conducting rails with armature that fits
between and closes the circuit between the two rails. A large current is sent
through the rails from one side creating a magnetic field that is perpendicular to
the current in the armature and forces the armature to move down the rails away
from whichever end the current is being applied. Since the force pushing apart
the rails is large as well, they must be built so as to withstand the force
developed during firing.

In order to enhance the magnetic field developed across the armature
several additions can be made to augment the classic configuration. A core with
high magnetic permeability may be used to increase the magnetic field density,
although this may not be helpful if the railgun operates well past the saturation
point of the material. A strong permanent magnet circuit may be used to provide
a magnetic field separate from that provided by the railgun, again that may not
help much if the railgun operates with a higher magnetic field than the permanent
magnets. Additional rails that do not make contact with the armature may be
placed near the main rails in order to increase the magnetic field density. Any of
these options could also be implemented with a completely separate
synchronized circuit.

The armature can also be the projectile, or the armature can act only as a
sabot or pusher for a projectile. The armature can be solid or plasma. The
plasma being supplied by a thin conductive metal that is heated rapidly by the
current through it. If a plasma armature is used the rails, and the structure
holding the rails together must provide a completely sealed barrel that closely fits
the projectile so that the plasma does not blow past the projectile.

The power supply to fire railgun must provide a very large current of short
duration. It must also operate at a high enough voltage to drive the required
current and to squelch any back emf from the armature. DC supplies such as
lead-acid supplies can deliver several thousand Amperes for short duration, but
are not practical for a large weapon since a large amount are needed to provide
the requisite voltage and current. capacitors and compulsators can store very
large amounts of energy and are capable of delivering hundreds of kilo-amperes.
Capacitors store energy via an electric field; compulsators store energy
mechanically in a flywheel. Compulsator stands for Compensated Pulsed
Alternator; a compulsator uses a very low inductance generator to allow for rapid
current rise and a high energy density flywheel to store energy. A compulsator
can store enough energy to fire a railgun several consecutive times where a
capacitor bank usually uses all of its energy on each shot and needs to be
recharged after each shot. Compulsators generally can store more energy per
unit volume than capacitors.
For any railgun the currents required will place large amounts of
mechanical stress on the current carrying parts. The rails current carrying bars
and connectors must be stiff and fastened into place. If the railgun uses a
plasma armature the armature/projectile will need to fit tightly into the barrel and
the barrel will need to be sealed to keep the plasma from escaping. For a solid
armature the surface area in contact with the rails will need to be maximized and
good contact maintained, this is necessary to reduce arcing, spot welding and to
allow for high current flow. The armature rail interface should be designed to
minimize gouging, and if any gouging is to occur it is clearly more desirable to
have the damage on the armature instead of the rails. So the rails should be as
hard as feasible and the armature as soft as possible. Many materials and
construction techniques have been tried to make long lasting rails. This is still an
area of significant research.

When the railgun is fired the armature/projectile should be injected at a
high velocity to overcome that static friction of the barrel and to prevent spot
welding. A fast injection also will spread out the heat generated across a greater
area again helping to prolong rail life. Unless the magnetic field is supplied
externally the armature should have a sufficient length of current carrying rail
behind before it contacts the rails it in order to allow strong a magnetic field to be
created. Even so it may be desirable to augment the magnetic field where the
armature first makes contact since the current will not immediately begin to
accelerate the armature.

Railgun Force and Magnetic Field Analysis

The general Lorentz force law describes the force on a moving charge in
magnetic and electric fields.

