ON THE PROBLEM OF THE ELECTRICAL EXPLOSION OF CONDUCTORS

Yu. A. Kotov, M. A. Melnikov, V. V. Nikitin

(Presented by the scientific seminar of the Scientific Research Institute of High Voltages at TPI).

The growing interest in the phenomenon of the explosion of conductors in recent years is explained by the possibility of using it both for practical purposes—light sources with a brightness temperature up to 60000°K, a source of shock waves, high-voltage fuses, obtaining large electron emission currents, etc.—and by the possibility of studying the behavior of a metal when a high energy density is dynamically achieved in it.

It has been shown qualitatively that the nature of the processes arising when a current pulse of large magnitude, but not causing its explosion, passes through a wire [1, 2], or the nature of the processes after the explosion of the wire—the duration of the “current pause” [3—5], the intensity of expansion of the vapor channel [5—7], the temperatures and energy densities attained [8—10]—is determined, on the one hand, by the current density reached and, on the other, by the time over which energy is supplied to the wire.

In other words, the nature of the behavior of a metallic conductor after the passage of a current pulse through it is determined by the value of the volumetric energy density achieved in it.

The present work is devoted to studying the character of the dependence of the maximum current density and the explosion time of wires on the initial voltage of the energy source and on the parameters of the wire itself: material and geometrical dimensions. The results obtained are compared with data from other authors.

The experimental procedure is described in [11]. The parameters of the discharge circuit were: \(C = 1.2\ \mu\mathrm{F}\), \(L_k = 0.28 \cdot 10^{-6}\ \mathrm{H}\), not including the inductance of the wire, which was determined by calculation. The voltage on the capacitor \((V_0)\) was varied from values at which a characteristic explosion of the wire was observed up to 15 kV. Copper, constantan, and nichrome wires were investigated, with lengths from 6.5 to 35 mm and diameters: copper—\(d = 0.06—0.15\) mm, nichrome—0.1 mm, constantan—0.09 mm.

By the explosion time of the wire we mean the time from the moment the current is switched on until its complete interruption after the first pulse (Fig. 1, a).

In the case when the “current pause” was absent, the explosion time was determined from the current oscillogram by extrapolating the current curve to zero, as shown in Fig. 1, b.

Simultaneously with the oscillographic recording of current and voltage, we carried out continuous photographic recording of the shadow pattern of the exploding wire [11]. Here we do not discuss the results of processing the shadow photographs. We note only that, according to our classification, the explosion time \((T_{\mathrm{v}})\) consists of three characteristic parts:

1. The time \((T_1)\) before the onset of slow expansion of the wire vapor. This is the time from the moment the current is switched on to the appearance of a step on the voltage oscillogram for copper (Fig. 1) and constantan wires. In nichrome wires, the onset of vapor expansion is manifested by an inflection in the current oscillogram.

2. The time \((T_2)\) of slow expansion of the wire vapor without formation of a shock wave, with a solid core of the wire material present inside the vapor channel. This time ends when the voltage begins to increase sharply (the beginning of the voltage peak).

3. The time \((T_3)\) of rapid expansion of the wire vapor—from the beginning of the voltage peak to the formation of the “current pause.”

A similar classification was carried out by the authors of [12], but they combine \(T_1\) and \(T_2\), although these instants are clearly recorded on the oscillograms and photographic time-resolved records of the explosion process.

Fig. 1. Oscillograms of current (above) and voltage during the explosion of a copper wire \(d = 0.12\ \text{mm}\): \(a\)—measurement of the explosion time in the presence of a “current pause,” \(V_0 = 5\ \text{kV}\), \(l = 20\ \text{mm}\); \(b\)—measurement of the explosion time in the absence of a “current pause,” \(V_0 = 7\ \text{kV}\), \(l = 10\ \text{mm}\)

Results and Discussion

The time of the first pulse was investigated in [13, 14], with which we compare the data of our experiments. It should be noted that in [13] the time up to the onset of rapid vapor expansion was studied, i.e., \(T_1 + T_2\). This time, of course, does not correspond to the time required for the wire material to receive all the energy; nevertheless, comparison with the data obtained in [13] is useful.

The dependence of \(T_{\mathrm{v}}\) on the cross section \(S\) of the wire (Fig. 2, curves 1, 2, 3) shows that, although in our experiments wires of smaller cross section were studied at energy input rates greater by a factor of 1.8 (at equal \(V_0\)) than in [13] (there \(d = 0.13\text{–}0.35\ \text{mm}\), \(l = 15\text{–}60\ \text{mm}\), \(L_k = 0.5 \cdot 10^{-6}\ \text{H}\), \(V_0 = 10\ \text{kV} = \text{const}\)) and at lengths smaller than in [14] for the same energy input rate (in [14] \(d = 0.08\text{–}1.16\ \text{mm}\), \(l = 71\) and \(151\ \text{mm}\), \(L_k = 0.3 \cdot 10^{-6}\ \text{H}\), \(V_0 = 10\text{–}20\ \text{kV}\)), the character of the curves \(T_{\mathrm{v}} = f(S)\) remains the same, namely: the explosion time increases linearly with the wire cross section. As \(V_0\) increases, the slope of \(T_{\mathrm{v}} = f(S)\) decreases.

Comparison with the data of [13] also shows that if the extrapolation \(T_{\mathrm{v}}=f(S)\) to the zero value of \(S\) in [13] (there \(T_{\mathrm{v}}=T_1+T_2\)) gives \(T_{\mathrm{v}}=0\), then in our case there is a different \(T_{\mathrm{v}}\), decreasing with increasing \(V_0\). It follows from this that the time of rapid expansion \(T_3\) must also be linear in \(S\), and the slope of \(T_3=F(S)\) must decrease with increasing \(V_0\).

