On the Question of the Energy Introduced into a Wire by the Time of the Onset of Electrical Explosion

T. N. Barchenko, M. A. Melnikov

(Submitted by the scientific seminar of the Research Institute of High-Voltage Engineering)

The study of phenomena occurring during the electrical explosion of conductors has recently been receiving considerable attention, because exploding wires are finding wide application. The revival of scientific interest in this phenomenon is evidenced by three international conferences on this question held in the USA.

However, the successful application of exploding wires has led, according to [7], to an unjustified reduction in theoretical studies from a purely scientific point of view and in quantitative investigation of the phenomenon itself of exploding wires.

In the present work, which is a continuation of [1], results are presented from studies of energy release in a nichrome wire when the generator voltage is varied.

Experimental results and their discussion

In an \(R, L, C\) circuit, in which the principal active resistance is the resistance of the wire, the greater part of the energy should be released in it; however, as is known, an unlimited amount of energy cannot be introduced into a wire. The obstacle to this is the destruction of the material of the wire, i.e., its explosion, which leads to a complete cessation of current flow in the circuit or to its interruption for some interval of time. Therefore it may turn out that, by the moment of explosion of the wire, a considerable part of the energy stored in the generating capacitance will not be used.

The amount of energy released in the wire by the moment of its explosion is, according to [2], equal to the sublimation energy; according to other data [3—6], it may considerably exceed the amount of energy necessary for complete evaporation of the wire.

In the present work, data are given on the dependence of the energy release in the wire on the current density by the moment of input of an energy equal to the sublimation energy.

The experimental conditions were as follows: the capacitance of the capacitor in the wire-explosion circuit remained constant and equal to \(1.2\ \mu\mathrm{F}\), the inductance was \(0.8\ \mu\mathrm{H}\), and the voltage was varied from \(4.5\) to \(17\ \mathrm{kV}\). A nichrome wire of diameter \(0.05\ \mathrm{mm}\) and length \(3\ \mathrm{cm}\) was used (its sublimation energy was \(W = 2.5\ \mathrm{J}\)). The energy stored in the capacitor, even at a voltage-

at voltage \(U=4.5\ \mathrm{kV}\) was considerably greater than was necessary for the explosion of the wire (\(W_{\mathrm{sub}}\)). The discharge regime was aperiodic, \(R(t)>2\sqrt{L/C}\).

The main results of the experiments are presented in Table 1.

Table 1

The current density \(j_a\) at the moment when the explosion of the wire begins (point \(a\), Fig. 1) was calculated as the ratio of the current at that moment to the diameter of the wire at \(t=20^\circ\mathrm{C}\). \(U_{\mathrm{p}}\) is the greatest voltage during the explosion of the wire. At

Fig. 1. Oscillograms of current (a) and voltage (b) during explosion of the wire.

\(U=17\) and \(14\ \mathrm{kV}\), the voltage peak did not fit on the oscilloscope screen; therefore it is indicated that it is greater than the value given in the table. \(P\) is the pow-

ness at point \(a\); \(t_0\) is the time from the beginning of current flow in the wire to the current pause; \(t_1\) is the time from the beginning of current flow to point \(a\); \(W_0\) and \(W_1\) are the energies introduced by the times \(t_0\) and \(t_1\); \(H^2/8\pi\) is the magnetic pressure on the surface of the wire; \(t_{\mathrm{sp}}\) is the time of the sharp drop in current.

Consideration of the results presented in Table 1 shows that the amount of energy released in the wire by the moment at which its explosion begins increases. At a voltage \(U = 17\) kV, 3 J had been released in the wire by the moment \(a\), while the sublimation energy of the wire is 2.5 J, i.e., the energy increases by 20%.

The above-mentioned increase in energy at high voltages (14 and 17 kV), compared with the sublimation energy in this work, is explained by the inertia of the wire.

According to the model of wire explosion developed in [2], the time of formation of the current pause is equal to:

1) the time for the discharge wave to pass from the surface to the axis of the wire, determined by the speed of sound in the heated metal under consideration (we shall denote it by \(t_3\)); 2) the time for the decollectivization front to traverse the same distance.

The loss of conductivity, according to [2], occurs during the passage of the electron decollectivization front.

Obviously, it may be assumed that during the time of passage of the discharge wave from the surface to the axis of the wire the conductivity will not change noticeably. The conductivity, apparently, also will not change until, at least in a small part of the wire, the distance between ions increases to such a value at which the process of electron decollectivization begins (we shall denote this time by \(t_a\)). Therefore, the time of passage of the sound wave from the surface to the axis of the wire and the time \(t_a\) may be taken as the inertia of the wire.

The time \(t_3\), if calculated from the speed of sound in the metal at \(t = 20^\circ\)C, is \(5 \cdot 10^{-9}\) s, but the speed of sound in that state of the wire material at which the loss of conductivity begins may change greatly. In addition, the diameter of the wire at the moment of explosion may be larger than at \(20^\circ\)C.

Consequently, the amount of energy introduced into the wire during the times \(t_3\) and \(t_a\) should characterize the excess energy. In our experiments the difference between the time \(t_1\) and the time at which the sublimation energy had been released in the wire should characterize the inertia time of the wire. For \(U = 17\) kV this difference is \(0.5 \cdot 10^{-8}\) s, i.e., an order of magnitude greater than the calculated value of \(t_3\).

It is natural to suppose that, with increasing voltage, the inertia time will not change appreciably, while the amount of energy introduced during this time will increase because of the growth of the current through the wire. In addition, as the voltage is raised, the fraction of this time in the total time \(t_0\) and \(t_1\) increases, since \(t_0\) and \(t_1\) decrease with increasing \(U\).