(1)
B
v
q
E
q
F
r
r
r
× + =

[1 pg 28]

where

is the force acting on
q
[N/C]
E</i>r
is the electric field intensity [N/m]
B</i>r
is the magnetic field intensity [T]
v
is the charge velocity [m/s]
q
is the charge [C]

In a rail gun the moving charge consist of a large electric current in a conducting
armature, the force to accelerate the projectile is given by the second term in (1).
Since current is defined as a movement of charge per second, a current over a
length l is equivalent to charge movement at a given velocity.
(2)
[ ]
m
l
s
C
i
s
m
v
C
q

=



]
[


(3)
B
l
i
B
v
q
F
r
r
r
× =
× =


Expressing the cross product in (3) using the angle theta between the current
and magnetic field gives (4)

(4)
( ) sin =
B
l
i
F
r
r


If the magnetic field is assumed to be perpendicular to the current then (4)
reduces to

(5)
B
l
i
F
=
r


Where
B
is the magnitude of the magnetic flux density.

A rail gun can provide the current and magnetic flux using a conceptually simple
circuit.



FIGURE 1. The basic railgun concept, two parallel rails and a sliding armature. The current in
the rails and armature creates the magnetic field indicated and a the desired force on the
Projectile/Armature.

For a given current the force along the armature varies along its length, and the
magnetic field varies according to the length of the rails.

The directed magnetic field of an infinitely long wire is
(6) 2
0
i
a
B
r
r =


where

0 is the permeability constant is the radial distance from the wire

The contribution from a single very long, thin rail would be ½ that of an infinitely
long wire

(7) 4
0
i
a
B
r
r =


The rails that create the magnetic field may not be well approximated by
assuming they are infinitely long. An expression for the magnetic field at a radial
distance from the end of a wire can be found using the Biot-Savart law.

(8)
(
)
'
4
,
,
2
0
3
2
1
dv
R
a
J
u
u
u
B
V
R × =
r
r
r

[1, pg 275]

where
J</i>r
is the current density [A/m
2
]
R
is the distance to the point of the contributed magnetic field

if it is assumed that the cross section of the wire is negligible compared to the
area to be integrated over then
dl
i
dv
J =
'
and (8) reduces to

(9)
(
) ×
=
l
R
R
a
dl
i
u
u
u
B
2
0
3
2
1
4
,
,
r
r


For the purposes of the following integration it assumed that the rails are very
thin.The rails have length L, the magnetic field at a distance d from the end of
one rail is then

(10)
( )
( )
=
L
R
dl
i
P
B
0
2
0
4
sin r


the distance from a segment of wire to the point P can be expressed as

(11)
2
2
d
l
R
+
=

the quantity
( ) sin
can be expressed as

(12)
( )
2
2
sin
d
l
d
R
d
+
=
=

the integral then becomes

(13)
( )
2
2
0
0
2
2
2
2
2
0
4
4
d
L
L
d
i
d
l
d
l
d
dl
i
P
B
L
+ =
+
+ = r


When L becomes much greater than d the equation reduces to (7).

Since the rails of a practical railgun cannot be infinitely long and a signifacant
magnetic field is desireable it is useful to know how long the rails must be to
reasonable approximate infinite rails. Expressing L as a multiple of d,
d
x
L =
,
setting d equal to 1 and dividing (13) by (7) gives a function describing the
fraction of magnetic field compared to a wire of infinite length from the beginning
of the rail.
The fraction of magnetic field produced by finite length wires compared to infinite
legnth wires as a function of the ratio x of wire length to the point where the
magnetic field is measured.

(14)
2
1 x
x
+


The relation shows that when the wire three times longer than the distance to the
point at which a given magnetic results, the field density is 95% of that of an ideal
infinite wire. Equation (14) assumes a very thin wire, this is not the case in a rail
gun, the dimensions of the projectile are of the same order of the rails carrying
the current. However, this relation does give some idea of the length of rail
necessary to set up a satisfactory magnetic field.
1
2
3
4
5
Ratio
0.2
0.4
0.6
0.8
1
FractionOfIdeal

FIGURE 2. The fraction of an ideal magnetic field created by finite rails in terms of the wire
length in multiples of the distance to the point the magnetic field is calculated.


Assuming that the rails are constructed such that the magnetic field created
closely approximates that of thin wires that extend to infinity, equations (5) and
(7) can be combined into an expression for the force on the projectile in terms of
current and length.

The magnetic field contribution from the first wire