Laser and Particle Beams (2009), 27, 49–55. Printed in the USA.
Copyright # 2009 Cambridge University Press 0263-0346/09 $20.00
doi:10.1017/S026303460900007X
Circuit simulation of the behavior of exploding wires
for nano-powder production
Z. MAO, X. ZOU, X. WANG, X. LIU,
AND
W. JIANG
Department of Electrical Engineering, Tsinghua University, Beijing, China
(RECEIVED 1 October 2008; ACCEPTED 15 November 2008)
Abstract
The electrical behavior of exploding wires was obtained by numerically solving the nonlinear differential equation describing
the discharge circuit. For metal wires of high conductivity and low sublimation heat, such as copper, aluminum, gold, and
silver, the circuit simulation can be well performed based on the resistivity model developed by Tucker in which the
resisitivity is expressed by the explicit functions of specific action, i.e., r ¼ f(g). For metals such as titanium and zinc
with anomalously changing resistivity, i.e., decreasing rather than increasing with the liquid heating, the circuit simulation
of the exploding wires can be performed using the implicit relationship between r and g that is read out point by point
from the experimentally measured curve. Using the circuit simulation, the rate of the energy deposition in the exploding
wires before the explosion can be obtained, which is helpful to choose the right experimental conditions for possible
overheating that is desirable for getting smaller nano-powders produced by exploding wires.
Keywords: Circuit simulation; Exploding wires; Nano-powder production
mechanism is the creation of a highly resistive channel of
dense metallic vapor, leading to the current to transfer to a
lower impedance load. Thus, one way to characterize the
fuses is to use their time-increasing resistance R(t). The
faster the R(t) increases, the better is the fuse. For wire
array Z-pinches in which the wire arrays are embedded in
vacuum, the parameters such as the current waveform, the
energy deposited resistively in the wire before plasma formation, and the peak voltage have much influence on the
initial explosion dynamics of wires (Hammer & Sinars,
2001; Politov et al., 2000). This initial wire dynamics may
affect the development of the instabilities, the formation of
precursor plasmas on the axis, and other aspects of the
later time behavior of the wire array implosions. For the production of nano-powders where the wires are embedded in
gases at a pressure from 0.1– 1 atm, the vapor produced by
EEW is cooled by collisions with the gas molecules,
forming nano-powders from the condensed vapor. The particle size depends on overheating of the wires, i.e., the ratio
of the deposited energy before explosion to the specific
energy of sublimation. The overheating results from a high
rate of energy deposition and an expansion lag of the
heated wires. In this article, a circuit model of EEW for the
nano-powder production was established in which the timevarying resistance R(t) of the wire was considered as a
INTRODUCTION
Exploding wires or electrical explosion of wires (EEW) is
performed by rapidly heating the wires to vaporization temperature with a high density (104 – 106 A/mm2) current pulse
flowing through the wires (Chace & Howard, 1959). EEW
has found many applications among which are the opening
switches (Chuvatin et al., 2006), named fuses, in the circuits
for inductive energy storage (Schoenbach et al., 1984;
Kolacek et al., 2008; Orlov et al., 2007; Sasaki et al.,
2006), the discharge loads of X-pinch (Liu et al., 2008a,
2008b; Li et al., 2008), or Z-pinches, namely wire array
Z-pinches (Sanfod et al., 1996), and the production of nanopowders (Kotov, 2003). Moreover, a large body of thermodynamic data in the liquid state, and evaluations of the
critical point resulted from traditional thermophysical
measurements, were obtained with “exploding wire” experiments (Lomonosov, 2007). Knowing the electrical behavior
of the exploding wires is important for the aforementioned
applications. For fuses in which the wires are embedded in
the dielectrics with low compressibility to prevent the expansion of the evaporated metallic vapor, the opening
Address correspondence and reprint requests to Xinxin Wang, Department
of Electrical Engineering, Tsinghua University, Beijing, China. E-mail:
wangxx@mail.tsinghua.edu.cn
49
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50
function of specific action (Tucker & Toth, 1975). The nonlinear differential equation describing the circuit was numerically solved and the electrical behavior, such as current and
voltage as well as the deposited energy of exploding wires
made from different materials under different experimental
conditions, was obtained. Although numerous papers concerned with EEW for nano-powder production could be
found, only a few of them are focused on the electrical behavior of EEW by circuit simulation that would be helpful to
the investigation of nano-powder production by EEW.
