Energy conversion efficiency of electrical exploding foil accelerators

An electrical exploding foil accelerator is a unique small-scale device for accelerating a thin dielectric foil to high velocities (∼20 km/s¹). Owing to developments and adaptations for high-energy sources, it has become increasingly useful for high-pressure and initiation devices^2,3 compared to other pulsed-power sources such as lasers and charged particle beams. Generally, an exploding foil accelerator comprises three layers: a substrate, a conductive layer with a small conversion section (often referred to as the “bridge”), and a dielectric flyer. Although the construction of an exploding foil accelerator itself is simple, its flyer acceleration mechanisms are complicated, and clarifying the entire process remains challenging because of the small size (typically of the order of 0.1 mm) and time scales (the flyer acceleration completes within several hundred nanoseconds). The flyer terminal velocity of a given material and/or electrical configuration and the conversion efficiency of input energy to material acceleration are essential considerations.⁴ Gurney⁵ developed an analytical model for estimating the initial velocity. Subsequently, several studies have modified the original Gurney model to improve its accuracy^6,7 with experimental validation.⁸ Rigorous modeling of the flyer acceleration by bridge explosion typically involves the use of a hydrocode with detonation equations.^9,10 The development of time-resolving velocimetry techniques, such as the velocity interferometer system for any reflector (VISAR)¹¹ and photonic Doppler velocimetry (PDV),¹² has facilitated direct comparison of the numerical results and experimental data. In contrast, few studies have quantitatively investigated the energy conversion efficiency of the flyer acceleration process.⁴ Because the effective mass ejected by the bridge explosion is of the same order as the designed-bridge mass (order of 10⁻⁹ kg), precisely measuring the difference in mass before and after ignition and correspondingly the kinetic energy of the flyer is exceedingly difficult.¹³ In this study, we estimate the effective mass indirectly based on a combination of the impulse produced by the flyer upon impact with a target and flyer velocity measurements. For the impulse measurement, we use a gravity pendulum-type impulse stand, which is typically used in space propulsion research.¹⁴ Combining this measurement with the velocity measurement obtained using a PDV system, the correlation between the specific input energy and flyer kinetic energy for two different bridge sizes is described and discussed with respect to the input electrical energy.

Figures 1(a) and 1(b) present the schematics of the flyer assembly and experimental setup for the flyer velocity measurement with its electrical circuit, respectively. The device comprised three layers: a 25-μm-thick substrate layer (polyimide), a 9-μm-thick conductive layer (copper), and a 25-μm-thick flyer (polyimide). In addition, a 300-nm-thick aluminum reflection layer was deposited on the flyer. The conductive layer included a converged square-shaped section (“bridge”), with a side length of l_b (0.2 or 0.4 mm). The designed-bridge mass (m_b) for l_b = 0.2 and 0.4 mm was 4.7 × 10⁻⁹ and 1.9 × 10⁻⁸ kg, respectively. The device was mounted on a mount plate supported by the XY, Z, and gonio stages and pressed using a presser plate with a visualization window (4 mm height and 6 mm width). The electrical circuit for device initiation is depicted in Fig. 1. The device was parallelly connected, and the capacitor was charged by the high-voltage DC power supply. After the condenser was charged by the power supply, the switch was turned on to serially connect the device for flyer initiation. The discharge current and voltage were measured using a voltage probe (THDP0100, Tektronix, Inc.) and two Rogowski coils (CWT UM/30 B/1/80, Power Electronic Measurements Ltd.). The flyer velocity was measured using a commercial modular PDV system (VH17, IDIL Fibres Optiques) with a band-pass filter (TBF-1550-1.0-FCAPC, Newport Corporation) having a center wavelength of 1550 nm and a bandwidth of 0.8 nm. A PDV probe with an outer diameter of 3.2 mm and a working distance of 50 mm was set in the PDV probe holder with two gauge pins for adjusting the probe center with respect to the bridge center. The distance between the probe head and bridge surfaces was 8 mm, which was within the probe working distance. To record the discharge current, discharge voltage, and PDV signal, two oscilloscopes (1 and 2) were used: oscilloscope 1 (MDO4024C, Tektronix, Inc.), with a sampling rate of 1.25 GHz, recorded the discharge current and voltage of the condenser, whereas oscilloscope 2 (SDA 820Zi-B, Teledyne LeCroy), with a sampling rate of 80 GHz, recorded the discharge current and the PDV probe signal. In both recordings, the discharge current was used as the trigger signal.

