High Temperature. Vol. 42, No. 4, 2004, pp. 539–544. Translated from Teplofizika Vysokikh Temperatur, Vol. 42, No. 4, 2004, pp. 538–543. Original Russian Text Copyright 2004 by A. S. Kravchenko, Yu. V. Vilkov, A. S. Yuryzhev, M. M. Saitkulov, and I. M. Brusnigin.
PLASMA INVESTIGATIONS
An Energy Source Based on a Spiral Magnetic-Flux Compression Generator with Simultaneous Axial Initiation of an Explosive Charge A. S. Kravchenko, Yu. V. Vilkov, A. S. Yuryzhev, M. M. Saitkulov, and I. M. Brusnigin Russian Federal Nuclear Center – All-Russia Research Institute of Experimental Physics, Sarov, Nizhni Novgorod oblast, 607190 Russia Received July 15, 2003
Abstract—A spiral magnetic-flux compression generator with simultaneous axial initiation of an explosive charge is treated. An external solenoid is used to develop the initial field in the generator. A theoretical model of this generator operating into inductive storage and to break point of the current circuit is described. Experimental results are given obtained in testing a generator with an inside diameter of the spiral of 115 mm and a working volume length of 330 mm. A 100-GW current pulse is obtained at a break point of electrically exploding copper conductors. The experimental results are compared with the results of numerical simulation of the generator operation.
INTRODUCTION One possible way of obtaining narrow (less than one microsecond) pulses of energy is its accumulation during the time of operation of a magnetic-flux compression generator (MFCG) in an inductive storage with its subsequent fast transfer to load with the aid of commutators such as electrically exploding breakers [1]. The power gain of current sources in the case of using break points of electrically exploding copper conductors is relatively low [2]. Therefore, highspeed MFCGs must be used to produce high-power energy pulses. One such generator is a spiral magnetic-flux compression generator with simultaneous axial initiation of an explosive charge – spiral AMFCG [3–6]. The investigation results given in [7] have demonstrated that spiral magnetic-flux compression generators with simultaneous axial initiation of a cylindrical explosive charge (SAMFCG) exhibit a high specific power in operation to load with a relatively high inductance. In an experiment with spiral AMFCGs, a generator with an inside diameter of the spiral of 115 mm and a working volume length of 320 mm in an inductive load of ~µH was used to produce a magnetic energy pulse of ~90 GW with a voltage pulse amplitude of ~350 kV. The deformation time of generator circuit
was ~10 µs. The specific power of this generator was ~10 GW/kg. The current gain of the investigated SAMFCG was limited by the value of residual inductance and did not exceed four [7]. Theoretical studies of an energy source based on such a spiral AMFCG reveal that voltage pulse above 1 MV may be obtained at a break point of electrically exploding copper conductors. In so doing, the maximum electric power at the break point, which reaches ~0.25 TW, significantly (by a factor of two-and-a-half to three) exceeds the power developed by the generator under load with optimal inductance without the sharpening of the energy pulse. The highest power and voltage may be produced in developing the initial field in an SAMFCG using an energy source whose effective operating time is comparable to the effective operating time of the power generator being investigated. An increase in the effective time of developing the initial field leads to a significant decrease in power and voltage of the output pulse. The calculated parameters of operation of a spiral AMFCG into load including an inductive storage and break point of the current circuit indicate that SAMFCGs may be used to generate powerful electromagnetic pulses. Given in this paper are the experimental results of investigation of such an SAMFCG in which the initial
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Fig. 1. Schematic of an SAMFCG: (1) copper tube, (2) spiral, (3) explosive charge, (4) organic-glass tube, (5) radial electric detonators, (6) inductive storage and the break point of current circuit, (7) external solenoid.
field was developed with the aid of an external solenoid by a scheme with flux interception [8, 9]. The generator operated into load including an inductive storage and a break point of current circuit. A diagrammatic view of this generator is given in Fig. 1.
CALCULATED MODEL The equivalent circuit of SAMFCG when powered from a capacitor battery by a scheme with flux interception is given in Fig. 2. The process of developing the initial field in an SAMFCG with such a method of powering is described by the equations dI 1 ------ dt dI 2 ------ dt dI 3 ------ dt dR --------f dt
dI 1 L 2 ), ------- = – ----2- ∫ I 1 dt/ ( L 1 L 2 – L 12 C dt L 12 dI 2 2 ), ------- = – -------- ∫ I 1 dt/ ( L 1 L 2 – L 12 dt C dI 3 ------ dt = 0.
