An experimental study of the explosion of cylindrical copper rods in the current skinning mode was performed using the GIT-12 high-current pulse generator with currents of amplitude up to 5 MA and rise time about 2 μs. It was observed that the waveforms of the electrical impedance of the exploded rods exhibited low-frequency oscillations with a period of several hundreds of nanoseconds. The results of the experiment with exploded solid copper rods carried out on the GIT-12 facility were interpreted by performing a series of magnetohydrodynamic calculations. It was shown that the experimentally observed low-frequency oscillations of the rod impedance were associated with a fast magnetosonic wave excited in the material of the rod.
Magnetic fields with inductions of more than 1 MG are usually called superstrong,1,2 since in these fields, the strength and thermal limits are exceeded for all known materials. Superstrong fields can be generated using pulsed power technologies, in particular, by initiating an electrical explosion of conductors (EECs) in the current skinning mode. Note that there is a wide variety of EEC modes3,4 associated with various technological applications. For instance, EEC in a medium is used to provide electric power peaking in pulsed power technologies5,6 and to produce nanopowders,7–9 and EEC in vacuum is used to create x-ray sources.10–12 Interest in EECs occurring in the current skinning mode,1,2,13–17 which is the subject of this paper, is also associated with various applications. It is worth mentioning some of them. First, this is the generation of strong magnetic fields, both by compressing metal liners18 and by exploding single-turn solenoids.13 Second, this is the compression of metal liners in the Z-pinch geometry to pressures of 1–100 Mbar to attain extreme states of the metal.19 Third, this is the electromagnetic acceleration of bodies,19 in particular, the acceleration of flat metal plates in experiments on studying shock waves (SWs). Finally, this is the transportation of electromagnetic energy through vacuum transmission lines in the currently developed multiterawatt generators with a current level of about 50 MA,20–22 which are supposed to be used to implement controlled thermonuclear fusion schemes based on Z-pinches.
The EEC in the skinning mode has attracted considerable attention in the context of research on implementation of controlled thermonuclear fusion in the Z-pinch geometry in the framework of the MagLIF (Magnetized Liner Inertial Fusion) concept23,24 proposing the compression of an initially heated deuterium–tritium mixture by a metal liner. Currently, the possibility of implementation of the MagLIF concept and the related problems are being actively investigated at Sandia National Laboratories (USA). These investigations involve the use of the Z pulsed power machine capable of generating currents of amplitude up to 20 MA with rise times of about 100 ns, which is the most powerful of the currently existing facilities of this type.25–27
The main processes involved in an EEC occurring in the current skinning mode are the shock and nonlinear magnetic diffusion waves (NMDWs) propagating in the conductor material and the formation of a dense low-temperature plasma at the conductor surface.28–30 The nonlinear diffusion is characterized by an anomalously high penetration rate of the electromagnetic field into the conductor compared to ordinary diffusion. The increase in the diffusion rate is associated with a decrease in the conductivity of the metal due to its heating by the current it carries. Nonlinear diffusion can occur only in a quite strong magnetic field (at high currents flowing through the conductor), the minimum value of which for most metals is several hundreds of kilogausses. This magnetic induction corresponds to a magnetic pressure of several thousands of atmospheres on the conductor surface. Therefore, the nonlinear diffusion wave usually propagates through the material together with the shock wave induced by the magnetic field pressure on the conductor surface.28,29 As the magnetic field is increased (to 2–4 MG), the conductor explodes and a dense low-temperature plasma is produced at its surface.17,31 When a nonlinear magnetic diffusion wave propagates through a conductor, a nonmonotonic distribution of current density j is established with its peak value shifted from the surface to the bulk of the conductor.28 For an exploding rod, this behavior of the current density wave should vary the rod impedance during the explosion.
This paper presents the results of experimental and theoretical investigations of the current–voltage characteristics of copper rods 3–5 mm in diameter exploded in the current skinning mode by a current pulse of amplitude 5 MA and rise time 2 μs. This current pulse provided a maximum magnetic field of induction 3.5–6.5 MG with a rise rate of (1.7–3.3)·1012 G/s at the rod surface.
