METHOD

OF I N C R E A S I N G

AN ELECTROMAGNET

THE

EFFICIENCY

INDUCTION

OF

CONDUCTOR

ACCELERATOR V.

P.

Gal'etov,

a n d E . N. I v a n o v

UDC 538.323:531.551

One of the methods of producing supersonic velocities in solids to investigate high-speed interactions is to acceler ate conductors in a pulsed magnetic field [1, 2]. In the induction acceleration of ring conductors of relatively low mass using capacitor bat t eri es the efficiency of the acceleration process is reduced due to the escape of the accelerated body from the inductor magnetic field [3]. The efficiency of the conversion of electromagnetic energy into kinetic energy of the accelerated conductor can be increased by forcingthe magnetic p r e s s u r e either by keeping the a c c e l e r at ed body in the initial position until the energy density of the field of the electromagnet supplied to the inductor system from the external source reaches a sufficiently high level corresponding to the required value [4], or by increasing the rat e at which the energy of the e l e c tromagnet is introduced into the inductor system. In this paper we consider a method of increasing the e fficiency of a high-speed induction a c c e l e r a t o r of ring conductors by increasing the rate at which the e l e c t r o magnet energy is introduced into the inductor system. In the practical application of a high-speed solid a c c e l e r a t o r the need a r i s e s to cover the maximum possible range of velocities of projected bodies of different mass. When the p a r a m e t e r s of the load and the energy source in the induction a c c e l e r a t o r are c o r r e c t l y matched one can obtain high conversion efficiency of the electromagnet energy into kinetic energy of the accel erat ed body [3]. However, when projecting r e l a tively small m a s s e s good matching and the achievement of high velocities a r e limited by the rate at which the energy can be t r a n s f e r r e d from the capacitive store to the inductor system. It is possible to increase the rate at which energy is introduced into the inductive load within certain limits by reducing the inductance of the discharge circuit and increasing the charging voltage of the capacitor battery. A large r at e of increase of the magnetic field in the inductor system can also be obtained by switching the energy store discharge c u r r e n t into parallel circuits (accentuating the current leading edge}. Figure 1 shows the equivalent circuit of an induction a c c e l e r a t o r of ring conductors with a curre n t leading edge accentnator in the inductor. While the capacitor battery is charging the switches S1 and $2 are open, and the switch S 3 is closed. After S1 is connected the battery discharges through the inductance L0, the resistance of the discharge c i r cult R0, and the r es i s t ance of the switch Rs. If at a certain instant of time the switch S 3 is opened and S2 is closed, then in the branch L1 t her e will be a sharp increase in the current, the rate of r i s e of which is higher than in the usual RLC circuit. We can use ordinary switching components used in high-current pulse techniques ( d i s c h a r g e r s and ignitrons) as the components $1 and S2. The switch $3, which must discharge higher current s at higher voltages, is a mo re complex component. A suitable type of switch exists, based on the principle of exploding metal wires or foil. Despite the large number of papers devoted to an investigation of the processes in a conducter when it is electrically exploded, so far t h e r e appears to be no generally accepted theory which would Ro

C

I

Lo

&

,9~

$3

R.

i Fig.

R~

M/r)

L'~LzR2~

L J 1

Istra. Tr ans l at ed from Zhurnal Pritdadnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 105-108, JulyAugust, 1979. Original article submitted July 25, 1978. 474

0021-8944/79/2004-0474507.50 9 1980 Plenum Publishing Corporation

T3

1o

o, 10-

9

6

o,4 5-

o,2

U

o

o;5

f,o

~

~o-~

w -~

Fig. 2

~o-~

~'

