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4hv.org :: Forums :: Electromagnetic Projectile Accelerators
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Lets's Play: Model the Multistage-Coilgun

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DerAlbi
Tue Feb 09 2016, 04:27AM Print
DerAlbi Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727
Hi guys. I need help with pseudo science smile
I want to model a multistage coilgun with LTSpice, so one could .step parameters and make efficiency and optimization predictions and stuff. LTSpice is used as a Solver for the involved differential equations while the fundamental mechanical interactions are simulated once by a FEMM LUA-script and exported as Spice-compatible dataset. So much for the introduction.

What I have done so far:

I wrote a LUA-Script that builds a parametrized geometry of 3 coils and one projectile.
Setup

It then sweeps the projectile position (pos) and the coil current (I, referenced to 1Amp/Turn). Thus creating a 2D output vector of the information:
  • Geometry inductance of the main/middle Coil in dependence of the projectile and current. Its similar to an AL-value of a transformer core, so lets just call it that: AL(pos, I)
  • It measures the force on the projectile
  • It measures the coupling factor to the adjacent coils. Thus outputting K_prev(pos, I) and K_next(pos, I)
  • I also saves the free air AL-Value of the coil geometry (like 12nH/N²) and the Simulation input data
  • All this data is then automatically exported to .subckt for spice, so I can access the information.

The exported data can then be accessed in LT-Spice like this:
1454986195 2906 FT0 Coildata
It takes the 2 inputs (projectile position, current*turns) and outputs the according AL-Value, force and the coupling factors dynamically.
The dependence of those things on the projectile position is obvious.. the current*turns I sweep so one can respect saturation effects.

Modeling stuff using differential equations
  • First the basic changing inductor. As learned reading the forum the complete differential equation is >> U_L = Turns^2 (AL . dI/dt + dAL/dt . I) <<.
    (I use L = AL . Turns^2 because it correspondents to the underlying data). This can be represented in LT-Spice with arbitrary behavioral sources:
    1454987049 2906 FT0 Coildiffeq
  • Interaction with the projectile is also modeled using the sources: By outputting the force as a current one can sum up the currents. The current over 1Ohm gives a combined force of all involved coils represented as a voltage: Acceleration=Force/Mass, Velocity=Acceleration.dt, Position = Velocity.dt. Et voila, you got the position you can put into the data set above. No problem there
  • Now the crazy shit: Coil coupling with changing inductances and changing coupling. First the basic model: The coupled inductors are "compensated" by the shared field introducing a voltage onto each other. This is modeled by a "mutual inductance" M=k.sqrt(L1.L2)
    1454988279 2906 FT0 Basiccoupling
    Now we add in the differntial equations for the changing inductors and the changing mutual inductance
    1454989029 2906 FT0 Fullcoupling
    I am not sure about those equations since i dont know if this preserves energy (i should actually measure that....)
  • For the sake of simplicity i model the 2 adjacent inductors with their mutual inductances. This makes a transformer with 6 voltage sources instead of 4, but the principle is the same.


What is working so far?
I have successfully verified that one magnetized coil can be simulated while interacting with the projectile. All the waveforms (Current, Projectile speed and Force) fit quite good to a reference FEM-Simulation which is known to be correct.

What does not work?
  • While force-waveforms qualitative shape is correct its quantitative values are wrong. (about 20% too high in LTSPice; and yes i used the BH-curves of FEMM for my reference FEM-Simulation). I figure the problem is, that there is different stuff going on. Even the first coil is inherently subject to a non DC-like current (100Hz..1kHz depending on the situation). While FEMM simulates at DC, the real FEM-Simulation takes everything into account, including eddy currents in the projectile (which reject the changing magnetic field of the coil depending on frequency).
  • The voltage induced into an adjacent coil is fundamentally wrong right now. Comparing it to the reference simulation, i cant even tell what the problem is... the coupling just seems way to strong thats simulated by FEMM. This is also an AC-effect that can not be neglected