An example plot of the time history of the discharge power (P_b, left axis) and flyer velocity (u, right axis) for l_b = 0.4 mm is displayed in Fig. 2. The input power (P_b) was calculated based on the discharge voltage and current data. At t = 73 ns, P_b reached the maximum value of 8.1 MW and, simultaneously, the bridge section exploded (burst time)¹⁵ after which P_b gradually converged to zero. The input energy (E_b) to the bridge section is defined as the result of time integration of P_b,

For the operation condition shown in Fig. 2, E_b was calculated to be 0.33 J. When the bridge exploded, the flyer accelerated. The measured history of u (gray line) was fitted using a sixth-order polynomial approximation (blue line). The rise time of u was adjusted to the burst time in order to cancel out the internal time delay in the PDV system. The flyer velocity increased rapidly, and at t = 335 ns (262 ns after the burst time), a terminal velocity of 2.9 km/s was reached. The flyer terminal velocity was evaluated and used for estimating the effective bridge mass and related parameters as described below.

To verify the flyer velocity measurement by PDV, we performed flyer visualization using a shadowgraph imaging technique, as depicted in Fig. 3. In the shadowgraph method, a 50-Mfps high-speed camera (Ultra-8, Hadland Imaging, LLC) and a pulsed diode laser (Cavilux Smart, Cavitar) with a wavelength of 640 nm and an emission time of 20 ns that was used as the light source were activated with an appropriate delay time using a delay generator (DG645, Stanford Research Systems), which was triggered by the discharge current signal from the Rogowski coil. A band-pass filter (PB0640-060, Asahi Spectra Co., Ltd.), with a central wavelength of 640 nm and a bandwidth of 10 nm, was installed immediately in front of the high-speed camera to remove the emissions caused by bridge explosion, thereby improving the contrast and resolution of the experimental features. The flap angle of the bridge surface with respect to the laser light was adjusted using the gonio stage.

The impulse produced by the flyer impact was measured using a gravity pendulum-type impulse stand. Figure 4(a) illustrates the pendulum and bridge with its fixture. The pendulum comprised a 381-mm-long pendulum arm supported by two knife edges at the fulcrum, a 3D-printed flyer catcher, and an eddy current damper. The knife edges were made of stainless steel (ISO 4401-316-00-I), with a width of 10 mm and an apex angle of 60°. Each knife edge was mounted on a V-shaped groove with a full apex angle of 120°. Each groove was formed by two blocks of stainless steel (ISO: 4301-304-00-I) such that the bottom was not rounded. The eddy current damper included an aluminum plate, and a neodymium magnet dampened the pendulum oscillation. Figure 4(b) displays a magnified image of the flyer catcher with a 20-mm diameter inner hole, 6-mm height, and a 6-mm entrance hole. Because the flyer moved with the compressing atmospheric air, the flyer catcher included 16 exhaust holes, with a diameter of 1.0 mm each, set every 45° on the side-surface. The bridge was mounted on an L-shaped plate in front of the flyer catcher. The flyer catcher and bridge were set coaxially, and the distance between the flyer catcher entrance and bridge surface was set to 3.5 mm, where the accelerated flyer reaches the terminal velocity. A laser displacement sensor (LDS) (IL-030, Keyence Corporation), with a resolution of 1 µm, was used to measure the pendulum displacement, which was recorded using an oscilloscope (DL850E, Yokogawa Test & Measurement Corporation) at a sampling rate of 500 kHz, triggered by the discharge current.

Figure 5 depicts the time history plot of the pendulum displacement (red line) for l_b = 0.2 mm and E_b = 0.33 J. The LDS signal was fitted using a dumping function (black dashed line) to calculate the envelope (black short-dash line). The fitted curve indicated that the natural oscillation period was 1.0 s, which was considerably larger than the timescale of bridge explosion (on the order of 100 ns). The pendulum was calibrated up to 97 µN s using an impact hammer (086E80, PCB Piezotronics, Inc.) supported by another knife edge, which is not shown in Fig. 4. The level of impact was varied by changing the angle of rise using metal washers, a pulley, an electromagnet, and a microstage. The impact point was the same as that of the flyer catcher. We applied the same calibration procedure described in Ref. 16. The calibration factor was 341 µN s/V and the regression correlation coefficient was 0.999. The uncertainty in the impulse measurement, which was caused by the repeatability and/or the curve fitting, was less than ±0.8 µN s, which was 1.2% of the minimum impulse value under the operation conditions.

