitude of the energy expended on one act of ionization. When the energy losses of electrons in the gas are large, the coefficient \(\alpha\) reaches its value at a higher field strength and, conversely, with small energy losses, at a lower field strength.

For solid dielectrics, depending on the form into which the energy of the electric field is transformed, one or another form of breakdown occurs (thermal, electrochemical, electrical). This, in turn, determines the breakdown voltage corresponding to one or another form of breakdown.

Mechanical destruction of a dielectric during breakdown is preceded by its loss of electric strength. After breakdown of the dielectric, a through current flows along the channel that has formed. Therefore the total energy expended during breakdown should be divided into three components:

\[ W_{tot}=W_{\phi}+W_{razr}+W_{tok}, \tag{1} \]

where \(W_{\phi}\) is the energy whose release in the dielectric causes it to lose electric strength; it may be called the energy of discharge formation. \(W_{razr}\) is the energy whose release leads to destruction of the dielectric and to formation of a through channel; it may be called the energy of destruction. \(W_{tok}\) is the energy absorbed in the dielectric due to the flow of the through current; it depends on the conductivity of the channel formed during breakdown and on the duration of the post-breakdown current flow.

The sum of the energies of formation \(W_{\phi}\) and destruction \(W_{\text{destr}}\) determines the energy whose release leads to breakdown and which may be called the breakdown energy: \(W_{\text{br}} = W_{\phi} + W_{\text{destr}}\).

The energy absorbed in the dielectric during breakdown is released, first, by the currents flowing through the dielectric and, second, by the release of the potential (electrostatic) energy of the field:

\[ W_{\text{abs}}=\int_0^t i u\,dt+\int_V \frac{\varepsilon E^2}{2}\,dV, \tag{2} \]

where \(i\) is the active current in the discharge circuit, \(u\) is the voltage across the dielectric, \(E\) is the field strength in the dielectric, \(\varepsilon\) is the dielectric permittivity of the dielectric, \(V\) is the volume of the dielectric between the electrodes, and \(t\) is the time of voltage application.

If breakdown of a crystalline dielectric is accompanied by destruction of the structure into separate ions, then the breakdown energy cannot be less than the lattice energy referred to the destroyed volume. For the thickness of specimens used in investigations of electric strength, of several hundred microns, the destroyed volume is of the order of \(10^{-7}\,\text{cm}^3\). The lattice energy of such a volume for NaCl is \(2.77\cdot 10^{-3}\,\text{J}\). The potential energy for this same volume under pre-breakdown conditions is \(5.58\cdot 10^{-8}\,\text{J}\). As is evident, rupture of the lattice solely at the expense of the potential energy of the field stored in the volume of the future breakdown channel is impossible if one takes as a guide the measured values of electric strength. However, during breakdown the potential energy stored in the entire volume of the dielectric between the electrodes is released. The magnitude of this energy is readily determined from the equation \(W=\dfrac{CU^2}{2}\), where \(C\) is the capacitance of the specimen. In this case

\[ \frac{CU^2}{2}=\int_V \frac{\varepsilon E^2}{2}\,dV. \tag{3} \]

The capacitance of specimens investigated for electric strength in a uniform field is of the order of \(5\div 10\,\text{pF}\). For \(C=5\,\text{pF}\) and \(U=15\,\text{kV}\),

\[ \frac{CU^2}{2}=0.563\cdot 10^{-3}\,\text{J}. \]

This amount of energy is sufficient to destroy a volume of \(0.2\cdot 10^{-7}\,\text{cm}^3\) of NaCl. The magnitude of this volume is comparable with the magnitude of the volume of destruction during breakdown. Apparently, as soon as the dielectric loses its electric strength and a conducting channel is formed, the released polarization energy, together with the energy of the current flowing through the dielectric, causes visible destruction of the dielectric.