EQUIVALENT CIRCUIT AND MODEL
OF WIRE RESISTANCE
Figure 1 shows our experimental setup of EEW as well as its
equivalent circuit for nano-powder production. Eight pieces
of metal wire are installed on a wire holder made from two
insulating discs inside the vacuum chamber. A controlling
handle for the wire holder comes out through the top plate
of the vacuum chamber. By rotating the handle in the horizontal plane, each wire could be moved, in turn, to the
right position to connect two electrodes inside the
chamber. In this way, eight wires could be exploded one
by one without opening the chamber.
In Figure 1b, C is the energy storage capacitance of 2 mF;
s is a gas spark gap switch; L1 and R1 are the equivalent
Z. Mao et al.
inductance and resistance of the external circuit outside the
vacuum chamber, respectively; L2 and R2 are those inside
the chamber. L and R(t) are the equivalent inductance and
time-varying resistance of the wire, respectively. Since the
resistive divider was installed outside the vacuum chamber,
the voltage across the wire could not be directly measured.
In order to compare the wire voltage obtained from simulation with the experimentally measured ones, we need to
know not only the values of the total inductance and the
total resistance of the external circuit, i.e., L1 þ L2 and
R1 þ R2, but also the values of L2 and R2. By replacing the
resistive divider with a thick aluminum rod and then
making short-circuit discharge, L1 and R1 were calculated
from the waveform of the measured discharge current to be
1.96 mH and 65 mV, respectively. By replacing the wire
inside the chamber by the aluminum rod and using the
same method, L1 þ L2 and R1 þ R2 can also be calculated.
Since L1 and R1 are known, L2 and R2 could be determined
to be 0.52 mH and 5.4 mV, respectively. The wire inductance
L was calculated using the following empirical formula
(Chen et al., 1992),
L¼
m0 l
2l 3
:
ln
2p
r 4
(1)
where m0 is the magnetic susceptibility of the vacuum; l and r
are the wire length and radius, respectively.
The only unknown parameter in the equivalent circuit is
the time-varying resistance R(t). It has been demonstrated
that R(t) can Ðbe expressed as a unique function of Ðeither
energy, W ¼ i2 . R(t) . dt, or specific action, g ¼ j2 . dt
(Tucker, 1961), where i and j are current and current
density, respectively. Energy, although perhaps physically
more meaningful, contains the resistance implicitly; on the
other hand, the specific action involves only current and time.
A simplification of the quasi-static theoretical model for
R(t) developed by Tucker and Toth (1975) was used. R(t)
was described in terms of two basic processes, the heating
process in a given phase and the phase change process.
Based on conservation of energy, the formulas of the resistivity were derived for these two processes.
For the heating process in a given phase the resistivity of
the wire can be expressed by
g(t)
r
r(t) ¼ ri exp
ln max
gmax
ri
Fig. 1. Experimental setup for nano-powder production by exploding wires.
(a) Schematic diagram. (b) Equivalent circuit.
0 g gmax
(2)
where ri is the initial resistivity at t ¼ 0, the time when
heating starts; gmax and rmax are the values of g and r at
the end points of the heating phase, i.e., the points of the
onset of melting or vaporization.
In the case of phase change processes in which the wire is
a two-phase material of solid and liquid or liquid and vapor,
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Circuit simulation of the behavior of exploding wires for nano-powder production
energy conservation, i.e., i2R dt ¼ 2H dM, yields
r1
r(t) ¼ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
r r2 g(t)
1 2 2 1
gmax
r2
gmax ¼
0 g gmax ,
Hd
(r þ r2 ),
2r1 r2 1
(3)
(4)
where H is the latent heat of fusion or vaporization; M is the
mass of phase 1; r1 and r2 are the resistivity of phase 1 and
phase 2 at melting temperature or vaporization temperature,
respectively; gmax corresponds to the point when the material
of phase 1 is completely transformed into that of phase 2, d is
the mass density of phase 1.