Fig. 2. Dependence of the explosion time on the wire cross section (curves 1–3) and on the initial capacitor voltage (curves 4–7): 1—copper, \(l=10\) mm, \(V_0=5\) kV; 2—the same, \(V_0=7\) kV; 3—the same, \(V_0=9\) kV; 4—nichrome, \(d=0.1\) mm, \(l=10\) mm; 5—nichrome, \(d=0.1\) mm, \(l=20\) mm; 6—copper, \(d=0.06\) mm, \(l=15\) mm; 7—copper, \(d=0.12\) mm, \(l=15\) mm.

The character of the change of \(T_{\mathrm{v}}\) with \(V_0\) (Fig. 2, curves 4–7) shows that, for materials sharply differing in physicomechanical properties, the curves \(T_{\mathrm{v}}=\varphi(V_0)\) have an analogous form: as \(V_0\) increases, \(T_{\mathrm{v}}\) rapidly decreases in the region of small \(V_0\), while \(\dfrac{dT_{\mathrm{v}}}{dV_0}\) increases. Starting from a certain \(V_0\), which is the larger the greater the length of the wire (curves 4, 5) and the cross section (curves 6, 7), the change in \(T_{\mathrm{v}}\) becomes practically linear in \(V_0\).

The influence of the wire length \(l\) on \(T_{\mathrm{v}}\) is presented in Fig. 3. It is seen that:

1. The dependence \(T_{\mathrm{v}}=\Psi(l)\) only for \(l\le 10\) mm has one and the same form at different \(V_0\) for all wires.

2. In the range \(l>10\) mm, the character of the curve \(T_{\mathrm{v}}=\Psi(l)\) is determined by the material (curves 2, 4 and 3, 6) and by the value of \(V_0\) (curves 1–3 and 4–5).

3. For each of the wires investigated there is such a \(V_0\) at which \(T_{\mathrm{v}}=\mathrm{const}\) is observed, independently of the length, for \(l>10\) mm.

(curve 6, and also for nichrome \(d = 0.1\) mm—\(V_0 = 16\) kV, for constantan, \(d = 0.09\) mm—\(V_0 = 12\) kV, for copper, \(d = 0.12\) mm—\(V_0 = 9\) kV, etc.).

Having investigated the influence of length on \(T_{\mathrm{v}}\) only at one voltage, the authors [13] concluded that \(T_{\mathrm{v}}\) does not depend on \(l\), which is true, as shown above, only in a special case.

Fig. 3. Dependence of the explosion time on the wire length: 1, 2, 3—nichrome, \(d = 0.1\) mm, \(V_0 = 5, 7\), and 12 kV, respectively; 4—copper, \(d = 0.12\) mm, \(V_0 = 7\) kV; 5—the same, \(V_0 = 12\) kV; 6—copper, \(d = 0.06\) mm, \(V_0 = 12\) kV

In [14] copper wire was investigated only at two lengths, 71 and 179 mm. It turned out that \(T_{\mathrm{v}}\) decreases linearly with the square root of the energy of the first pulse per unit length. This confirmed the hydrodynamic theory of wire explosion, which the authors of [14] use to explain their results. The energy measurements and calculation carried out showed that in our case such a relation between \(T_{\mathrm{v}}\) and the energy of the first pulse indeed holds for all wires in the range \(l > 15\) mm (Table 1), but in the region of smaller lengths an increase in the energy per unit length leads to an increase in \(T_{\mathrm{v}}\), i.e., if the hydrodynamic theory is valid, then only within certain limits of variation of the wire parameters.

The dependence of the maximum current density (\(jm\)) on the wire length is presented in Fig. 4. It is seen that decreasing \(V_0\) leads to a reduction in the \(jm\) attained. The decrease of \(jm\) at all lengths, materials, and cross sections of the wires is less than follows from the proportional dependence of \(jm\) on \(V_0\). All curves \(jm = \gamma(l)\) show that, beginning with some length, the smaller the larger the wire diameter (curves 2 and 4), a further decrease in \(jm\) ceases. Obviously, this will occur only within some range of lengths.

At equal voltages, a higher current density is attained in thinner wires (curves 2, 3, 4), but \(j_m\) increases less than would follow from an inversely proportional dependence of \(j_m\) on \(S\).

Table 1

Apparently, the average value of \(j_m\) is affected not only by the parameters of the explosion circuit and of the wire, but also by the uniformity of the current distribution over the cross section, discussed in [15].

Fig. 4. Dependence of the maximum attained current density on wire length: 1—copper, \(d = 0.008\) mm, \(V_0 = 12\) kV; 2—copper, \(d = 0.06\) mm, \(V_0 = 7\) kV; 3—copper, \(d = 0.08\) mm, \(V_0 = 7\) kV; 4—copper, \(d = 0.12\) mm, \(V_0 = 7\) kV; 5—constantan, \(d = 0.09\) mm, \(V_0 = 7\) kV.

The assumption of the authors of [12] that wires always explode at one and the same current density is not confirmed.

In conclusion, it should be noted that at present there is no theory of the explosion of conductors that satisfactorily explains all aspects of the phenomenon; therefore, further accumulation of experimental material will make it possible to test and determine the limits of applicability of existing theories, and also to create a general theory of the explosion of conductors.

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