It is clear from equations (2) and (3) that, in order to determine the energy absorbed during breakdown of a solid dielectric, it is necessary to know the magnitudes of the current and voltage across the dielectric (which can be obtained by oscillographic recording of the breakdown process), as well as the capacitance of the specimen. And if, from the oscillograms of current and voltage, the moments of the beginning and end of the individual stages of breakdown are determined, it is possible to calculate the magnitudes of the energies expended on the development of these stages. To elucidate the mechanism of breakdown of solid dielectrics, it is of interest to determine both the magnitude of the energy leading to the loss of electric strength and the magnitude of the energy causing mechanical destruction of the dielectric under the action of a high-intensity field. We measured, by a calorimetric method, the total energy absorbed during pulsed breakdown of solid dielectrics.

Measurement Method

Figure 1 shows the electrical circuit for generating voltage pulses. The capacitor \(C\) is charged through resistances \(R_1\) and \(R_2\) to a known voltage \((10 \div 20\ \text{kV})\). The charging voltage is determined by an electrostatic voltmeter \(V_1\) on the low-voltage side of the high-voltage transformer. Closing the switch \(S\), which is achieved by bringing the spheres into contact, causes the discharge of capacitor \(C\) through the resistances \(R_2 = 10\ \text{M}\Omega\) and \(R_3 = 154\ \Omega\). In series with the resistance \(R_3\), the dielectric (specimen) could be switched out by opening switch \(K\). In the absence of the dielectric, all the energy stored in capacitor \(C\) is released in the resistance \(R_3\). The fraction of energy released in the resistance \(R_2\), in a first approximation, is equal to the ratio

Fig. 1. Circuit of a pulse generator for measuring energy losses during breakdown of solid dielectrics.

of the resistances \(R_3/R_2\) and, consequently, is very small. When the specimen breaks down, part of the energy stored in the capacitor is absorbed in the specimen. The remaining part of the energy is still released in the resistance \(R_3\). The difference between the energy \(W_1\), released in the resistance \(R_3\) in the absence of the specimen, and the energy \(W_2\), released there during breakdown of the specimen, characterizes the magnitude of the energy absorbed in the specimen:

\[ W_{\text{loss}} = W_1 - W_2 = \Delta W . \tag{4} \]

In addition to losses in the dielectric, losses occur in the lead wires and in the capacitor \(C\). However, these losses, for a given voltage, are the same in both cases and, with the difference method of determining losses in the dielectric, are eliminated.

To record the energy released in the resistance in the form of heat, a semiconductor thermistor MMT—1 of \(24\ \text{k}\Omega\) was used as the heat-sensitive element.

Changes in the temperature of the heat-sensitive element were calibrated as a function of the quantity of heat supplied. The heat-sensitive element was included in a bridge circuit (Fig. 2). To improve the symmetry of the bridge and increase its sensitivity, two identical thermistors (\(T_1\) and \(T_2\)) were connected into two of its arms, and identical KMS—6 resistance boxes were connected into the other two arms. The bridge was adjusted by means of one of the resistance boxes (\(R_{\text{reg}}\)). An alternating voltage with a frequency of \(50\ \text{Hz}\) was applied to one of the diagonals through a balancing transformer (S. Tr.) with a transformation ratio of 1/1. A tube amplifier \(G\) was connected into the other diagonal of the bridge.

Heat transfer from the resistance \(R_3\) to the heat-sensitive element was carried out as follows. The resistance \(R_3\) was made in the form of a coil \(0.5\ \text{cm}\) in diameter from manganin wire of diameter

0.008 cm. Inside this coil, as shown in Fig. 2, one of the thermoresistors ($T_2$) was placed, and the entire system was filled with paraffin, which served both as an electrically insulating medium and as a heat-conducting medium between the coil and the thermoresistor. The outer layer of paraffin, together with the walls of the Plexiglas box into which the paraffin was poured, eliminated the influence of the external environment (fluctuations in the external temperature, etc.). To eliminate the influence of the external environment on the other thermoresistor ($T_1$), the latter was also embedded in paraffin. The entire measuring circuit and the calorimeter were carefully shielded. With a bridge voltage of 5 V, the scale division of the output instrument of the measuring circuit was equal to $4.1 \cdot 10^{-4}$ J, or $0.00035^\circ$C.