Verified problems
  • Force is known to be dependendt on projectile position change. afaik F= ~ dL/dx, so simulation the right inductance in FEMM is crucial. I verified that there is allready a problem concerning AC and DC-Simulation: Just putting a projectile into my coils and measuring the inductance with the LCR-Meter is frequency dependent. Changing the test frequency from 100Hz to 120Hz is allready an influence, changing it to 1kHz is allready close to a 50% reduction of inductance which tells the that a coilgun with LC-resonance of 1kHz puts less force on the projectile just due to eddy currents. The eddy currents in that case do not create important losses, but they just inhibit the magnetic field penetration imho.
  • I also experimented with the coupling between coils. I found the maximum coupling at 100Hz was about 0.45 while FEMM predicted upto 0.87 (at DC) which is a pretty crazy deriviation. Its all about the AC-vs-DC-problem.


Not sure
I want to test my coupled transformer model with the changing inductance. This could be done in Spice by building a perfect LC-Circuit using the transformer as L. When tracking the overal energy it should remain constant during oscillation. But what happens if i change the inductance or coupling over time? Should the overal energy remain constant? Usually changing inductance or coupling need energy (like removing a core from a coil) so i am not sure what i should observe and what is right or wrong. I also dont know what i should add up. E=C/2*V^2 is obvious, but adding up the energy in the transformer...?
(L1+M)/2*I1^2 + (L2+M)/2*I2^2 ??? It just feels wrong. It counts the mutual inductance twice and.. awww just nooh. But thats whats in the model.

I like also to dicuss..
How to get the AC-phenomenon into the model? Its clear to me that i can not afford changing the frequency in FEMM. It makes a 3dimensional parameter set which explodes the computational effort and negates the elegance of the LT-Spice method. I also would not know how to chose the frequency parameter in spice, since in time domain, you do not have access to frequencies.
In my head i want to solve it with the coupling factor. One should actually be able to get a coupling factor calculated representing the coupling of the coil to the projectile. The projectile is then a short-circuit winding which should change the effective inductance of the coil massively. If that is truely AC-dependent i dont know. i figure calculating the coupling at DC will still result in a wrong result. so how to fix this?
There is also the question on how the projectile should interact with the underlying data... we know it should reduce the fore... but how? (maybe subtracting the current in the proectile from the current in the coil?)


There are other problems left to discuss (like superposition of 2 adjacent coils (force, coupling)) but for now i like to get the most basic single-coil case working "good enough". At some point it must become pseudo science to reduce the model complexity...
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2Spoons
Tue Feb 09 2016, 09:39PM
2Spoons Registered Member #2939 Joined: Fri Jun 25 2010, 04:25AM
Location:
Posts: 615
I'd start by taking AC out of the system as much as possible: i.e. assume a laminated projectile (no eddy effects) and Litz wire coils (no skin / proximity effects) and test your model that way.
I think you also need to calculate the EMF resulting from a moving, magnetised projectile.
Adding AC back in is going to be hard - I'm thinking you may have to extract a harmonic series out of LTspice and feed that back into Femm. Then feed that to LTspice again. Each iteration would (hopefully) reduce the residual error, which you'd need to track so you could decide when to move on to the next timestep.
Or another idea: model the projectile as an LCR network in LTspice to account for eddy effects? I have no idea how you'd do that though.

Or finally you could go down the ethically dubious route of getting a hot copy of Ansys Maxwell with the transient simulator (unless you have a spare $200k), and letting that do all the hard math. You'd also need an i7 with 32Gb+ ram, and a week to run the sim.
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DerAlbi
Tue Feb 09 2016, 10:16PM
DerAlbi Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727
I am currently working on getting the projectile coupling factor into the mix.
The coupling factor is between 0.0046 (far away) and 0.076 (center of the coil). This could actually be enough to make a difference, since given that short circuit coupled "winding" does not attract the projectile (it should actually push it away!). I also increased mesh resolution to get more usefull results. If that works, i can optimize the FEMM-comutation further to reduce simulation time. Edit: i just halved the computational effort of FEMM.
I didnt expect the projectile coupling so loose but well.. that how it is.

Maybe thats enough to take the AC into account. I mean.. i could build a transformer out of that coupling factor using a frequency dependen resistor to simulate energy disipation.

