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Forumula for calculating the force on a coilgun's projectile?

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LAN_403

BigBad

Mon Mar 11 2013, 11:29PM

Registered Member #2529 Joined: Thu Dec 10 2009, 02:43AM
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Posts: 600

Yanom wrote ...

For electromechanical solenoids (of which coilguns are a type), Wikipedia says:

wrote ...

The force applied to the armature is proportional to the change in inductance of the coil with respect to the change in position of the armature, and the current flowing through the coil (see Faraday's law of induction). The force applied to the armature will always move the armature in a direction that increases the coil's inductance.

does this mean that for a powered coil with an iron element,
Force = (dL/dX)*I
? (note that the number of turns of the coil is already handled by the inductance L)

(dL/dX) is the derivative of the function correlating the inductance with the location of the projectile at the current location. (correct me if I'm wrong, I haven't taken Calculus yet)

does this formula (Force = (dL/dX)*I ) work, or am I missing a constant in there somewhere?

Off hand:

energy in coil = LI^2/2 is the energy in the coil

Force x distance is work done.

Neglecting resistance losses and assuming that the capacitance isn't gaining energy when the projectile is moving then the total energy is constant.

Differentiating the energy in the coil by x we get:

d/dx(0.5 * LI^2) = 0.5 dL/dx I^2 + dI/dx LI (where the 'd' is a partial derivative)
= 0.5 dL/dx I^2 + di/dt dt/dx L I
= 0.5 dL/dx I^2 + V I/v

But since for the projectile force x distance = work done d/dx (Fx) = F, and they must be equal, so:

F = 0.5 dL/dx I^2 + IV/v
Fv= 0.5 dL/dx I^2v + IV

But force x speed is mechanical power

power = 0.5 dL/dx I^2 v + IV

which looks interesting, but then again, I never knowingly get my maths right; fixing it up left as an exercise to the reader

Yandersen

Tue Mar 12 2013, 12:54AM

Registered Member #6944 Joined: Fri Sept 28 2012, 04:54PM
Location: Canada
Posts: 340

The place for this art is in a Hollywood sci-fy movie in a background. Typos in formulas, no punctuation and messy non-descriptive appearance will ensure the desired science-atmospheric effect.

Yanom

Tue Mar 12 2013, 10:28PM

Registered Member #4659 Joined: Sun Apr 29 2012, 06:14PM
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Posts: 158

We also have to deal with heat losses, which is where 90% of the energy goes in a non-superconducting coilgun.

DerAlbi wrote ...

Nature will not work like this.
The issue is respected by Âµ_r allready.
It should not be constant of course.

As mentioned before... its highly nonlinear thing.

anyway, what does DerAlbi mean by Âµ_r? I haven't encountered this before....

Barry

Wed Mar 13 2013, 04:48PM

Registered Member #90 Joined: Thu Feb 09 2006, 02:44PM
Location: Seattle, Washington
Posts: 301

DerAlbi wrote ...

Nature will not work like this.
The issue is respected by Âµ_r allready.
It should not be constant of course.

As mentioned before... its highly nonlinear thing.

anyway, what does DerAlbi mean by Âµ_r? I haven't encountered this before....

He means "relative permeability" which is just plain hard to type sensibly.
Read more on Wikipedia Permeability (electromagnetism)

Barry
Reasons I check my voicemail: 1% to hear the message. 99% to get rid of that annoying stuttered dialtone.

Yanom

Thu Mar 14 2013, 04:50PM

Registered Member #4659 Joined: Sun Apr 29 2012, 06:14PM
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Posts: 158

oh yes, Permeability. That's what that means.

But, permeability is not a factor in the formula we were discussing!
(Force = dL/dx * I^2)

Adrenaline

Thu Mar 14 2013, 07:18PM

Registered Member #235 Joined: Wed Feb 22 2006, 04:59PM
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Posts: 80

Isn't L a function of geometry and permeability?

Yandersen

Thu Mar 14 2013, 07:55PM

Registered Member #6944 Joined: Fri Sept 28 2012, 04:54PM
Location: Canada
Posts: 340

Yanom, Force = f(coil shape, projectile shape and position and magnetic properties, total magnetic field energy). Magnetic field energy is L*I*I. Add any external iron in the formula also if one presented (shape and magnetic properties).

2Spoons

Thu Mar 14 2013, 09:41PM

Registered Member #2939 Joined: Fri Jun 25 2010, 04:25AM
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Posts: 615

Yanom wrote ...

oh yes, Permeability. That's what that means.

But, permeability is not a factor in the formula we were discussing!
(Force = dL/dx * I^2)

Permeability is a factor if you are operating anywhere over ~ 60% (very rough estimate) of saturation, where the B/H curve starts to become noticeably non-linear - L becomes dependent on I as well as on geometry. Your formula is still true, its just that some of the dependence on I is hidden inside the dL/dx.

BigBad

Mon Mar 18 2013, 05:47AM

Registered Member #2529 Joined: Thu Dec 10 2009, 02:43AM
Location:
Posts: 600

I've never really thought about it, but these iron projectile guns are a type of switched reluctance motor; whereas if you use a hard permanent magnet as a projectile with no iron it's just a much more conventional PM 'DC' type motor.

The field maths for switched reluctance motors are covered briefly here:

pages 10-13, notably you should read the top of page 12 which translated into English I think says something like 'in motors where there's no electric current in the rotor' i.e. your coil guns 'then there's energy given to the projectile as it falls towards the coil, and the iron gets more magnetised'; type thing. And then it gives a tensor on the next page; which I think is basically just adding up the forces kinda like you're doing with FEMM.

WaveRider

Mon Mar 18 2013, 03:20PM

Registered Member #29 Joined: Fri Feb 03 2006, 09:00AM
Location: Hasselt, Belgium
Posts: 500

I had a look at this problem in some detail a few years ago. Here's a plot that may give some insight of armature force. (Note that the "force" in the plot is actually a "pressure", i.e. force per unit of armature cross-sectional area.)

The "perfect" non-saturable material exhibits a force that is proportional to the square of the field strength, as expected. Steel transitions to a linear force relationship between 1 and 2T. Ferrite saturates very quickly and the force appears linear over most of the field strength range.
Comes from this paper

Cheers!

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