The parameters, such as ri, rmax, r1, r2, and gmax in
Eqs (2) and (3) were given by Tucker and Toth (1975) and
are listed in Table 1 for some materials used in our circuit
simulation. As well-known, the region of vaporization is
characterized by a very rapid increase of resistance associated
with a decrease in wire cross-section. If the voltage across the
wire is sufficiently high, the resistance rise will be terminated
by an arc breakdown through the wire vapor, resulting in a
resistance maximum designated the “burst.” Hence, EEW
in our simulation was ended by a vapor breakdown,
forming plasma with a resistivity of empirical values.
RESULTS AND DISCUSSIONS
With the model of the wire resistance described above, the
nonlinear circuit shown in Figure 1b was numerically
solved using a circuit analysis code called Simplorer from
ANSOFT. In order to check the validity of the circuit simulation, the simulation results of exploding copper, titanium
and zinc wires are compared with our experimental results
for these materials.
Exploding Copper Wires
The copper wires used in our experiment are 196 mm in
diameter and 85 mm in length, which yields an equivalent
inductance of 0.11 mH calculated using Eq. (1). Figure 2 presents the waveforms of the discharge current and voltage
51
measured in the experiment in which the energy storage
capacitor was charged to a voltage of 20 kV. It shows a
typical picture of exploding wire. The voltage begins to
rise strongly when the current reaches its maximum of
about 10 kA and then falls down, which means that the
vaporization of the wire begins, leading to a rapid increase
of the wire resistance. After the current falls down to a
value of 3 kA, it rises up again, which represents an arc
breakdown through the wire vapor, a shunting of the
current by this low resistance arc.
It should be noted that the experimentally measured
voltage shown in Figure 2 is not the voltage of the wire resistance but rather the voltage across L2 and R2 in addition to the
wire voltage. The voltage of the wire resistance is important
for us to make an estimation of the energy deposition in the
wire before the explosion. It was calculated by
VR (t) ¼ Vm (t) R2 i(t) (L2 þ L)
di(t)
:
dt
(5)
Since Vm(t) and i(t) are the measured voltage and current, we
called VR(t) obtained using Eq. (5), the experimentally
obtained VR(t). The experimentally obtained VR(t) together
with the measured current are compared to those obtained
from the circuit simulation, as shown in Figure 3.
It is obvious that the waveforms of the current in the experiment and the simulation are quite similar in shape, but a little
bit different in amplitude, about 10 kA occurred in the experiment in contrast to 12 kA in the simulation. The reason for
this difference may be that the inductance of the circuit
used in the simulation was experimentally determined a
little bit lower than the real one. As we know, the current
in EEW is mainly dependent on the inductance rather than
on the resistance of the circuit before the vaporization of
the wire. Concerning the voltage, the waveform in the experiment is wider in the pulse width and much lower in the
amplitude than in the simulation, about 95 kV in contrast
to 180 kV. Except that the higher current the simulation
would cause a higher voltage in the simulation. The main
reason for this big difference in the voltage pulse obtained
in the experiment and the simulation, may be caused by
the fact that the time response of the resistive divider, we
used to measure the voltage in the experiment is not fast
Table 1. Parameters used in simulation for some metals
Solid heating
Metal
Copper
Aluminum
Silver
Zinc
Titanium
Melting
Liquid heating
Vaporization
ri,
mV . cm
rmax
mV . cm
gmax
A2 sec/mm4
r1
mV . cm
r2
mV . cm
gmax
A2 sec/mm4
ri
mV . cm
rmax
mV . cm
gmax
A2 sec/mm4
r1
mV . cm
r2
mV . cm
gmax
A2 sec/mm4
1.77
2.82
1.59
5.8
41
9.9
11.2
8.6
16.0
156.0
80492
25238
61682
11260
3034
9.9
11.2
8.6
16.0
156.0
18.9
23.1
15.9
31.6
163.5
13736
6797
10089
3224
1129
18.9
23.1
15.9
31.6
163.5
26.3
41.5
27.3
29.4
158
29780
16616
18361
5506
2911
26.3
41.5
27.3
29.4
158
620
393
859
925
613
48992
17215
22158
18955
12187
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52
Fig. 2. Waveforms of discharge current and voltage measured in the
experiment.