I do have access to a computer running COMSOL 4.3 which is the program of choise for me (thats doing my reference calculation). A 10 coil shot takes about 2h to calculate on an i7 overclocked to 4.2GHz and 32GB ram. But thats about it. A bigger model will be more complicated and wont fit into memory. And i really do like an open source solution. Such an LT-Spice solution should be good for everyone.

I think you also need to calculate the EMF resulting from a moving, magnetised projectile.
As far as i know thats allready taken into account by the differential equation for the inductance. U=L*dI/dt + I*dL/dt

Edit: Some progress.....

1455072100 2906 FT175359 Matching

I got the projectile coupling factor in and the general matching increased by a great deal. I desreased the effective Amop*Turns by a factor of [1-sqrt(Kp)] and increased the Inductance of the Coil by a factor oif (1+Kp) with Kp beeing the Coil<->Projectile Coupling factor.
This is desperate trial&error and i dont know why it is so, but now all the small things are actually in the current and voltage wafevorm now. The force plot matches also and the final velocity error between the 2 simulations is only 0.5m/s.

What is not matching is still the induced voltage in the 2nd stage. This is a consequence of 2 problems:
The FEMM dataset still predicts a peak coupling factor of up to >0.8 which i feel (and measured) this is unrealistic. This cold still be a AC-problem.
The other thing is that during the low current (zero corssing of the green line) the FEMM-Dataset predicts a quite high inductance due to the lack of iron saturation. I dont think that is true. Infact, the reference FEM-Simulation shows around 2.9ms that most parts of the projectile are still at 2T-2.5T saturated while the coil current is near zero. This dynamic iron effect is not accounted for in FEMM.

Field
FEMM would never give such a result at near zero current. If the projectile would still count as saturated, the inductance of the coil wouldnt shoot up that much and thus the induced voltage to the adjacent coil would not show that plateau-behavior around 3ms.

What really bothers me is why the iron is actually behaving that way. Is it really only because the eddycurrent in the iron keep flowing? Is it the imaginary part of the complex µr = µr' + iµr''?

Maybe i need another simulation basis for the projectile. Maybe one could see it as a current carryiung inductor. (but based on what maths?)

So to summarize the open problems and questions are:
  • Transformer model of changing inductors with changing coupling factor
  • The projectiles AC-Behavior


And where the hell is Uspring when I need him amazed
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Uspring
Wed Feb 10 2016, 10:46AM
Uspring Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711
Sorry for the delay, DerAlbi, I had to do some thinking before I could utter anything sensible. I haven't gone through all the details of your posts, but let me mention, what I've come up until now pertaining to your simulation. In the most simple case of 2 coils and a projectile somewhere in the vicinity, the induced voltage in the first coil would be:

V1 = L1 * dI1/dt + d(mu1 * L1 * I1)/dt + d(M * I2)/dt

where L1 and L2 are the bare inductances of the coils, I1 and I2 the respective currents and M the mutual inductance. mu1 lumps the permeability of the projectile together with its position. Basically this equation is Faradays law (=voltage equal to time derivative of flux), applied to the sources of flux. These are the flux by the inductance L1 itself (first term), the flux caused by the projectile (second term) and the flux caused by the second coil. mu1 is a function of projectile position x and also M. You should be able to extract these functions from static simulations, i.e. for several positions x with FEMM.

You can get the voltage in the second coil by just exchanging 1 and 2 in the upper equation. The mutual inductance is the same. For more than 2 coils generalisation should be straight forward. There should be some way to obtain the force on the projectile from this equation by considering the energy contained in the coupled transformer, but at the moment, I don't know. HTH.
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DerAlbi
Wed Feb 10 2016, 11:18PM
DerAlbi Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727
Thx Uspring, i will keep that in mind smile I decided to give that a lower priority right now. Why? Because up on closer inspection of my waveform-comparison above, i noticed, even if the waveforms match nicely in a qualitative way, the quantitive representation is still nor right.
In detail: just compare the capacitor voltage (red graph). I notive that the reference simulation is way more damped and the second oscillation peaks only at 60V while LTSpice predicts 100V.
I count that as eddy current effect. I know the ESR of both assumed coils is equal, so it must be some AC-Effect again.
There is either eddy currents involved (comparable to core losses) or the fact that the projectile is still magnetized (why? is that eddy current too?) which has extracted energy from the capacitor and stored it in magnetic form.
So how to model such behavior? awww frown

