Modeling and simulation of an output electric current from a flux compression generator coil.

Dobref, Vasile ; Sotir, Alexandru ; Tarabuta, Octavian 等

1. INTRODUCTION

Following the trail of new applications development regarding the modern warfare information strategy and tactics, the technologies based on the high energy electromagnetic pulse evolved continuously (Abrams, 2003). The newly developed research have been leading to the so called electromagnetic bombs (projectiles), E-bombs, capable of taking out enemy's C4I (command, control, communication, computers & intelligence) equipment and systems with no collateral casualties.

The first damaging effects of the electromagnetic pulse (EMP) have been noticed during the high altitude nuclear weapons tests, when intense, extremely brief EMPs (hundreds of nanoseconds) have been generated and transmitted over some distance according to Maxwell's equations of the electromagnetic field.

2. MODELING AND SIMULATION OF THE OUTPUT CURRENT DUE TO THE EXPLOSION

Physically, an FCG is built from a copper cylinder (the armature) which holds inside the explosive charge. A copper thick wire helical coil, standing for the stator is placed around with a sufficient air gap needed for the explosive's expansion during the explosion. The initial current is produced through the discharge of an external capacitor battery.

In order not to prematurely destroy the generator, a thick protective non-magnetic material wrapping (fiber glass or Kevlar) is placed on the coil's external surface - see Figure 1.

The simulation of the final current in the coil, estimated to be produced by the explosive's detonation, has been done by the means of a MATLAB program called PSCF-1.

The modeling of the coil current during the shortcircuit took into account a linear evolution of the inductivity and resistance of the RL series circuit, that is they are functions of time as variable (Fowler et al., 1993). The solution of the differential equation depends in this case by the integrals emerging in the computing program.

[FIGURE 1 OMITTED]

Thus, we considered:

R(t)=[R.sub.0](1-kt), [R.sub.0]=R and [R.sub.T](t)=R(t)+ [R.sub.b] (1)

where:

R=0,00155 [OMEGA] is the overall resistance of the coil, with the loop and the armature; [R.sub.b]=3,6298 x [10.sup.-7] [omega] is the resistance of the current (load) loop considered to be constant.

k = 1/[t.sub.i] = 1/52.14 x [10.sup.-6] = 1,9178 x [10.sup.4] ([s.sup.-1])

and [t.sub.i] = l/[v.sub.C] = 0,365/7000 = 52,14 x [10.sup.-6] (s) = 52,14([mu]s)

is the explosion propagation duration. On the other hand,

L = [L.sub.0](1 - kt), [L.sub.0] = L and [L.sub.T](t) = L(5) + [L.sub.b] (2)

There have been studied two cases (Dobref et al., 2008):

[1.sup.0] The real (measured) electrical parameters of the coil and the current loop and the coil has an interior cylinder. In this case, the values are: [L.sub.0]=42[mu]H, [R.sub.0]=0,097[OMEGA], [L.sub.b]=0,1498[mu]H, [R.sub.b]=0,006466 [OMEGA], [L.sub.p]=2 nH.

[2.sup.0] The real (measured) electrical parameters of the coil and the current loop and the coil doesn't have an interior cylinder. For this case, the values are: [L.sub.0]=22[mu]H, [R.sub.0]=0,097[OMEGA], [L.sub.b]=0,1498;[mu]H, [R.sub.b]=0,006466[OMEGA], [L.sub.p]=2 nH.