Fig. 3

enable one to explain and give a quantitative d e s c r i p t i o n of t h e s e p r o c e s s e s . In [5], using a n u m b e r of s i m plifying a s s u m p t i o n s , a m a t h e m a t i c a l s i m u l a t i o n of the f i r s t c u r r e n t pulse which o c c u r s in the e l e c t r i c a l e x plosion of conductors is given, and it is shown that a c o m p u t e r calculation of the m a i n p a r a m e t e r s c h a r a c t e r izing the e l e c t r i c a l explosion of conductors a s a function of the quantity II = GU60CsI/L2S 7 gives good a g r e e m e n t with e x p e r i m e n t a l r e s u l t s . H e r e U0 is the initial voltage a c r o s s the c a p a c i t o r b a t t e r y , L0 is the inductance of the d i s c h a r g e c i r c u i t , l and S a r e the lengths and a r e a of t r a n s v e r s e c r o s s section of the exploding conductor, GCu = 6 . 7 2 . 1 0 .60 kg" m15/A 8. s e c 6, and GA1 = 1 . 2 5 . 1 0 -5~ kg" mlS/A 8- sec 6. A s i m i l a r s e t of p a r a m e t e r s for the d i s c h a r g e c i r c u i t and the g e o m e t r i c a l d i m e n s i o n s of the exploding w i r e was used in [6] to c a l c u l a t e c e r t a i n p a r a m e t e r s of the e l e c t r i c a l explosion of conductors. T h e quantity H can be r e p r e s e n t e d in t e r m s of the s i m i l a r i t y c r i t e r i a ~1 = poll Z S and ~2 = W U ZS2 introduced in [6] as I] = G~l~r2, w h e r e W 0 is the e n e r g y s t o r e d in the c a p a c i t o r s , Z = ~-L0/C is the wave impedance of the d i s c h a r g e circuit, and P0 is the initial r e s i s t i v i t y of the exploding wire. When investigating the c i r c u i t we used the m a t h e m a t i c a l m o d e l of an e l e c t r i c a l l y exploded conductor p r o p o s e d in [5], and in addition we m a d e the following a s s u m p t i o n s : 1) The r e s i s t a n c e s R0, R!, and Re, and the inductances L0, L1, and L2 of the c i r c u i t r e m a i n constant during the d i s c h a r g e , 2) the switches S 1 and S2 a r e ideal, i.e., the switching o c c u r s instantaneously and without l o s s , 3) the switch S2 c l o s e s a t the instant c o r r e s p o n d i n g to m a x i m u m voltage on the switch $3, and 4) the r e s i s t i n g f o r c e s acting on the a c c e l e r a t e d ring are small. T h e s e t of equations d e s c r i b i n g the p r o c e s s e s in the a c c e l e r a t o r h a v e the following d i m e n s i o n l e s s f o r m : djo/d~ + ro]o -~ r s (~)(]0 -- ]1) + ~0 = 0;

(1)

d~o/d~ = J0;

(2)

d]l , Lo

.

Lo

.

d

~- L-T ~ j t ~ - ~ rs ( T ) ( h - - j 0 ) T ~ [ ~ ( e ) ] ~ ] = 0 ; dj~

,

Lo

(3)

d

d2s

t

d~t(e)

. .

d~

-5

~

]d2.

(5)

T h e d i m e n s i o n l e s s and d i m e n s i o n a l quantities a r e connected b y the following r e l a t i o n s : ] ~ i/i:nd; U/Und = cp; r =R/Rnd; ~ =t/tad s =X/Xnd: ~ =M/Mnd; ~ = m/mnd

We took the following a s the b a s i c quantities: Rnd=]/Lo/C, Lad = n0, Mnd= L1, i = u o V c ~ o ,

Und= Uo,

tad = ~/~oC,

mnd: C~U~Lo/D ~, Xnd= D,

w h e r e m is the m a s s of the a c c e l e r a t e d ring, x is the coordinate of the d i s p l a c e m e n t , and D is the m e a n d i a m e t e r of the inductor and the body. In this c a s e , we t~ke a s the b a s i c inductance (unlike [3]), the inductance of the s t o r e L0, and not L~, since in this c a s e the p r o b l e m of d e t e r m i n i n g the m a s s of the conductor for which a c c e n t u a t i o n of the c u r r e n t leading edge leads to an i n c r e a s e in its final v e l o c i t y is m a d e e a s i e r , T h e s y s t e m of d i f f e r e n t i a l equations (1)-(5) w a s i n t e g r a t e d on the ES-1020 6omputer using the R u n g e Kutta method with a v a r i a b l e step. T h e e r r o r in calculating the v a r i a b l e s was 0.1~:' In the integration we neglected the heating of the inductor and of the a c c e l e r a t e d ring, and t h e i r mutual inductance was (~lculated

475

6 ]

/0

2 r5 "s

163

~52

d.

Fig. 4

!

