Energy and Efficiency

[RobertL]

I'm having fun building a multistage coil gun.   Using low voltage, and high current.

In reading some of the canon websites (eg. Barry's Coil Gun size - which is excellent, btw).  theres a lot of talk about determining energy from the capacitor, and comparing with the kinetic energy of the projectile to determine efficiency.

However, it occurred to me that it was more useful to look at the energy stored in the inductor, rather than the capacitor.  As its the device directly affecting the projectile.

Energy in an Inductor is

    E_L = 1/2 L I²

The inductance (L) can be measured, calculated etc.  and has a direct bearing on the construction of the coil.  For example, in a solenoid type coil:

    L = u N²A / l

Where A is the cross section area of the inductor, and l is the length.  Other formula exist for disk/pancake coils.

The current (I) can be measured too, and is more "important" as the energy is proportional to the square.  The field strength is proportional to current also:

    B = u N I / L

So my plan is to use super high current, several hundred to kilo Amps, in relatively low value inductors.   Switched with ganged MOSFETs.

I'm hoping this will prompt a discussion.
cheers!


*edited, based on Twospoons comments

[Twospoons]

The inductance (L) can be measured, calculated etc.  and has a direct bearing on the construction of the coil.

    L = u N²A / l

Where A / l represents the volume of the inductor.

Eh ... its a bit more complicated than that. Your equation is based on the long solenoid approximation  -  the length must be longer than the diameter of the hole.  Other solenoid shapes are possible, and may be more efficient in terms of  copper use, and inductance vs size.  This is why finite element field solvers exist.  Have a play with FEMM 4.2 - its reasonably easy to understand , and its free.

"Where A / l represents the volume of the inductor."  Not even close. It has units of length, not volume.

[RobertL]

Eh ... its a bit more complicated than that. Your equation is based on the long solenoid approximation  -  the length must be longer than the diameter of the hole.  Other solenoid shapes are possible, and may be more efficient in terms of  copper use, and inductance vs size.

thanks, I've edited the head post.

This is why finite element field solvers exist.  Have a play with FEMM 4.2 - its reasonably easy to understand , and its free.

thanks I'll take a look

[T3sl4co1l]

You will find that the number of turns is not actually important to the energy storage of an inductor.  After all, you cannot get as many turns in the same space.  The limiting factor is actually the resistivity of copper, which limits you to a certain total amp-turns, in a given winding volume, for a given power dissipation.

Under pulsed conditions, this can be quite high, but the efficiency drops substantially as most of the inductor's voltage drop is due to I^2 R losses rather than L dI/dt electromotive force.

The efficiency drops further when considering the lack of pole pieces (in a typical coilgun design), so that the amount of field seen by the armature is a tiny fraction of the total field.  You're wasting a lot of electrical energy building fields that will never perform mechanical work.

If this is a pulling type coilgun, there's the added limit that the armature and pole pieces saturate at 1.5T or so.  This corresponds to a best-case pressure of about 10 atm -- a respectable "potato cannon" level of performance, but embarrassingly pitiful compared to powder propellant types.

This limit can be overcome by pushing a conductive slug instead, in which case the flux density is unlimited, but material resistivity dominates quite seriously.  On top of coil efficiency, you're also induction heating the slug, which increases its resistance, which increases its heating...  At some point, it will be so hot that acceleration can't overcome drag, and it'll be at top speed.

One bonus to the pushing type launcher is, if the barrel is filled with ferrofluid, the inductance of coils away from the armature can be made relatively high, and we can wire them in parallel, in sequence.  The barrel would not be cylindrical, but a channel which guides the armature and which otherwise allows free flow of the ferrofluid around it.  Thus we can make a brushless linear motor.  And as an induction type motor, the frequency isn't too important, as long as the force is adequate (high slip condition).

The full build of such a thing would look like, probably a stack of pancake coils alternating with wedge shaped ferrite pole pieces, soaked in a bucket of ferrofluid, with a guide channel in the middle.  The armature is copper, and the drive signal is a specialty VFD (voltage-frequency drive), a rising voltage and frequency (chirp) shaped to match the armature's acceleration (the coils can also be spaced differently towards the muzzle end to ease the frequency chirp rate).

This is also a ferrofluid fountain or splasher, so have fun with cleanup.

Tim

[klugesmith]

>> This is also a ferrofluid fountain or splasher, so have fun with cleanup.

How 'bout a mercury fountain?   Cross between disc launcher and "repulsion" coil gun.  Or, with electrical contacts to the mercury, a cross between MHD pump and rail gun.


On the original topic of energy: 
If you don't count losses from coil resistance, and losses in the projectile,
the peak energy in inductance (L*I^2/2) is automatically the same as initial energy in capacitor (C*V^2/2).  You can check that using Barry's "simulator" by making R very small.

[edit] For given coil dimensions, as you trade off the wire size and number of turns, the L/R time constant is invariant.  But thicker wire & fewer turns reduces the oscillation period, which is proportional to sqrt(LC). That means a larger fraction of initial C energy will get to be L energy at the end of first 1/4 cycle (if we ignore the fixed L and R of capacitor bank and interconnections).  Not necessarily better for coilgun efficiency, which depends on matching between pulse width and projectile moving time. 

As Tim said, most of the energy in typical coil guns goes into I^2 R heating of the coils.
Whether the damping is heavy or light (but finite), you can figure the temperature rise per shot by dividing capacitor energy by copper mass and heat capacity.