Optimised Coil Geometry
1,2,3
Let's
look at how the thickness of the coil affects the field strength at its centre.
The basic coil parameters are illustrated below.
Fig
1
Firstly
let's define some basic geometrical relationships
It
can be shown that the field, Ho, at the centre of the solenoid is
where
N is the number of turns, I is the coil current, and F(a,b)
is the 'field factor' defined as
We
can use these equations in conjunction with the coil resistance equation to plot
Ho as a function of Ro for any combination of L, Ri and wire diameter. Fig 2 shows
the field strength for three different wire diameters using a coil with L=26mm
and Ri=7mm.
Fig
2. Field strength vs outer radius for various wire diameters
The
field strength plots are based on a source of 1V. The particular voltage level
doesn't matter, we're only interested in the relative effect of wire diameter
and outer radius. It's obvious that for any given wire diameter there is an optimal
outer radius which maximises the field strength. Why do we want to maximise the
field strength? Well the central field strength is probably the most important
factor governing the muzzle energy of the projectile.
Need
some convincing? Fig 3 shows the results of a simple experiment which involved
measuring the muzzle speed from a coil with varying values of outer radius. Coil
B (1mm wire) was used for this. As each speed measurement was completed a layer
of was removed until the coil was 2 layers thick. 'Suckback' is reduced by using
the multidiode commutation arrangement. The projectile energy is plotted along
with the calculated central field strength.
Fig
3. Correlation between central field and projectile energy.
There
is a good degree of correlation between the energy and the central field. Perhaps
the most notable aspect of these curves is that the peak energy and peak field
occur at the same coil outer radius. The correlation between these peaks is very
striking and reflects the likelyhood that central field strength will play a key
role in determining the best coil geometry.
It's
perhaps worth commenting on the fact that there isn't a 1:1 correspondence on
all points of the curve. The first thing that comes to mind is the source resistance
value. This is based on the battery source internal resistance, the mosfet module
resistance, the current sensor resistance and the wiring resistance. Apart from
the current sensor none of these resistances are known to a high degree of accuracy
so the absolute magnitudes of Ho could be off by some amount. Another reason could
simply be that the the field strengths involved in this experiment are insufficient
to induce saturation magnetisation in the projectile so it is still subject to
the nonlinear portion of its BH curve. There's also suckback acting as an uncontrolled
factor which will no doubt vary with the changes in coil radius. I suspect that
running a similar experiment with a higher currents, and hence field strengths,
will produce a tighter correlation.
This
experiment is by no means definitive, additional experiments need to be run with
a selection of 'base geometries'  length and inner radius combinations  in order
to determine if the outer radius can be confidently specified using the peak Ho.
Also, the Ho analysis is based on the pulse width being at least 6 time constants
 steady state approximation. The Coil B configurations satisfied this limitation
but obviously not all coils will. Some type of dynamic factor is required to refine
the Ho analysis.
One
last point, if you examine the 1.5mm wire curve in fig 2 you'll notice that a
coil using this wire diameter has a peak field when Ro is 19 mm. This peak is
significantly larger than the 1.0mm peak so does a coil based on this geometry
perform better? We can answer this by looking at Coil D since it more or less
conforms to these dimensions. From previous experiments using identical multidiode
commutation we can see that Coil D produces a muzzle energy of about 970mJ which
is obviously larger than the 790mJ maximum generated by the 1mm wire. If we calculate
the energy ratio and field ratio we get 1.23 and 1.20 respectively.
Sources:
Magnetism and Magnetic
Materials, David Jiles.