This, apparently, can explain the results of works [3–6], in which an increase occurred in the energy released in the wire with an increase in the current density in it.

On the basis of the foregoing, it may be considered, in agreement with [2], that the explosion begins when an energy equal to the sublimation energy has been introduced into the wire.

The total amount of energy released in the wire before the current pause increases with increasing voltage. At \(U = 17\) kV, \(W_0 = 4\) J, and at \(U = 4.5\) kV, \(W_0 = 2.6\) J. This occurs because, although the time of the drop decreases with increasing voltage, the value of the current and voltage during the drop increases. Moreover, the energy increases

proportional to the square of the current, while the decay time changes insignificantly.

Therefore, the amount of energy released in the wire during the current decay increases with increasing voltage. The energy released during the decay time will go into overheating the wire vapor. Let us consider the current-decay time \(t_0—t_1\).

To determine the influence of the circuit parameters on the front and decay of the current pulse, a calculation was performed.

The change of resistance in the \(R, L, C\) circuit with time is specified by curve 1 (Fig. 2).

The initial change of resistance from infinity to the established value simulates the process of switching by spark gaps.

It is known that the change in the resistance of the spark of the switching gap from infinity to tenths of an ohm occurs in a time of the order of \(5 \cdot 10^{-8}\) sec. After \(5 \cdot 10^{-8}\) sec, the principal active resistance of the circuit, for \(R > 2\sqrt{1/C}\), becomes the resistance of the wire (it is taken equal to 21 ohms). The change in the resistance of the circuit at \(t > 0.5\ \mu\text{sec}\) simulates the explosion of the wire, during which the active resistance changes from 21 ohms to infinity.

The explosion time, for convenience of comparison, was taken equal to the switching time. The results of calculating the change of current with time for this specified circuit resistance are given in Fig. 2 (curves 2 and 3).

Fig. 2. Influence of the circuit parameters \(R, L, C\) on the rise time and current-decay time for a prescribed change of active resistance with time:
1 — \(R(t)\); 2 — \(I(t)\) for \(L = 5\ \mu\text{H}\), \(C = 1\ \mu\text{F}\); 3 — \(I(t)\) for \(L = 1\ \mu\text{H}\), \(C = 1\ \mu\text{F}\).

As can be seen from the figure, the current rise is determined not by the commutator, but by the parameters of the circuit \((L, C, R)\); the current-decay time, upon a sudden increase of the resistance from the established value to infinity, depends only on the rate of change of this resistance.

Thus, the current-decay time during the explosion of the wire characterizes the rate of change of its resistance during expansion, i.e., only the processes inside the wire. According to [2], this, apparently, will be the transit time of the decollectivization front from the surface to the axis of the wire. As shown in Table 1, the decay time decreases with increasing voltage on the capacitor. When the voltage changes from 4.5 to 17 kV, this time decreased from \((2.5 \div 3)\cdot 10^{-7}\) sec to \(1 \cdot 10^{-7}\) sec, respectively.

Such a decrease may be associated with:
1) a faster expansion of the wire vapor; 2) the fact that, by the moment of explosion, the wire has different diameters at different voltages.

It is seen from Table 1 that, as the voltage is decreased, the time to explosion of the wire \(t_1\) increases and the magnetic pressure \((H^2/8\pi)\) decreases. Therefore one may expect that, at low voltages, the wire has greater opportunities to expand than at high voltages.

A characteristic indicating greater expansion of the wire at low voltages is the resistance measured at the moment the explosion begins. The table shows that this resistance is larger at low voltages, but the difference (18.5 ohms at \(U = 4.5\) kV versus 14.4 ohms at \(U = 17\) kV) is small.

On this basis it may be considered that the decay time is determined not so much by the diameter of the wire as by the expansion rate of the wire vapor. Consequently, as the voltage is increased, the expansion rate of the vapor increases and the current decay time decreases. Thus, one may expect that at higher voltages the velocities of the shock waves should be higher, as has been noted repeatedly in the literature.

As the voltage is increased, the current density and the power at point \(a\) increase: the density is proportional to the voltage to the first power, and the power to the square of the voltage.

Conclusions

1. The explosion of the wire occurs not at the moment of maximum current in the circuit, but when an amount of energy equal to the sublimation energy has been imparted to the wire.

2. As the current density at the moment the explosion begins is increased, the amount of energy introduced into the wire may exceed its sublimation energy. This excess occurs as a consequence of the inertia of the wire.

3. The total amount of energy introduced into the wire before the current pause may considerably exceed the energy necessary for the explosion of the wire, and the increase will be the more significant the higher the current density at the onset of explosion of the wire.

4. As the voltage is increased (more precisely, the current density at point \(a\)), the current decay time decreases, which apparently occurs because of the more rapid expansion of the wire vapor. This should lead to the formation of more powerful shock waves.

References

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2. N. A. Protopopov, V. M. Kulgavchuk. ZhTF, 31, 5, 557, 1961.

3. I. F. Kvartskhana, V. V. Bondarenko, A. A. Plyutto, A. A. Chernov, ZhETF, 31, 5 (11), 745, 1956.

4. Exploding Wires. Ed. A. A. Rukhadze, IIL, Moscow, 1963.

5. Electrical Explosion of Conductors. Ed. A. A. Rukhadze, IIL, 1965.

6. T. J. Tucker. J. Appl. Phys. 32, 10, 1894, 1961.

7. Y. Chase. UFN, vol. 85, 2, 381, 1965.