As long as this basic property is not represented, its useless to care about adjacent coils. However i have the feeling that i need changing transformers to model the issue.. soo lets see.
Edit: Ok, i just read a little bit about eddy currents. they have a decay time. given such time one could actually build a model for the magnetisation of the projectile.
Edit2: I read a lot now about this eddy current decay and its simply not it. Link2 on top of page 38 calculates the time constant to be in the lower µs-range for my geometry which is simply not the right order of magnitude to explain the problem.
grrrrr

Edit3: ok. pushing through the night (its currently 9:00am cheesey ) let me made some discoveries.
1) my reference simulation had bad precission. (which is interesting) Increasing it accuracy hasnt made any difference to the quantitative or qualitative mismatch of the LTSPice simulation but at least without any premagnetisation the results are now perfectly matchiung (namely the first half wave of the current of the circuit above)
2) Energyconservation. Its a Bitch.
I had a static projectile (no movement assumed) that goes though saturation and "desaturation" during the ringing of the LCR-Circuit causing a current dependend inductance change. Should this conserve energy at all times?? Not sure.
Curiously changing the diff.equation for timedependend inductors from "U=L*dI/dt + dL/dt*I" to "U=L*dI/dt + dL/dt*I/2" leads to energy conservation in the case. Should it?
In my optinion changing an inductance is allways an "energy-transfer-event" so it should not conserve the energy, as long as you only look at the L- and C-energy-content of an oscillator circuit since you put an external force on the system.
On the other hand thinking about a non saturated coil with core.. again conserves the energy during the excact same scenario. So the projectile magnetisation is part of the L/2*I^2 term allready and should NOT suck energy out of the system. After all a saturated inductor is still just an inductor.

If inductance change should lead to energy conservation how is then the energy transfermechanism to the projectile? Its also just an inductance change.
I think 2Spoons had a point. Currently there is only energy conservation described, so my statement above that the projectile energy trasfer is allready formulated by the diff.equation of the inductor is WRONG.
AwwLink2 I am too stupid for this. cry
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Uspring
Thu Feb 11 2016, 02:17PM
Uspring Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711
Edit2: I read a lot now about this eddy current decay and its simply not it. on top of page 38 calculates the time constant to be in the lower µs-range for my geometry which is simply not the right order of magnitude to explain the problem.
There is a diagram with time constants of about 700us. Did you do the calculation from the equations in the appendix? The time constant is an L/R effect similar to that you have in your acceleration coils. There are some differences: One is, that copper resistance is smaller, leading to a larger L/R in the coil. In the projectile there is a ur, leading to a higher inductance and therefore to a bigger L/R. Coil and projectile time constants might be not so different.

I had a static projectile (no movement assumed) that goes though saturation and "desaturation" during the ringing of the LCR-Circuit causing a current dependend inductance change. Should this conserve energy at all times??
Yes, excepting the losses due to the R in the LCR model.
Curiously changing the diff.equation for timedependend inductors from "U=L*dI/dt + dL/dt*I" to "U=L*dI/dt + dL/dt*I/2" leads to energy conservation in the case. Should it? So what is right and what is wrong?
In my optinion changing an inductance is allways an "energy-transfer-event" so it should not conserve the energy, as long as you only look at the L- and C-energy-content of an oscillator circuit since you put an external force on the system.
Exactly. The second equation conserves electrical energy. But since mechanical energy is also involved, electrical energy can't be conserved. The first equation is correct.

2spoons wrote:
I think you also need to calculate the EMF resulting from a moving, magnetised projectile.
That is already included in V=L*dI/dt + dL/dt*I.
Or another idea: model the projectile as an LCR network in LTspice to account for eddy effects? I have no idea how you'd do that though.
That's possible. You'd have to find out the inductance and resistance of the projectile and its (position dependent) coupling to the coils.