II. EXPERIMENTAL ARRANGEMENT
An experiment aimed to study EEC in the current skinning mode was carried out on the GIT-12 terawatt pulsed power generator (Fig. 1). It consists of 12 Marx generators connected in parallel.32 The capacitor bank of the generator is capable of storing energy of 5 MJ at a maximum charging voltage. However, in the mode in which the experiment was carried out, the energy stored in the capacitor bank was 2.6 MJ. With this stored energy, current pulses of amplitude above 5 MA and rise time about 2 μs could be produced in the short circuit mode.
During the experiment, oscilloscope recording of current and voltage waveforms was carried out to obtain the time dependence of the rod impedance. Measurements of the current through the load were carried out (with an accuracy of 5%) using an inductive groove sensor.33 Voltage measurements were performed with an inductive voltage divider. The arrangement of the central unit of the GIT-12 current generator and the locations of the current and voltage sensors are shown in Fig. 2.34
The central unit of the GIT 12 generator is a current collector to which the transmission lines of the 12 Marx generators are connected. The collector rests on a short-circuited coaxial line (coax C1), and another coaxial line (coax C2) serves to transfer the collector current to the load.
Inductive groove sensors33 are used to measure the total current at the input to the load and the current in the load, . Sensor G1 comprises an inductive groove of diameter 700 mm and width 5 mm with an inductance of 0.074 nH. Sensor G2 comprises an inductive groove of diameter 370 mm and width 5 mm with an inductance of 0.053 nH. Sensor G2 has three leads spaced evenly in azimuth. This makes it possible to control the symmetry of the current flow in the load. The load current is determined by averaging the readings from the three outputs. To calibrate current sensors G1 and G2, we used a 75-Ω high-voltage cable as the load of the GIT-12 generator. The cable was terminated on a matched resistive voltage divider (75 Ω) with a built-in 0.075-Ω shunt. The impedance of the voltage divider and the resistance of the shunt were measured with an accuracy of 10−2%.
To measure the voltage across the rod load, we used an inductive voltage divider (see Fig. 2). The inductive voltage divider is based on a short-circuited section of coax C2, on which the current collector is mounted. The lower arm of the divider is the inductive groove sensor (GV) of inductance 0.21 nH and the upper arm is the inductive coax C2 (631.8 nH). The Rogowski coils (RC) and the inductive groove sensor (GV) installed on the bottom flange of coax C2 are intended to measure the current in coax C1 and the derivative of this current, respectively. The signal of GV is the voltage across the lower arm of the inductive voltage divider. Thus, the voltage drop across the load, V(t), is proportional to the GV signal with a division factor of 3008.6. Calibration of the RC and GV sensors was carried out similarly to the calibration of the G1 and G2 sensors.
The plasma formation and the development of large-scale instabilities at the rod surface were investigated by directly observing the behavior of the plasma–magnetic field interface. The frame-by-frame shooting of the processes involved in the explosion was performed in the optical range using an HSFC Pro four-frame optical camera with a frame exposure time of 3 ns.
Figure 3 shows schematically the arrangement of the load unit with an exploded rod and the high-voltage electrodes of the GIT-12 generator. In designing the load unit, special attention was paid to ensuring good contacts between the exploded conductor and the GIT-12 coaxial electrodes (see Fig. 3). To do this, the following measures were taken. On the side of the GIT-12 high-voltage electrode (cathode), the rod was welded to a steel disk, which, in turn, was welded to the coaxial cathode. On the anode side, a steel flange was used to provide sliding contact between the rod and the steel flange. The sliding contact on the anode side was not in the plane of the anode but was recessed 1 cm deep into the anode plane; the recess diameter was 1 cm. Thus, we increased the contact area and removed the contact from the region of the high electric field. In addition, a recess was made in the anode body for centering the rod and inserting it into the sliding contact of the anode electrode.