(1)
Equations describing the electrotechnical processes in a system during the generator operation have the following form:
2 ) + b 2 ( ( L 3 + L 4 )L 12 – L 13 L 23 ) + b 3 ( L 13 L 2 – L 12 L 23 ) b 1 ( ( L 3 + L 4 )L 2 – L 23 = -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, 2 ( L 1 L 2 – L 12 ) ( L 3 + L 4 ) + ( L 12 L 13 – L 1 L 23 )L 23 + ( L 12 L 23 – L 2 L 13 )L 13 2 ) + b (L L b 1 ( ( L 3 + L 4 )L 12 – L 13 L 23 ) + b 2 ( ( L 3 + L 4 )L 1 – L 13 3 12 13 – L 1 L 23 ) = -----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------, 2 )(L + L ) + (L L ( L 1 L 2 – L 12 3 4 12 13 – L 1 L 23 )L 23 + ( L 12 L 23 – L 2 L 13 )L 13 2 ) b 1 ( L 2 L 13 – L 12 L 23 ) + b 2 ( L 12 L 13 – L 1 L 23 ) + b 3 ( L 1 L 2 – L 12 -, = ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------2 )(L + L ) + (L L ( L 1 L 2 – L 12 3 4 12 13 – L 1 L 23 )L 23 + ( L 12 L 23 – L 2 L 13 )L 13
= f ′ ( t ), HIGH TEMPERATURE
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(2)
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I1
in which r(t) is the radius of the inner tube, and l is the length of the generator working volume. Then,
I3
L12(t) I2
C
L1
L3
L4
L 2 ( t)
L13 L23(t) Fig. 2. Equivalent electric circuit of an SAMFCG: C, capacity of the source of initial energy; L1, total inductance of the external solenoid developing the initial field within the generator and the powering line; L2(t), inductance of the inner tube; L3, inductance of the SAMFCG spiral; L4, sum of the storage inductance and the inductance of the break point; L12(t), mutual inductance of the external solenoid and inner tube; L13, mutual inductance of the external solenoid and SAMFCG spiral; L23(t), mutual inductance of the SAMFCG spiral and inner tube; Rf (t), active resistance of the break point; I1, current in the powering circuit; I2, current in the inner tube; I3, current in the SAMFCG spiral.
b2
dL 2 4×10 –6 r ( t )v ( t ) ( 0.9r ( t ) + 2l ) --------- = -------------------------------------------------------------------------- , dt ( 0.9r ( t ) + l ) 2 where v(t) is the inner tube expansion velocity.
R f( t)
b1
541
dL 12 1 = I 2 ----------- – ---- ∫ I 1 dt, dt C
dL 12 dL 2 dL 23 = I 1 ----------- – I 2 --------- – I 3 ----------- , dt dt dt dL 23 b 3 = –R f ( t )I 3 – I 2 ----------- . dt
The inductance of external solenoid developing the initial field within the generator and the inductance of the SAMFCG spiral may be represented as [10] µ0 L i = ------ ω i2 d i Φ i . 4π Here, µ0 is the magnetic constant, di is the average diameter of the spiral, ωi is the number of spiral turns, and Φi is a coefficient whose values depend on the ratio between the spiral length and diameter.
The expression for mutual inductance of the external solenoid and inner tube may be written as [10] ω1 L 12 ( t ) = µ 0 -----2- Φ 12 ( t ) . l Here, ω1 is the number of turns of the external solenoid; r2 ( t ) -q ; Φ 12 ( t ) ≈ πr 1 r 2 ( t ) q 112 + ---------- 8r 12 312 1 q 112 = –1 + ------- ; γ 12
3 ; q 312 = 1 – γ 12
r1 γ 12 = --------------------- ; r 12 + l 2 and r1 is the average radius of the external solenoid. As a result, we have dL 12 ω dΦ 12 ---------- = µ 0 -----1- -----------, dt l 2 dt dΦ 12 r2( t ) ----------- = 2πr 1 r ( t )v ( t ) q 112 + ------------ q 312 . dt 4r 12 One can similarly write the expression for mutual inductance of the SAMFCG spiral and inner tube, ω3 L 23 ( t ) = µ 0 -----2- Φ 23 ( t ) , l where ω3 is the number of turns of the SAMFCG spiral; r2 ( t ) -q ; Φ 23 ( t ) ≈ πr 3 r 2 ( t ) q 123 + ---------- 8r 32 323 1 q 123 = –1 + ------- ; γ 23
3 ; q 323 = 1 – γ 23
The inductance of the inner tube may be found from the expression [11]
r3 γ 23 = --------------------- ; r 32 + l 2
2×10 –6 r ( t ) L 2 ( t ) = ----------------------------------- , 0.45 + l/2r ( t )
and r3 is the average radius of the SAMFCG spiral [10]. Then,
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KRAVCHENKO et al. dI/dt,
GA/s
I,
4.5
kA 250
4.0 200
3.5
Fig. 3. External view of the generator.