Considering that

[R.sub.c]=R+[R.sub.b]; [L.sub.c]=L+[L.sub.b] (3)

The equations (1) and (2) become:

[R.sub.T](t)=[R.sub.C]-kR(t) and [L.sub.T](t)=L-kL(t) (4)

As we analyze the transitory process starting with the moment of coil's short-circuiting, the differential homogenous equation of the RL series circuit has the well-known form:

d/dt [L.sub.T] (t) i(t)+[R.sub.T] (t) i(t) = 0 (5)

In the same time, we know that:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

and

d/dt [L(t) + [L.sub.b]=-k[L.sub.0] (7)

Equation (5), amended by the equations (6) and (7), is now:

([L.sub.c] - k[L.sub.0])/di/dt - k [L.sub.0] i + ([R.sub.c] - kRt)x i = 0 (8)

or

([L.sub.c] - k[L.sub.0]t) di/dt + [([R.sub.c] - k[L.sub.0] - kRt] i = 0 (9)

Equation (9) can be expressed as:

di/dt + ([R.sub.c] - k [L.sub.0]) - kRt/[L.sub.c] - k[L.sub.0]t i = 0 (10)

or:

di/dt + [[R.sub.c] - k [L.sub.0])/[L.sub.c] - k [L.sub.0]t - kRt/[L.sub.c] - k[L.sub.0]t] x i = 0 (11)

If we equalize the expression in the brackets with P(t), we obtain:

P(t) = [R.sub.c] - k [L.sub.0]/[L.sub.c] - k [L.sub.0]t - kRt/[L.sub.c] - k [L.sub.0]t (12)

Therefore, equation (11) will be written as:

di/dt + P(t) x i = 0 (13)

After solving the equation (13), the formula of the transitory current through the coil, during the explosion, has the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

The computing program has numerically solved the equations above and has two reverse applications.

a) If we want to find out the initial currents needed to obtain imposed output currents (for example, 15 kA;10 kA; 5 kA), according to a known elapsed time of the explosion, from equation (14) we will have:

[I.sub.o] = t ([t.sub.i]/exp[el([t.sub.1])] (15)

Where, again, we cose the values i(ti)=15kA; 10kA; 5kA. Accordingly, we have found the values:

[I.sub.01] = 15000/exp[e1([t.sub.i])] = 226,163916A)

[I.sub.01] = 10000/exp[e1([t.sub.i])] = 150,77594455A)

[I.sub.01] = 5000/exp[e1([t.sub.i])] = 75,387972A)

b) On the basis of equation (14) we can find out the output current through the FCG coil with parametric resistance and inductivity, for the two cases mentioned above: the coil with or without an interior cylinder.

[1.sup.0] The coil with an interior cylinder.

We adopted in the computations an initial current I1 = 551A. The diagram of the function i(t) for the respective time interval is presented in Figure 2. The computations have resulted into an output current [I.sub.out] = 128,158.84 A.

[FIGURE 2 OMITTED]

[2.sup.0] The coil without an interior cylinder.

The starting current was [I.sub.1]=571 A. The evolution of the output current during the explosion is presented in Figure 3. The final value of the current was [I.sub.out] = 61,385.62 A.

[FIGURE 3 OMITTED]

3. CONCLUSION

By comparing the two output currents subsequent to the short-circuiting process, we notice that the presence of an internal metallic cylinder (steel) doubles the size of it.

The results of the modeling and simulation have been used in further experimentation of a real FCG prototype and proved to be very close to the experimental data. On the other hand, the results of the present research could be used reversely, in order to take proper measures of protecting electrical devices from the electromagnetic pulse damaging effects.

4. REFERENCES

Abrams, M. (2003). The dawn of the e-bomb, IEEE-Spectrum

Bykov, A.I.; Dolotenko, M.I. & Kolokol'chikov, N.P. (2001). Achievements on Ultra-High Magnetic Fields Generation, Physica B, 294-295, p.574-578

Dobref, V. et al., (2008). Research report: Modeling and design of a conventional payload for an electromagnetic pulse generation destined to block C4I systems, "Mircea cel Batran" Naval Academy, Constanta, Romania

Fowler, C.M.; Caird, R.S. & Garn, B. (1993). An introduction to explosive magnetic flux compression generators. Abstract, Los Alamos

Johns, D. (2004). Analysis of EMI/E3 Problems in Defense Applications, Flomerics

Knoepfel, H. (1970), Pulsed High Magnetic Fields, Nord Holland