In Sec. III A, the experimental results of flyer velocity validated by high-speed imaging and impulse measurement are presented. All experiments were conducted under atmospheric pressure. The effective flyer mass and energy conservation efficiency were evaluated based on these results.

We measured the flyer velocities at different E_b values using high-speed imaging as well as PDV. Figure 6 shows the shadowgraph images of bridge explosion with l_b = 0.2 mm and E_b = 0.33 J. All images were acquired perpendicular to the discharge current direction. The flyer was expected to be at the top of the particle cloud, which consisted of vaporized copper and polyimide flyer. However, unlike x-ray imaging,¹⁷ the shadowgraph imaging used in this study cannot visualize the internal structure of the particle cloud because the incident 640-nm light was reflected and/or refracted at the cloud. Figure 6 shows the corresponding projection profile of the cloud. When the bridge exploded, a hemispherical particle cloud was generated. The particle cloud expanded with time, maintaining a hemispherical shape, and the cloud radius (indicated by the white arrow in Fig. 6) gradually increased. In Ref. 17, detailed 3D in-flight flyer images, in which the flyer departs from the substrate, were reported; the radius of the flyer shape was consistent with that of our experiment. The cloud radius in each frame was estimated by counting the number of pixels, where a single pixel corresponded to 8.0 ± 0.05 µm, and calibrated using a glass-made reference scale. The estimated cloud radius in each frame is also depicted in Fig. 6, and based on the variation of the radius with time, the expansion speed of the cloud head was calculated to be 4.0 ± 0.15 km/s.

Figure 7(a) shows the flyer velocity (u) measured by the PDV system for different l_b values. The input energy E_b was in the range of 0.15–0.33 J for l_b = 0.2 mm and 0.13–0.33 J for l_b = 0.4 mm. For both l_b values, the measured u increased almost linearly with the increase in E_b, and at E_b = 0.33 J, u was measured to be 3.9 ± 0.1 and 2.9 ± 0.07 km/s for l_b = 0.2 and 0.4 mm, respectively. The open symbols shown in Fig. 7(a) at E_b = 0.33 J were obtained through shadowgraph imaging. For the case with l_b = 0.2 mm, the open and closed symbols overlapped. For both bridge sizes, the velocity measured through the PDV system agreed well with that of the shadowgraph imaging. The l_b = 0.4 mm case exhibited a higher rate of increase in u with increasing E_b values; within the operating conditions, the velocity increase rates were 2.8 and 6.7 km/s/J for l_b = 0.2 and 0.4 mm, respectively.

Figure 7(b) depicts the induced impulse (I) at the flyer catcher for different l_b values. Overall, the measured impulse increased linearly with an increase in E_b. However, in contrast to the velocity values, a smaller bridge size resulted in a smaller impulse; at E_b = 0.33 J, I was measured to be 45 and 67 µN s for l_b = 0.2 and 0.4 mm, respectively. The slope of the line from the origin represents the momentum-coupling coefficient (C_m ≡ I/E_b);¹⁴ hence, a higher C_m indicates more efficient energy conversion from the input electrical energy to the kinetic energy of the flyer. The maximum value of C_m was obtained at E_b = 0.33 J, and the values of C_m for l_b = 0.2 and 0.4 mm were 136 and 200 µN s/J, respectively. The energy conversion efficiency is discussed in Sec. III B.

We estimated the flyer effective mass based on the u and I values measured in the aforementioned procedures and evaluated the energy conversion efficiency for conversion from the input electrical energy to flyer kinetic energy.