III. EXPERIMENTAL RESULTS
Before the experiment with copper rods, we obtained experimentally the current–voltage traces for a 36-mm diameter steel rod. Figure 4 shows the waveforms of , , and obtained in the experiment with the 36-mm diameter steel rod.
As can be seen from Figs. 4(a) and 4(b), high-frequency oscillations with period T = 14.1 ns and frequency MHz are present in several sections of the waveform. Most likely, these oscillations were excited due to sparking in the region where the lower arm of the inductive voltage divider was located and due to the buildup of voltage oscillations in the stray capacitance circuit of the inductive groove sensor. Despite the fact that the stray oscillations have a well-defined frequency, they cannot be merely subtracted from the useful signal, as the stray signal has complex amplitude modulation. Based on these considerations, in order to smooth out the stray oscillations, we splined the voltage values over the time interval equal to two oscillation periods (28 ns). This splining interval was chosen for the following reasons. On the one hand, it was long enough to provide a reasonable degree of smoothing and, on the other hand, it was more than an order of magnitude less than the characteristic period of low-frequency oscillations in the voltage waveforms obtained using numerical simulation of the EEC process (see Fig. 5). The splined waveforms of and are shown in Fig. 4(c). The same figure shows the values of the voltage determined using Eq. (3). As follows from the measurements, the value of for the 36-mm diameter steel rod is significantly less than the measurement error for (5%).
Figure 5 shows the results of determining the waveforms of the voltage for rods 5, 4, and 3 mm in diameter. We have splined all experimental waveforms given in Fig. 5. The same figures (dashed lines) show the waveforms of obtained by numerical simulation of the EEC process (see Secs. IV and V). It should be stressed that, according to Eq. (2), when , the values of and may become negative if . As can be seen from Fig. 5, there are oscillations of various frequencies in the experimental voltage waveforms. Both relatively high-frequency oscillations and low-frequency oscillations with periods close to the periods of oscillations of the calculated voltage waveforms (several hundreds of nanoseconds) are present in these waveforms. Below, in Sec. VI, we will discuss the cause for the occurrence of low-frequency oscillations with the use of a Fourier transform to analyze the measured and calculated waveforms of the voltage across the exploded rods, . The measured amplitudes of ranged between 20 and 30 kV. They showed a slight tendency to increase with a decrease in rod diameter, which is more pronounced for the calculated curves.
Figure 6 presents optical images of exploded copper rods at various points in time, obtained using an HSFC Pro four-frame optical camera with a frame exposure time of 3 ns. It is well known that during the explosion of a conductor in a megagauss magnetic field, large-scale instabilities develop on the conductor surface.17,35–37 The initial stage of development of these instabilities can be seen in Figs. 6(d) and 6(g). The images shown in this figure indicate that by the 1300th through the 2000th nanosecond (depending on the diameter), the conductor still retained its cylindrical shape; large-scale instabilities showed up clearly at later times. Since low-frequency oscillations on the waveforms (see Fig. 5) had already formed when the conductor was still cylindrical, it can be said with confidence that large-scale instabilities are not the cause of their occurrence. As for the later times, it cannot be ruled out that the growth of these instabilities might affect the voltage waveforms.
The results of the experiment performed on the GIT-12 facility allow the following conclusions:
When the copper rods exploded, the waveform of the GIT-12 current was in fact independent of the rod diameter, which indicates a relatively low energy input into the rod material.
In the EEC modes implemented on the GIT-12 generator, a shunting breakdown, which is characteristic of the explosion of microconductors in vacuum at lower currents did not develop along the surface of the exploded rods.4
The time dependences of the voltage and electrical impedance obtained in experiments on the GIT-12 generator showed the presence of oscillations of different frequencies. Relatively high-frequency oscillations are apparently related to the measurement error, and the cause for the occurrence of low-frequency oscillations (with periods of several hundreds of nanoseconds) will be discussed as follows.