enough for the measurement of a narrow pulse of about 50 ns
in full width at half magnitude (FWHM) as shown in
Figure 3b. Indeed, the voltage pulse of EEW in experiments
should be such a narrow pulse (Tucker, 1961). It is well
known that a voltage divider of lower time response will
Z. Mao et al.
change a narrow pulse to a wider in pulse width and
smaller in amplitude output one. We had calibrated the resistive divider with a quasi-square wave of about 7 ns in rise
time, to our surprise; the output pulse from the divider
became 20 ns in rise time, 13 ns longer than the real
one. Furthermore, it should be indicated that the results
from the circuit simulation were based on a computation
with 0.1 ns time step that means with quite a high time resolution. If the time step of the computation was changed to
10 ns to reproduce, to some extent, the effect of the slow
time-response divider, the amplitude of the voltage in the
simulation was decreased from 180 kV to 140 kV. Based
on these analyses, we believe that the waveform of the
measured voltage was much distorted by the divider and
the voltage obtained from the simulation may be more
believable.
The wire resistance and the energy deposition in the wire
could be readily obtained by the calculation using VR(t)
and i(t). It was found that the wire resistance behaves
almost in the same way as VR(t) does, i.e., a narrow pulse
with its maximum of about 16V, appearing at the time
when VR(t) is maximum, which means that the current is
mainly determined by the rapidly increasing wire resistance
after the vaporization of the wire, begins. The energy deposition in the wire can be calculated by
ð Td
Wd ¼
VR (t) i(t) dt,
(6)
0
Fig. 3. Comparison of the current and the voltage of the wire resistance. (a)
Experimental results. (b) Simulation results.
where Td is the time at which the energy deposition stops
usually by an arc breakdown through the wire vapor.
For an exploding wire, the wire may be under heated, i.e.,
the arc breakdown through wire vapor occurs before the wire
is fully vaporized, or overheated, i.e., one has a high rate of
the energy deposition in contrast to the expansion lag. Wd is
an important parameter for the application of EEW in the
production of nano-powders since the particle size usually
decreases with the increase of the overheating of the wires,
i.e., the ratio of Wd to the specific energy of sublimation.
Unfortunately, Td is a parameter difficult to be predetermined
due to the complicated process of the arc breakdown through
the wire vapor. In the model, it was assumed that the arc
breakdown happens as soon as the deposited energy
reaches Wvapor, a value just enough to vaporize the whole
wire, 135 J for one copper wire we used. As a result, Wd unfavorably becomes a constant for a definite wire and no overheating or under heating could be predicted from this
model. Even so, the rate of the energy deposition obtained
from the circuit simulation based on this model could be
helpful to choose the right experimental conditions to
increase the possibility of overheating.
It is expected that a high rate of energy deposition can be
realized with a high rate of current rise. There are several
ways to increase the rise rate of the current, including
raising up the charging voltage of the energy storage
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Circuit simulation of the behavior of exploding wires for nano-powder production
53
capacitor, reducing the capacitance of the energy storage
capacitor while keeping the stored energy unchanged, lowering the inductance of the external circuit outside the vacuum
chamber. Figure 4 shows the corresponding results of the
simulations.
From Figure 4 it can be seen that during the first microsecond, the deposited energy is very small due to the small
wire resistance, no matter which circuit parameters were
chosen in the simulations. Hence, we defined the averaged
rate of energy deposition as
Rd ¼
Wvapor
,
Td
(7)
where T*d is the time interval from t ¼ 1 ms to the time instant
corresponding to Wvapor, the effective time of the energy
deposition.
Since Rd is inversely proportional to T*d, we need only to
pay attention to T*d that was listed in Table 2. Figure 4a
yields a group of shortest T*d, but it was obtained at the cost
of consuming much more energy in one discharge. While
consuming a lower energy compared to Figure 4a,
Figure 4b yields a group of a little bit longer T*d. It seems
that lowering L1, the equivalent inductance of the external
circuit outside the vacuum chamber, is a good choice.
However, when you put this method into practice you
would find that the equivalent inductance of a circuit
working at a high voltage could be limitedly reduced. For
Figure 4c, the shortest T*d of 2.08 ms was obtained by reducing the capacitance of the energy storage capacitor while
keeping the stored energy unchanged at 225 J. It is probably
the most practicable method for reducing T*d.