10

t0 ~

103 I7

Fig.

as in [7]. T o m o n i t o r the c o r r e c t n e s s of the p r o g r a m compiled and the reliability of the r e s u l t s obtained we calculated the e n e r g y balance. The e r r o r in calculating the total e n e r g y in the s y s t e m v a r i e d f r o m 0.1% to F i g u r e 2 shows the r e s u l t s of a calculation of the a c c e l e r a t i o n t r a n s i e n t of a ring conductor of c i r c u l a r c r o s s section when the d i a m e t e r of the inductor and of the a c c e l e r a t e d ring a r e the same in an a c c e l e r a t o r with accentuation and without accentuation (the continuous and dashed curves, respectively) of the c u r r e n t leading edge in an inductor with r 1 = r 2 = 0. H e r e and l a t e r the relative initial gap between the inductor and the body e 0 = 0.01. F i g u r e 3 shows the r e s u l t s of a calculation of the final relative velocity of the conductor as a function of the p a r a m e t e r a for II = 10, k = L 0 / L I = 3, and rl = r~ = 0. It can be seen f r o m a c o m p a r i s o n of the c u r v e s that accentuation of the c u r r e n t leading edge in an inductor when projecting bodies the r e l a t i v e m a s s of which is less than a c e r t a i n c r i t i c a l value a , leads to an i n c r e a s e in the velocitSes obtained. The e f f i c i e n c y of the accentuation i n c r e a s e s as a is reduced. T h u s , for a = 10 -4 the i n c r e a s e in the velocity due to c u r r e n t a c cerSuation is 72% c o m p a r e d with 36% for a = 10- s and 6% for a = 10 -2. The r e l a t i o n between the c r i t i c a l relative m a s s of the a c c e l e r a t e d conductor and the p a r a m e t e r k for II = 10 is shown in Fig. 4. In zone I a c centuation of the c u r r e n t leading edge in the inductor leads to an i n c r e a s e in the final velocity of the conduct o r . H e r e we show the effect of the relative r e s i s t a n c e s of the inductor r 1 and the a c c e l e r a t e d ring r 2 on the change in the zone I [1) r 1 = r 2 = 0; 2) r 1 = r 2 = 0~ 3) r i = r 2 = 0.1; 4) r 1 = r 2 = 0.2]. The v a r i a t i o n of the final velocity of the p r o j e c t e d conductor a s a function of the quantity [I for different r e l a t i v e m a s s e s [1-3) ff = 10-4; 4 - 6 ) a = 10 -S] and different r a t i o s of the internal inductance of the s o u r c e to the inductance of the inductor [1, 4) k = 3; 2, 5) k = 5; 3, 6) X = 7] is shown in Fig. 5. To obtain the m a x i m u m value of the final velocity of the conductor it is n e c e s s a r y to choose II in the r a n g e f r o m 1 to 10. This is p a r t i c u l a r l y important for the c o r r e c t choice of the t r a n s v e r s e c r o s s section of the exploding w i r e of the switch $3. LITERATURE 1.

2. 3. 4.

5.

6. 7.

476

CITED

V. F. A g a r k o v et al. " T h e a c c e l e r a t i o n of c o n d u c t o r s to h y p e r s o n i c velocities in a pulsed magnetic field," Zh. PriM. Mekh. Tekh. Fiz., No. 3 (1974). V. N. Bondaletov and E. N. Ivanov, " C o n t a c t l e s s induction a c c e l e r a t i o n of conductors up to h y p e r s o n i c velocities," Zh. Prikl. Mekh. Tekh. Ftz., No. 5 (1975}. A. N. Andreev and V. N. Bondaletov, "Induction a c c e l e r a t i o n of conductors and a h i g h - s p e e d actuator," l~.lektrichestvo, No. 10 (1973). V. T ; C h e m e r i s and S. A. Gavrilko, "Diffusion of an e l e c t r o m a g n e t field into a moving conducting piston and the s y s t e m of s e c o n d a r y c i r c u i t s of a pulsed e l e c t r o m e c h a n i c a l e n e r g y c o n v e r t e r , " P r e p r i n t 155 II~D Akad. Nauk UkrSSR, Kiev (1978). E. N. Ivanov, " M a t h e m a t i c a l simulation of the f i r s t pulse in the e l e c t r i c a l explosion of conductors," in: E l e c t r i c a l P r o c e s s e s in a Pulsed D i s c h a r g e [in Russian], Chuvashsk. Univ., C h e b o k s a r y (1976}. E. I. A z a r k e v i c h , "The use of the t h e o r y of s i m i l a r i t y to calculate c e r t a i n c h a r a c t e r i s t i c s of the e l e c t r i c a l explosion of conductors," Zh. Tekh. Fiz., 43, No. 1 (1973). P. L. Kalantarov and L, A. Tseitlin, Inductance Calculation [in Russian], i~nergiya, Leningrad (1970.