IV. MAGNETOHYDRODYNAMIC MODEL
For the calculations, we used wide-range semi-empirical equations of state (EOSs) developed at the Joint Institute for High Temperatures.40 These EOSs take into account the effects of high-temperature melting and evaporation and, in addition, provide for the possibility of the existence of metastable states near the liquid and gas parts of the phase diagram.
When calculating the electrical characteristics of the metal and the thermal conductivity coefficients, we invoked the data on copper conductivity given in tables compiled using the calculational-experimental BKL method.38,41,42 This method assumes that the conductivity of a metal parametrically depends on the form of its equation of state. In compiling the conductivity tables, the variable parameter was the conductivity at the critical point, that is, at the point at which the condensed, gaseous, and two-phase regions (vapor-drop mixture) of the metal phase diagram converged. We chose the value of the conductivity at the critical point so that the results of the MHD simulation were in the best agreement with experimental data. To do this, we used the data obtained for microconductors exploded in water.4,38 The BKL method also assumes that a metal-to-dielectric transition occurs in a material when its density becomes equal to that at the critical point. At this density, the dependence of conductivity on temperature changes. When the density of a material is greater than its critical density, the conductivity is metallic in nature, that is, it decreases with increasing temperature, which is typical for metals in condensed states; otherwise, it increases with temperature, which is typical for both dielectric materials and plasmas.
V. RESULTS OF THE MHD SIMULATION
As can be seen in Fig. 7, the waveforms of calculated in the MHD simulation, as well as the experimental voltage waveforms (see Fig. 5), indicate the presence of low-frequency oscillations. The amplitude and frequency of these oscillation increase with a decrease in the initial radius of the rod, as in the experiment performed on the GIT-12 facility. Note that, in addition to the qualitative agreement between the experimental and calculated data, there is also a satisfactory quantitative agreement between them. Thus, the calculated amplitude values of the voltage across the rod differ from the experimental values by no more than twice. The same takes place for the voltage oscillation frequency, found both in the simulation and in the experiment performed on the GIT-12 facility.
To understand the nature of the detected low-frequency oscillations, let us turn to the results of the MHD simulation of the electrical explosion of a copper rod with a diameter of 3 mm and height of 22.4 mm, which are presented in Figs. 8–14. Figure 8 shows plots of the time dependences of various types of energy accumulated in the material of the rod. As can be seen from Fig. 8, the full energy deposited in the rod material, , is about 200 kJ (calculation for the rods with diameters 4 and 5 mm gave lower values of this energy), which makes an insignificant fraction of the energy stored in the capacitor bank of the GIT-12 generator.
Figures 9–11 show plots of the calculated radial dependences of thermodynamic parameters [Figs. 9(a)–11(a)] and electrophysical quantities at different times [Figs. 9(b)–11(b)]. The radial dependences of thermodynamic parameters were calculated for times approximately corresponding to one of the times at which the images shown in Fig. 6 (see Sec. III) were taken. Thus, Fig. 9 ( ns) approximately corresponds to Fig. 6(a) (sh. 3034), Fig. 10 ( ns) to Fig. 6(c) (sh. 3034), and Fig. 11 ( ns) to Fig. 6(d) (sh. 3034). Letters A, B, C, and D in Figs. 9–11 mark the spatial arrangement of points with a fixed mass coordinate. Point A is on the axis of the conductor, point D is on the outer boundary of the plasma, and points B and C occupy intermediate positions. Point A, due to the boundary condition , is always in region I; point D, due to the boundary condition , is always in region III, and points B and C, initially located in region I, move to regions II and III over time.