Exploding Titanium Wires
While the above model of wire resistance works well in the
circuit simulation for the metals of high conductivity and
low sublimation heat, such as copper, aluminum, gold, and
silver, it introduces a big error into the calculated voltage
of the exploding wires made from titanium and zinc. The
reason for this may be that the resistivities of titanium and
zinc anomalously change with the joule heating. As we
know, the resistivity of many metals may be expressed by a
linearly increasing function of temperature, i.e.
r(T) ¼ ri [1 þ k(T Ti )],
(8)
where T is the temperature; ri and Ti are the initial resistivity
and temperature, respectively; k is the linear factor depending
on the metal material.
The model of wire resistivity we used for the heating
process, Eq. (2), was derived based on energy conversation
and Eq. (8). However, as was shown in italics in Table 1,
the resistivity of titanium and zinc are decreased rather than
being increased during the process of liquid heating, which
means that their resistivity may not be expressed by Eq. (8).
Fig. 4. Rate of the energy deposition as function of the circuit parameters.
(a) L1 ¼ 1.96 mH, c ¼ 2 mF. (b) Ucharge ¼ 15 kV, c ¼ 2 mF. (c) L1 ¼
1.96 mH, Wstorage ¼ 225 J.
It should be noted that some other metals such as palladium,
vanadium, and scandium, also show a decreasing resistivity
during the process of liquid heating.
Although the resistivity of titanium and zinc may not be
expressed by the explicit functions of specific action g, i.e.,
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54
Z. Mao et al.
Table 2. The effective time of the energy deposition T*d as a
function of the circuit parameters
Curve No.
Figure No.
4a
4b
4c
1
2
3
2.08 ms
2.08 ms
2.87 ms
1.36 ms
1.68 ms
2.52 ms
0.97 ms
1.22 ms
2.08 ms
Eqs (2) and (3), the relationship between r and g can be read
out, point by point, from the experimentally obtained curves
provided by Tucker (1975). With this method, which we
called resistivity acquisition method (RAM), the nonlinear
circuit shown in Figure 1b may also be numerically
calculated.
Figure 5 shows the comparison of the results of exploding
titanium wires obtained by the experiment and by the simulation with RAM. The titanium wires used in our experiment
are 234 mm in diameter and 85 mm in length. By making the
Fig. 6. Dependence of the wire resistivity on the specific action.
circuit outside the vacuum chamber more compact, L1 was
reduced from 1.96 mH to 0.94 mH. The energy storage
capacitor of 2 mF was charged to a voltage of 20 kV.
It can be seen that the experimentally measured current
and voltage agree well with those obtained in the simulation
until t ¼ 1.45 ms, which means one has more or less the same
deposited energy or specific action in the wire in both cases
before t ¼ 1.45 ms. After that time, the wire in the simulation
first begins to burst, a voltage peak occurs, and the experimental results become quite different from the simulation
ones. While the burst happens at the time t 1.57 ms and
with the amplitude of 34 kV in the simulation, it appears at
the time t 1.98 ms and with the amplitude of 21 kV in
the experiment. Even with more or less the same specific
action before the burst, the wire bursts behave differently in
the experiment and the simulation. It was suggested by this
phenomenon and confirmed by Figure 6 that during the
burst, the dependence of the wire resistivity on the specific
action g, i.e., r ¼ f(g), which we used in the simulation is
different from the experimental one derived from
Figure 5a. It is this difference that results in the differences
between Figures 5a and 5b.
CONCLUSIONS
Fig. 5. Comparison of the results of exploding titanium wires. (a)
Experimental results. (b) Simulation results.
The electrical behavior of exploding wires of high conductivity and low sublimation heat, such as copper, aluminum,
gold, and silver, can be well simulated based on the resistivity model developed by Tucker in which the resisitivity
is expressed by the explicit functions of specific action,
i.e., r ¼ f(g). For metals such as titanium and zinc with
anomalously changing resistivity, i.e., decreasing rather
than increasing one at liquid heating, the circuit simulation
of the exploding wires can be performed using the implicit
relationship between r and g that is read out point by point
from the experimentally measured curve. Using the circuit
simulation the rate of the energy deposition in the exploding
wires before the explosion can be obtained, which is helpful
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Circuit simulation of the behavior of exploding wires for nano-powder production
to choose the right experimental conditions for possible
overheat that is desirable for getting smaller nano-powders
produced by exploding wires.
ACKNOWLEDGEMENT
This research was supported by National Natural Science
Foundation of China under contracts 50677034.
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