3.0
150
2.5
100
2.0
50
1.5 1.0
dL 23 ω dΦ 23 ----------- = µ 0 -----3- -----------, dt l 2 dt dΦ 23 r2( t ) ------------ = 2πr 3 r ( t )v ( t ) q 123 + ------------ q 323 . dt 4r 32 The mutual inductance of the external solenoid and SAMFCG spiral is found from the expression ω1 ω3 - Φ 13 , L 13 = µ 0 -----------l2 where r 32 Φ 13 ≈ πr 1 r 32 q 113 + -------2- q 313 ; 8r 1 1 q 113 = –1 + ------- ; γ 13
3 ; q 313 = 1 – γ 13
r1 γ 13 = --------------------- . r 12 + l 2 Semiempirical relations obtained as a result of experiments in electric explosion of conductors (for example, [1, 2, 12]) may be used to describe the variation of active resistance Rf (t) of the break point. So, by solving the sets of differential equations (1) and (2) in view of the equations of motion for liner given in [7], one can get an idea of the electrotechnical processes occurring in the system whose equivalent electric circuit is given in Fig. 1.
EXPERIMENTAL RESULTS The basic geometric dimensions of the generator under study were as follows: the spiral length of 330 mm, the inside radius of the spiral over insulation of 57.5 mm, 24 spiral turns, the outside radius of the inner tube of 36 mm, and the outside radius of the explosive cylinder of 32.5 mm. The initial inductance
0 0
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40
60
80
100
t , µs
Fig. 4. The current (dashed curve) and current derivative (solid curve) in the external circuit.
of the generator was ~15 µH. The external view of the generator is given in Fig. 3. The parameters of the external solenoid developing the initial field within this generator were as follows: the length of 330 mm, the average radius of 75 mm, and 14 turns. The inductance of such a solenoid was ~8 µH. A turn emerging from the middle of the generator spiral served the function of inductive storage. Fifty conductors of copper wire 0.12 mm in diameter were used for the break point of the current circuit. Conductors with a length of 1100 mm were wound on a frame 110 mm in diameter over a length of 850 mm. The total load inductance was ~2.8 µH. In so doing, the inductance of the break point is estimated at 0.4–0.43 µH. The initial value of the active resistance of copper conductors was 35 mOhm, with a total mass of 5.54 g. A 900-µF capacitor battery with a charging voltage of 35 kV was used to develop the starting field; the inductance of the powering line was 1 µH. An oscillogram of the current derivative and the current curve in the external circuit obtained from this oscillogram are given in Fig. 4. The current in the external solenoid by the instant of the beginning of generator operation was approximately 250 kA, with the powering time of 82 µs. Figure 5 gives the experimentally obtained and calculated curves of the load current. The maximal experimentally obtained value of load current is estimated at ~160 kA, with the calculated value of 170 kA. HIGH TEMPERATURE
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dI/dt,
kA
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TA/s
0.1
180 150
0
120 0.1
90 60
0.2
30 0.3
0 0
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t, µs
Fig. 5. Load current, solid curve – experiment, dashed curve – calculation.
Analysis of the experimental data and the calculation results lead one to assume that the electric strength at the generator output was disturbed at a voltage of 250–300 kV. Figure 6 gives an oscillogram of the load current derivative along with prediction curves obtained for two limiting voltage values at which an insulation breakdown might occur at the generator output. The calculation was performed for the instant of time before the inner tube reached the spiral turns (before insulation). The time was reckoned from the instant of the beginning of liner motion. The maximal value in the positive part of the load current derivative was ~3.6×10 negative part was 29×10
10
10
0.4 0
11
12
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Fig. 6. The load current derivative: solid curve, experiment; dashed curve – U = 250 kV and dotted curve – U = 300 kV, calculation.
ed value in the positive part of the load current deriv10
ative was ~4.5×10 A/s for both values of U = 250 and 300 kV, and those in the negative part were 29.5×10
10
A/s and 31.5×10
10
A/s, respectively.
Figure 7 gives an oscillogram of voltage on the break point, along with calculated curves. The maximal value of voltage on the break point was 810 kV, and that on the break point active resistance was 930 kV. The maximal power on the break point is estimated at 90–100 GW. A curve of power on the break point is given in Fig. 8.
A/s, and that in the
CONCLUSIONS
A/s; the maximal calculat-
The experimental results on the shaping of a highvoltage pulse on the break point of electrically exploding conductors confirm the possibility of using spiral
U, kV 800
P, GW
600
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400
60 200
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t, µs
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t, µs
Fig. 7. Voltage on the break point (designations are the same as in Fig. 6). HIGH TEMPERATURE
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Fig. 8. Power curve on the break point
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AMFCGs to develop high-power high-voltage sources of energy for diverse physical research and engineering applications.
7.
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