As can be seen from Figs. 9–11, in the experiment carried out on the GIT-12 generator, the conductors exploded in the current skinning mode. The main processes involved in the EEC in this mode are a shock wave (SW) propagating in the conductor material and a nonlinear magnetic diffusion wave (NMDW).28,29 A nonlinear diffusion of an electromagnetic field into a conductor is characterized by an anomalously high penetration rate compared to ordinary diffusion. The increase in the diffusion rate is associated with a decrease in the conductivity of the metal due to its heating by the current it carries. When the magnetic induction rise rates are as high as close to 1014 G/s, the NMDW propagates with a velocity close to that of the SW.28,37
In the conductors previously exploded on the GIT-12 facility, just as in those exploded on the MIG facility,14,35–37 an NMDW formed at magnetic field rise rates of 2–5 × 1013 G/s. However, in the experiment under consideration, due to the low rise rate of the magnetic field (2–4 × 1012 G/s), the shock-wave processes ceased as early as 200–300 ns after the onset of current flow. According to the simulation, the rod material was subsequently compressed in a quasi-adiabatic mode. In such a compression mode, which lasted 400–600 ns (depending on the rod diameter), the magnetic pressure on the outer surface of the rod was approximately equal to the thermal pressure in its center (see Fig. 9); that is, a Bennett equilibrium was established. The quasi-adiabatic compression stopped when the temperature of the material at the rod surface approached the temperature corresponding to the critical point of the phase diagram (for copper, the critical temperature is 0.72 eV). After that, a plasma was produced at the rod surface, during the expansion of which a rarefaction wave formed (see Figs. 10 and 11).
In the final phase of rod explosions that occurred in the modes characteristic of the experiments on the GIT-12 facility, the NMDW propagated through the material together with the rarefaction wave. In this case, the rod material split into a dense, relatively cold core and a low-density, relatively hot peripheral “coat” [see Figs. 10(a) and 11(a)]. These regions can be distinguished from each other using a parameter , where is the thermal pressure and is the magnetic pressure. In Figs. 9–11, the densest part of the core corresponds to region I, for which , and the low-density peripheral “fur coat” corresponds to region III, for which . In addition, in these figures, transition region II is highlighted, for which . Note that Figs. 9–11 show that the current density maximum lies near the inner boundary of transition region II, and the magnetic pressure maximum lies near its outer boundary. Since the density at the outer boundary of the transition region is only two times less than the density of the solid body, region II is part of the dense core. The time dependences of the thermodynamic parameters of the metal at points A, B, C, and D will be discussed in Sec. VI.
The core material, due to its high conductivity, carried a significant proportion of the current that passed through the rod. On the contrary, the conductivity of the coat was several orders of magnitude lower than that of the core, and correspondingly, the current density in it was vanishingly small [see Figs. 10(b) and 11(b)].
VI. DISCUSSION OF THE SIMULATION RESULTS AND INTERPRETATION OF THE EXPERIMENTAL DATA
To elucidate the excitation mechanism of the low-frequency oscillations observed in the voltage across the rod and in the electrical impedance of the rod in our experiment (see Fig. 5), we consider the time dependences of thermodynamic quantities at fixed mass coordinate points. In the case under consideration, the mass coordinate is associated with the layer of the exploded rod material moving along the radius. The spatial arrangement of fixed mass coordinate points at different time points is indicated by letters A, B, C, and D in Figs. 9–11. As mentioned above, point A is on the axis of the rod, point D is on the outer boundary of the plasma, and points B and C are at intermediate positions. Points A and B refer to the parameters of the material in the dense core and points C and D to those in the peripheral low-density coat.
Figure 15 shows the amplitude–frequency characteristics of the voltage waveforms for rods 3, 4, and 5 mm in diameter and the AFC of the waveform of recorded for a steel rod 36 mm in diameter [see Fig. 4(c)].
As can be seen from Fig. 15, all AFCs of the calculated waveforms of have pronounced maxima at frequencies of 7–15 MHz, indicating the formation of a magnetosonic wave. For the 3-mm diameter rod, the maximum is at a frequency of 15.1 MHz, which corresponds to the oscillation period ns. For the 4-mm diameter rod, the maximum is at a frequency of 10.4 MHz, which corresponds to ns. For the 5-mm diameter rod, the maximum is at a frequency of 7.1 MHz, which corresponds to ns. This correlates with the calculated waveforms of shown in Fig. 7.
