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Coilgun modeling - help required

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LAN_403

inversed

Mon Oct 20 2008, 09:29PM

Registered Member #1770 Joined: Mon Oct 20 2008, 08:57PM
Location:
Posts: 3

Hi all,
I'm trying to write a coilgun simulation using Lagrangian approach. It has been done like 100 times before and I've read the papers, but I just can't make the bloody thing work.
System I'm trying to model:
1 coil, 1 capacitor, no current switching, ideal ferromagnetic core without saturation.
First, one should write Lagrangian: LL = T + Wm - We
Kinetic energy T = (1/2)m*v^2
Solenoid energy Wm = (1/2)L(x)*I^2
Capacitor energy We = (q^2)/(2*C)
Then by applying equation (d/dt)(dLL/dz'i) + dD/dz'i - dLL/dzi = 0, I derive equations of motion:
mv' = (1/2)L'x*I^2
I'*L(x)+I(R+L'x*v)+q/c = 0
Then I numerically integrate this stuff with Runge-Kutta and get a (wrong) solution: no conservation of energy; with some parameters I get an efficiency more than 100%. What am I doing wrong?

WaveRider

Wed Oct 22 2008, 08:13AM

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

We need more information.

Check the signs of the terms in your equations.
How are you modeling the inductance?
What order RK integration?
How are you doing the time stepping? (Explicit stepping will probably not work.)
The equations are non-linear. What solver are you using?

It is not a trivial exercise to make this work, even for the simplest case. You need to understand the limitations of the numerical methods you are using as well as the physics of the system.

Note also that the Lagrangian only gives a convenient route to developing the equations of motion. It has nothing to do with the numerical method as you describe it. You could have just as well used another method.

Cheers and good luck!

inversed

Wed Oct 22 2008, 12:30PM

Registered Member #1770 Joined: Mon Oct 20 2008, 08:57PM
Location:
Posts: 3

Probably figured out what was wrong. My math was correct. The problem lies in peculiar behavior of L and L'x during integration: I was using a function that returns L(x) and L'x(x), but when I tried to integrate L'x*v*dt, it turned out to be different from L(x). This "desynchronisation" was the cause of enegy issues (magnetic energy was wrong). Now I'm using integrated L(x) and it kinda works (energy conservation holds).

What order RK integration?
How are you doing the time stepping? (Explicit stepping will probably not work.)

RK4, explicit time stepping. Sure, implicit methods have advantages. But it's not like the explicit method will absolutely not work. It will just give less accuracy, right?

How are you modeling the inductance?

For given core position, I integrate magnetic energy over volume, increasing it mu-fold inside of the core. Then I tabulate it for different core positions with a fixed resolution dx and numerically differentiate it to get L'x. This kind of inductance representation seems to be related to the energy conservation problems, so I'm still working on it.
And it's not that I confuse Lagrangian mechanics with numerical methods. I also derived equations of motion without using Lagrangian and got the same result.

WaveRider

Wed Oct 22 2008, 05:10PM

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

RK4, explicit time stepping. Sure, implicit methods have advantages. But it's not like the explicit method will absolutely not work. It will just give less accuracy, right?

RK4 is a pretty good workhorse method. Just watch out for instability. If it happens, reduce time step. Implicit and semi-implicit methods are immune to this, but tend to be more complicated to implement. Also, simple explicit methods tend not to be "flux-conservative", meaning that over time, errors creep into things like charge, momentum and energy balance....

For given core position, I integrate magnetic energy over volume, increasing it mu-fold inside of the core.

I am not sure what you mean here. In order to know the magnetic field energy inside the core, you need to apply a boundary condition at the surface. The magnetic flux density B perpendicular to the core surface will be continuous and the tangential field H will also be continuous. In magnetic "circuit" terms, within the highly permeable ferromagnetic core, Vm = int( H . dl) and therefore H will essentially vanish. Energy is int(B . H dV) and will be very close to zero inside of the core material. The bulk of magnetic energy is found (counterintuitively) in the air surrounding the core. It is probably simplest to assume an infinitely permeable core (not a bad approx for non saturated iron) and work from there.

Have fun!

inversed

Wed Oct 22 2008, 09:48PM

Registered Member #1770 Joined: Mon Oct 20 2008, 08:57PM
Location:
Posts: 3

RK4 is a pretty good workhorse method. Just watch out for instability. If it happens, reduce time step.

It seems to be pretty accurate and stable with dt up to 1000 microseconds, and it's simple, so for now I'll stick with it.

The bulk of magnetic energy is found (counterintuitively) in the air surrounding the core.

O_o shocked. Since inductance of a solenoid filled with magnetic is increased mu times (assuming no saturation), I thought that increasing energy inside of the core mu times is the way to go. I'll examine calculating magnetic field energy in more details.

Thanks for the help!

WaveRider

Thu Oct 23 2008, 08:08AM

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

I should say that the bulk of the energy is found in the air-gaps of the magnetic circuit. If the core completely encloses the coil (like in a transformer) then this argument does not work (and in fact mu does become a simple multiplying factor)

Steve Conner

Thu Oct 23 2008, 02:57PM

Registered Member #30 Joined: Fri Feb 03 2006, 10:52AM
Location: Glasgow, Scotland
Posts: 6706

It seems counterintuitive, but magnetic energy is the product of B and H, and a magnetic core means less H is required for a given B. Hence as Waverider says, in a mixed magnetic circuit of air and iron, most of the energy is stored in the air gaps, and only about 1/mu worth of it in the iron.

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