Open Loop Pulse Triggering
Results
Coil A
First let's look at the current behaviour during a 'dry' firing,
i.e. without a projectile. Fig 1 shows a typical exponential growth with the current
reaching a plateau. On turn off the current decays exponentially with a similar
time constant. Since the current sensor has a sensitivity of 10mVA^{1}
we can say that the current reaches a maximum of 25A (the horizontal voltage markers
give a dV of 250mV). A simple reality check says this is reasonable  the open
source voltage is about 33V and the coil resistance is 1.35,
with a total circuit resistance of around 1.45(estimated)
this gives a theoretical maximum current of 22.8A. Why the difference? well there
are many potential sources of error; the oscilloscope isn't the most accurate
instrument for measuring voltage levels (especially since I did the calibration);
the current sensor may read a bit high; the estimated circuit resistance may be
too high; etc. The actual value of the error is fairly small in percentage terms
 the difference is 2.2A which is 8.8% of 25A, it's not that terrible. We can
translate this current value to a current density. The data for Coil
A shows a wire diameter of 0.8mm giving a wire current density of 49.7Amm^{2}.
If we assume a filling factor of /4
then the coil current density becomes 39Amm^{2}.
Fig 1. Coil current during a
dry firing
Fig 2 shows how the current curve is modified by the interaction
of the projectile and the coil. As the projectile accelerates into the coil it
increases the flux linking the coil, this has the effect of inducing a voltage
in the coil which opposes the current, the result is that the coil current is
driven down a bit as the projectile approaches the middle  we can term this part
of the interaction as the 'Conduction Phase'. Let's consider what happens as the
projectile begins to leave the coil, the mosfet current is now switched off. As
the projectile exits the coil it still carries a magnetisation due to the current
which is decaying in the commutating loop, this means that the flux linkage contributed
by the projectile is decreasing. If the projectile were stationary the result
would be an exponential decay with a slightly longer time constant but the projectiles
motion induces additional voltage and creates the little bump in the current decay
curve. Note that the voltage induced by the projectiles motion changes from negative
to positive as it passes the middle of the coil. Since the energy which drives
the extra current comes from the kinetic energy of the projectile, the interaction
with the commutating current slows the projectile down a bit. We can term this
interaction as the 'Commutating Phase'.
Fig 2. Projectile interaction
with coil
The current pulse length is a little harder to define when the
coil is firing a projectile but there is usually a fairly distinct change in current
gradient as the mosfet switches off. The time base markers in fig 2 give the pulse
width as 12.50ms. It's best to get as much of the curve on the screen as possible
in order to maximise the accuracy of the reading  to make the actual measurement
in fig 2 I set the time base to 2ms.
The graphs below show the data for each of the mild steel projectile
types solid, 4mm cored and 6mm cored. A 3rd order polynomial trend line is
fitted to the data points, the error bars are estimated on the uncertainties of
the various measuring instruments used.
The open loop pulse testing was used to get a 'feel' for the sensitivity
of the speed to the pulse length. The graphs in figs 3  5 show that pulse timing
doesn't need to be 'super critical' to achieve speeds very close to the optimal.
While this is true in this instance, we must remember that these are low power
tests with slow projectiles. In this case we can allow , say, +/0.25ms either
side of the optimal pulse length without much change in the resulting speed, however,
in a high power supersonic coilgun we may need to tighten this to +/5us or less
 a much more critical setup. Finally fig 6 illustrates a comparison between the
speeds of the three projectiles.
Fig 3. Solid projectile No.1
speed and energy data
Fig 4. 4mm cored projectile No.1
speed and energy data
Fig 5. 6mm cored projectile No.1
speed and energy data
Fig 6. Projectile Speed Comparison
Coil Efficiency
Let's look at the efficiency of the coil at the maximum projectile
speeds. I'll just concentrate on the coil copper (resistive) losses and ignore
the mosfet conduction/switching losses, the commutating diode conduction loss,
and power supply losses. These will be investigated in the optical triggering
experiments. Fig 7 shows the near optimum current curve for solid projectile No.1.
Fig 7. Optimal current pulse
for solid projectile No.1
In order to determine the energy lost as heat we need to integrate
the power dissipation function over the length of the pulse. The power dissipation
function is simply i(t)^{2}R and we can determine the equation by importing
the DSO data (text format file) into a spreadsheet package. Fig 8 and 9 show the
conduction and commutating phases as separate curves with their own trend lines
fitted.
Fig 8. 6th order polynomial trend
line fitted to conduction phase
Fig 9. 5th order polynomial trend
line fitted to commutating phase
If we integrate the trend line functions over the appropriate
time span we can determine the losses. The conduction and commutating phase trend
lines yield losses of 8413mJ and 501mJ respectively giving a total loss of 8914mJ.
Looking at fig 3 we can see that the peak projectile energy occurs at the 6th
data point and has a value of 298(+/15)mJ, this means that the efficiency of
the coil is 3.23(+/0.17)%. Obviously the efficiency of the whole system will
be a bit less than this. In a similar way we can determine the efficiency for
the 4mm and 6mm cored projectiles. The data is tabulated below.
Projectile 
Peak Power (W)

Conduction Loss (mJ)

Commutating Loss (mJ)

Total Loss (mJ)

Kinetic Energy (mJ)

Efficiency (%)

Solid No.1 
8.15E+2 +/ 1.5E+1

8.413E+3

5.01E+2

8.914E+3

2.98E+2 +/1.5E+1

3.23E+0 +/ 1.7E1

4mm Cored No.1 
8.15E+2 +/ 1.5E+1

7.922E+3

5.10E+2

8.432E+3

2.84E+2 +/1.5E+1

3.26E+0 +/ 1.8E1

6mm Cored No.1 
8.15E+2 +/ 1.5E+1

7.121E+3

5.05E+2

7.626E+3

2.56E+2 +/1.5E+1

3.25E+0 +/ 2.0E1

Table 1. Energy and efficiency
data
What can we conclude from all this? It's certainly obvious that
the cored projectiles attain a higher speed than the solid projectile, the reason
behind this may be due to several effects 

The dynamic flux penetration into a ferromagnetic material is
severely limited by the induced eddy currents so the solid projectile may only
attain a fraction of its steady state induced magnetisation in a dynamic field.

Since the cored projectiles have less bulk material the flux penetration
will give an greater average magnetisation.

The geometry of the cored projectile may also increase the magnetisation
under given external (coil) field conditions by reducing the internal demagnetising
field of the projectile.
This is all speculation since the system operates under coupled
nonlinear interactions making these effects very difficult to analyse in isolation.
Note that the efficiencies are quite similar. This is reasonable if you consider
that although the cored projectiles carry less energy, they are fired from the
coil with a shorter current pulse so the coil losses are also less.
On a last point, it's pretty clear that the coilgun is a very
inefficient machine, the reason being that the current is primarily limited by
the coil resistance. This is much like the early stage of a DC motor spinning
up under load after initial switch on. The current will quickly rise according
to the time constant of the windings and reach a limit due to resistance. As the
motor speeds up the induced voltage in windings reduces the current until the
motor reaches its operating speed and current. Electric motors operate in a region
where the current is limited primarily by the induced voltage. In this condition
most of the energy is transferred to mechanical work and little is wasted in heating
the copper. It is quite possible that a multistage coilgun will have improved
overall efficiency due to the fact that the projectile will induce an increasing
voltage in each successive coil as it accelerates.