Fig. 2. Measuring circuit for determining energy losses during breakdown.

Since the amounts of energy to be measured are very small, it was also necessary to have a small reserve of energy in the capacitor. For this purpose it was necessary to have a capacitor with very small capacitance and a sufficiently high operating voltage. We made a capacitor of organic glass with capacitance $C = 32$ pF for an operating voltage of more than 20 kV. The energy stored in it for the case $U = 24$ kV was $9.22 \cdot 10^{-3}$ J.

Results of measurements

Figure 3 gives the calibration curve of the calorimeter—the readings of the measuring instrument as a function of the amount of energy released in resistance $R_3$. The calibration was carried out on alternating voltage. The same figure gives the values of the readings of the measuring instrument caused by the discharge of capacitors with different energies. As can be seen from the graph, the average values of the instrument readings for a given value of the energy sent into the calorimeter fall quite well on a straight line. The fact that both the points obtained on alternating current and the points obtained during capacitor discharge fall on this straight line indicates that all the energy stored in the capacitors is released in the receiver, while the energy losses in the discharge circuit are small and lie within the limits of the accuracy of measurement. The scatter of the instrument readings for one and the same energy is associated with the influence of high-voltage fields on the measuring circuit.

We measured the energy absorbed during breakdown of NaCl and KJ single crystals. The thickness of the samples being broken down was $100 \div 300$ µm. Breakdown was produced both in uniform and in nonuniform fields.

Figure 4 gives the measured values of the losses $\Delta W$ as a function of the magnitude of the destroyed volume $V$ of NaCl samples. The destroyed

the volume \(V\) was determined by measuring under a microscope the length and the mean diameter of the breakdown channel. As is seen from the graph (Fig. 4), the destroyed volume of the specimen is, on average, greater for a smaller value of the expended energy and smaller for a greater expenditure of energy. The straight line drawn from the origin divides the points into two regions. In the upper region are located the points for which the ratio of the energy absorbed during breakdown to the energy of the destroyed volume is less than unity: \(\dfrac{\Delta W}{W_{\mathrm{resh}}} < 1\); in the lower region this ratio is greater than unity: \(\dfrac{\Delta W}{W_{\mathrm{resh}}} > 1\). The straight line is the geometric locus of the points for which \(\Delta W = W_{\mathrm{resh}}\). For rock salt the value of the ratio \(\dfrac{\Delta W}{W_{\mathrm{resh}}}\) was found to range from 0.1 to 2.5. For

Fig. 3. Calibration curve of the calorimeter.

Fig. 4. Energy losses during breakdown \(\Delta W\), joules, as a function of the destroyed volume \(V\), cm\(^3\).

○—in a uniform field,
●—in a nonuniform field.

in the majority of cases, \(\dfrac{\Delta W}{W_{\text{latt.}}} < 1\). Apparently, in this case the lattice structure is not destroyed down to individual elements; only fragmentation of the substance occurs. When \(\dfrac{\Delta W}{W_{\text{latt.}}} > 1\), there evidently occur additional energy losses due to the flow of the post-breakdown current and other secondary effects. For KJ the ratio \(\dfrac{\Delta W}{W_{\text{latt.}}} = 0.15 \div 1.5\), which agrees with the results for NaCl.

The measured energy is the total energy absorbed during the breakdown of a solid dielectric. As the experiments have shown, the energy expended on the breakdown of NaCl and KJ single crystals is, in order of magnitude, equal to the lattice energy referred to the destroyed volume of the corresponding salt.

LITERATURE

1. Edited by Prof. L. I. Sirotinskii. High-Voltage Engineering, Part I, Gostekhizdat, p. 37, 1951.
2. Ibid., p. 14.
3. B. M. Gokhberg and E. Ya. Zandberg. DAN SSSR, 53, 515, 1946.
4. B. M. Gokhberg. Izv. AN SSSR (ser. fiz.), 10, 125, 1946.

Tomsk Polytechnic Institute