Figure 8 shows the dependence of the normalized flyer effective mass m_b,eff/m_b on E_b. The effective mass m_b,eff was calculated using the relationship m_b,eff = I/u. For both l_b values, m_b,eff/m_b increased linearly with an increase in E_b and exceeded 1.0 at E_b = 0.15 J for l_b = 0.2 mm and E_b = 0.26 J for l_b = 0.4 mm. This indicates that the exploded area was larger than the designed-bridge area. Lei et al.¹⁸ presented several bridge images after explosion, where the exploded area increased with an increase in input energy and was larger than the designed-bridge area. In our experiment, the minimum distance (d) between the non-exploded electrodes was measured using a laser microscope (VK-X3000, Keyence Corporation). At E_b = 0.33 J, d/l_b was found to be 3.8 for l_b = 0.2 mm and 1.5 for l_b = 0.4 mm, which is consistent with m_b,eff/m_b = 2.5 and 1.2 for l_b = 0.2 and 0.4 mm, respectively.

Information on the input energy, effective flyer mass, and flyer velocity enabled the calculation of the energy conversion efficiency at the bridge section. Figure 9 displays the specific kinetic energy (u²/2) as a function of the specific input energy (E_b/m_b,eff) for various l_b values. The slope of the line from the origin represents the energy conversion efficiency [η_b ≡ (u²/2)/(E_b/m_b,eff)]. For l_b = 0.4 mm, E_b/m_b,eff ranged from 9.4 to 15 MJ/kg, and u²/2 increased with the increase in E_b/m_b,eff from 1.1 MJ/kg at E_b/m_b,eff = 9.4 MJ/kg (η_b = 12%) to 4.3 MJ/kg at E_b/m_b,eff = 15 MJ/kg (η_b = 30%). For l_b = 0.2 mm, E_b/m_b,eff and u²/2 were larger with comparable η_b; at E_b/m_b,eff = 29 MJ/kg, u²/2 = 7.7 MJ/kg (η_b = 27%), whereas at E_b/m_b,eff = 34 MJ/kg, u²/2 = 5.7 MJ/kg (η_b = 17%). Although both l_b values exhibited comparable η_b, the E_b/m_b,eff dependency of η_b exhibited contrasting behavior in the operation range examined. As shown in Fig. 8, the rates of increase in (m_b,eff/m_b)/E_b are 11 and 1.4 J⁻¹ for l_b = 0.2 and 0.4 mm, respectively. This result indicates that compared to the l_b = 0.4 mm case, not only the designed-bridge area but also the surrounding materials exploded for the l_b = 0.2 mm case owing to the excessively high input energy density. Therefore, E_b/m_b,eff decreased with the increase in E_b. Thus, the quantitative evaluation of the effective flyer mass helped us infer that the specific input energy should be optimized in order to maximize the energy conversion efficiency.

In this study, we quantitatively investigated the energy conversion efficiency of an electrical exploding foil accelerator for two different bridge sizes (0.2 and 0.4 mm square) based on the flyer velocity and induced impulse measurements. For measuring the flyer velocity and impulse, a PDV system validated using high-speed imaging analysis and a gravity pendulum-type impulse stand, respectively, were used. With an increase in the input energy at the bridge section, the flyer velocity as well as the impulse increased almost linearly. The maximum flyer velocity was 3.9 km/s with an impulse of 45 µNs and 2.9 km/s with an impulse of 67 µNs for the 0.2 and 0.4 mm square bridges, respectively. The effective exploded-bridge mass calculated from the velocity and impulse data increased with input energy and was up to 2.5 times greater than that of the designed-bridge mass. The energy conversion efficiency from the electrical input energy to the flyer kinetic energy was evaluated based on the input energy, flyer velocity, and effective exploded bridge mass. Regardless of the bridge size, the maximum values of η_b were comparable (27% and 30% for the 0.2 and 0.4 mm square bridges, respectively), whereas the specific input energy dependency of the efficiency was contrasting. A higher specific input energy resulted in higher efficiency for the 0.4 mm bridge at least up to 15 MJ/kg, whereas the 0.2 mm bridge exhibited a contrasting dependency above 30 MJ/kg owing to the excessively high energy density; therefore, the effective flyer mass increased rapidly by including the surrounding materials. These results indicate that the specific input energy should be optimized for maximizing the efficiency and requires further investigation.

This research was supported by Japan Society for the Promotion of Science Grant-in-Aid for Early-Career Scientists Grant No. JP20K14950.

The data that support the findings of this study are available within the article.