Use of the Thrust Constant
in the Design of the Coil Gun

(Reproduced from "Zero to Eighty" pp.298-303 by EF Northrup)

From the narrative, the length of the gun is taken as 1000
meters, the inside diameter as 40.0 centimeters, and the
outside diameter of the projectile as 37.0 centimeters, the
length of the projectile exclusive of nose and tailpiece
being 500 centimeters. Selecting a thrust constant of 13 x 10^{-6}
grams, we find from experimental data that the following coil and
cylinder proportions must exist:

The phase difference between unit coils must be sixty
electrical degrees.

The physical coupling, or the relative cross-sectional area of
the projectile and the inside of the coil, must be about 86%.

The length of the magnetic travelling wave must be about 2.6
times the inside diameter of the coil. Six unit coils, sixty electrical
degrees apart, constitute three hundred and sixty degrees or one
complete cycle, or in this case one wave. The wave-length is thus
six times the distance between centers of adjacent phase coils.

For maximum thrust, the effective a.c. resistance and the reactance
of the induced current paths in the projectile must be equal.

From the foregoing proportions, together with other data, we
arrive at the following specifications for the one-kilometer-long
electric gun:

Length of gun = 1000 meters 40.0 cms.

Inside diameter of gun = 40.0 cms.

Outside diameter of copper conductor tubing = 1.0 cms.

Wall thickness of tubing = 0.34 cms.

Number of phase coils = 5880

Length of phase coils = 14.4 cms

Space between phase coils = 2.6 cms

Wave-length = 102 cms

Turns per phase coil =12

Mean length of copper tubing in phase coil = 15.5 cms

Frequency = 2000 cycles

Wave speed = 2040 m. per sec.

A.c. resistance of one phase coil at approximately 15 x 10^{-4} ohm per meter = 0.023 ohm

Approximate inductance of phase coil = 0.000007 henry

Approximate empty reactance at 2000 cycles = 0.88 ohm

Approximate weight of projectile = 100 kilograms

Average turns per centimeter, taken over wave-length of 102 centimeters = 0.705

From the equations given under the General Laws of Motion it
is shown that the acceleration of a projectile in the gun is expressed
by the formula:

where a is the average acceleration due to magnetic thrust, v the
muzzle velocity in meters per second, and l the length of the gun in
meters. For the projectile to attain a muzzle velocity of 1000 meters per second in a gun 1000
meters long we have:

From the same general equations it can be shown that the force
in kilograms necessary to impart an acceleration of 500 meters persecond per second to a projectile
is given as:

where T is the thrust in kilograms, w the
weight of the projectile in the same units, a the
acceleration of the projectile in the gun, in
meters per second per second, and g the acceleration of gravity. Substituting
numerical values in (2):

When the gun is in a vertical position, an additional thrust must
be exerted to balance the weight of the projectile, besides the thrust
necessary to give vertical speed. Formula (2) is then modified as follows:

Having determined the total thrust necessary to give the
desired acceleration and velocity, we approach the problem of
obtaining this thrust with polyphase magnetic waves.

As shown in the previous section, the magnetizing force inampere turns per centimeter is
given by the formula

where T is the total thrust on the projectile in grams, K the thrust
constant, A the area of the projectile on which the thrust is
effective, in square centimeters, and the product (in) is equal to the
ampere turns per centimeter.

From (3) we have found the total thrust to be 5200 kilograms
or 5.2 x 10^{6} grams; the maximum value of K has been found by
experiment to be 13 x 10^{-6} gram; the active area of the projectile is
taken as the area of a cylinder 500 centimeters long and 37
centimeters in diameter, which calculates out as approximately 58,200 square centimeters. Thus,
by (4),

As there is only an average value of 0.705 turn per centimeter
length of the gun, the current per phase necessary to produce 2620
ampere turns (found by dividing the average number of turns per
centimeter of length into the required ampere turns) is 3720 amperes.

Figure 1. Coil dimensions used in the Utah gun

As mentioned elsewhere, the maximum thrust constant cannot
be realized except under certain set conditions, the main condition
being that the frequency of the currents induced in the cylindrical
shell of the projectile shall be such as to make the electrical
resistance and the reactance of the induced current paths
substantially equal.

Under the influence of the travelling magnetic wave produced
in the electric gun, the projectile is started from rest and gains
speed. In general, the velocity of the magnetic wave remains
constant, whereas the speed of the projectile varies from zero to a
theoretical value known as the synchronous speed, which is the
same as the speed of the wave.

In practice, the projectile speed is less than synchronous speed
by an amount known as the “slip.” The slip is the difference between
the speed of the projectile and the synchronous speed, and is given
in terms of percentage of synchronous speed. Thus, one hundred
per cent slip means that the projectile is at a standstill; fifty per cent
slip indicates half of synchronous speed; while at zero slip the
projectile is the same velocity as the wave.

It is possible to show that the frequency of the currents induced
in the projectile is a direct function of the slip. This means that the
relation of resistance to reactance constantly changes as the
projectile gains velocity — which in turn affects the value of the
thrust constant K. By suitable design of the projectile the maximum
thrust constant can be made to come at any convenient value of
slip.

Certain experimental and theoretical considerations show that
if the maximum or muzzle velocity of the projectile is equal to a fifty
per cent slip, and the resistance and reactance factors are balanced
to give the maximum thrust constant with this slip, then the thrust
constant will not decrease more than twenty per cent under
stationary conditions. Hence, in starting the projectile, the thrust
constant will not be less than eighty per cent of the maximum value
which will occur at maximum velocity.

By (4), the ampere turns per centimeter necessary to maintain
a constant total thrust varies as the inverse square root of the value
at K. Hence, with K at eighty per cent of its maximum value the
impere turns will be increased by a factor of one over the square root
of eight-tenths, or 1.12. The current per phase at the start should then be about twelve per cent
higher than the value calculated with
the rise of the maximum value of K. The starting current per phase
is thus about 4170 amperes.