The AFCs of the experimental voltage waveforms contain a large number of maxima at different frequencies. However, in the low-frequency region, each of the experimental AFCs has maxima that can be associated with magnetosonic oscillations. In Figs. 15(a)–15(c), these maxima are marked with vertical dotted lines. The positions of these maxima are as follows. For the 3-mm diameter rod (sh. 3036), the maximum is at a frequency of 12.5 MHz, which corresponds to the oscillation period ns. For the 4-mm diameter rod (sh. 3031), the maximum is at a frequency of 12.5 MHz, which corresponds to ns. For the 5-mm diameter rod (sh. 3028), the maximum is at a frequency of 9.1 MHz, which corresponds to ns. For the remaining experimental waveforms of shown in Fig. 5, the frequencies related to the maxima, , which can be associated with magnetosonic oscillations, are given in Table I. Thus, the experimental data show a tendency, although weak, for the frequency of low-frequency oscillations to increase with a decrease in diameter of the exploded conductor, which is more pronounced in the AFCs for the calculated curves of .
|Diameter (mm) .||(MHz) .|
|Calculation .||Experiment .|
|Diameter (mm) .||(MHz) .|
|Calculation .||Experiment .|
Figure 15(d) shows the AFC obtained by processing the waveform of for the 36-mm diameter steel rod. As can be seen from this figure, the voltage across the rod at frequencies ranging between 7 and 20 MHz was not above 0.7 kV in amplitude, which is much lower than that for copper rods (∼4 kV) [see Figs. 15(a)–15(c)]. This suggests the absence of magnetosonic waves in the shot with the 36-mm diameter steel rod.
In an experiment with exploded metal rods, carried out on the GIT-12 facility (up to 5 MA current amplitude, about 2 μs current rise time), it was found that the waveforms of the voltage across the rod experienced low-frequency oscillations with a period of several hundreds of nanoseconds.
The MHD calculations performed to interpret the experimental results have shown that the main processes that occurred during the explosion of the rods at magnetic field rise times of 2–5 × 1012 G/s were the propagation of a nonlinear magnetic diffusion wave in the material and the formation of a rarefaction wave.
The rarefaction wave caused the rod to split into a dense, relatively cold core and a low-density, relatively hot peripheral “coat.” The parameter , which is the ratio of thermal pressure to magnetic pressure, was found to be for the core and for the coat.
The conductivity of the low-density coat was several orders of magnitude lower than that of the core; therefore, the core carried the main fraction of the current passed through the conductor.
During the explosion of a rod, magnetosonic waves were excited in the core material, the oscillation period of which was several hundreds of nanoseconds. The low-frequency oscillations of the voltage across the exploded rod and its electrical impedance observed in the experiment were associated with the excitation of a fast magnetosonic wave in the rod material. Since magnetosonic waves were excited in the core, electrical measurements provided information about the state of the dense axial plasma, in which the most extreme states of the metal might occur.
Conflict of Interest
The authors have no conflicts to disclose.
Vladimir Ivanovich Oreshkin: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Rina Baksht: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Methodology (equal); Project administration (equal); Software (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Stanislav Anatol'evich Chaikovsky: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Funding acquisition (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). Rustam Cherdizov: Conceptualization (equal); Data curation (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Project administration (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). V. A. Kokshenev: Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). N. E. Kurmaev: Data curation (equal); Investigation (equal); Methodology (equal); Writing – original draft (equal). Gennady A. Mesyats: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Validation (equal). Evgeny V. Oreshkin: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Writing – original draft (equal). Nikolai Ratakhin: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Writing – original draft (equal). Alexander Gennadievich Rousskikh: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Investigation (equal); Methodology (equal); Resources (equal); Validation (equal); Writing – original draft (equal). Andrey Zherlitsyn: Data curation (equal); Investigation (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (equal). Alexander S. Zhigalin: Data curation (equal); Investigation (equal); Methodology (equal); Resources (equal); Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).
The data that support the findings of this study are available from the corresponding author upon reasonable request.