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DerAlbi
Thu Feb 11 2016, 02:23PM
DerAlbi Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727
Aww Uspring, i just edited my post while you were writing. i am finally awake again so i corrected some babbeling.

I would still like to stick with dL/dt*I/2 to respect saturation effect correctly (which does not transfer energy). The energy transfer can be done in simulating a voltage soruce thats allways such that I*U = F*v to extract the projectile energy out of the system.
I allready have the projectile coupling, however how can i get the projectile inductance? The inductance of a solid iron piece? how? Is it just a matter of assuming a current density in FEMM like with coils? Couldnt i just use any inductance as long as the time constant fits?
If i had the inductance of the projectile i could calculate the resistance accorting to the time constant of the eddy currents by the equation of the PDF above.

Edit: with the mentioned method (using U=L*dI/dt + dL/dt/2 - F*v/I) i have not complete energy conservation. however its questionable if thats backed up by physics. Inserting a constant-power-sink just feels like chating. But i dont see much other way to respect inductance change due to saturation (without energy transfer) and Inductance change due to projectile movement (with energy transfer).
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Uspring
Thu Feb 11 2016, 04:03PM
Uspring Registered Member #3988 Joined: Thu Jul 07 2011, 03:25PM
Location:
Posts: 711
I'm not sure if you actually need the projectiles inductance. Possibly L/R and coupling suffices. The equations for power loss in a primary coil due to a (stationary) projectile coupled to it don't seem to involve the projectiles inductance explicitly.

If you want to include saturation, the concept of inductance becomes questionable, since we're getting non linear. The approach with fluxes and Faradays law as I outlined in a previous post will hold irrespective of non linearity.
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DerAlbi
Thu Feb 11 2016, 04:24PM
DerAlbi Registered Member #2906 Joined: Sun Jun 06 2010, 02:20AM
Location: Dresden, Germany
Posts: 727
Saturation is allready automatically included in my Dataset that comes out of FEMM...
I put in the Amp-Turns and Projectile position in spice and get the corresponding force and coupling and inductance and stuff out of the exported dataset
So inductance change due to saturation can not be distinguished from inductance change due to projectile position (in time domain).
When i think about it, i become more confident that the Back-EMF as U=F*v/I should be ok, because since F is a function of the inductance-change it should be correct.

I try the eddy current coupling now..
Edit: naaah this coupling to the projectile is not right. Or at least its not the basis for the eddy current loss imho.
The coupling arries between 0.04 and 0.08 or somnething. .short circuiting such a small portion of the field does not extract enough energy.
There is a fundamental problem with this solution. its either in the FEMM dataset (bad coupling extraction?) or whatever.
*help*

Edit2: Look at the projectile... after the LCR circuit oscillated for a little more than one period.

1455215578 2906 FT175359 Magnetic Remanence
The B=0 lines are created by the remagnetisation of the iron. The first cycle is overwritten by the 2nd currentpeak (the negative swing of the oscillation period). Since the -I creates a -B and only |B| is displayed you get a superposition of the old +B and the new -B thus creating the B=0 lines.
The interesting stuff is why the projectile is still magnetizes as if there was +I in the coil while where is infact -I.
This has to be eddy current.
Maybe i am concentrating on the wrong stuff. I though eddy current are just responsible for losses. But maybe thats neglectible! WHat the eddy currents really so is that they inhibit the outside field from penetrating.

So while the coil has I(t) in it the projectile is magnetized by a low pass filtered (delayed) current. This lag in B creates less force obviously. It all makes sense (to me cheesey )
So the question is... how to approximate it?

Lets just assume we are below saturation: the Force is not ~I^2, but I_Coil * I_Projectile. Kind of. Anyone know where i am going?
I_projectile beeing the Coil-Current that would create the equivalent B-Field magnitude.
The more i think about it, the more it comes down to the B-Field being Low-pass filtered due to eddy currents. i mean.. thats what eddy currents do.. right?
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2Spoons
Thu Feb 11 2016, 11:35PM
2Spoons Registered Member #2939 Joined: Fri Jun 25 2010, 04:25AM
Location:
Posts: 615
DerAlbi wrote ...

WHat the eddy currents really so is that they inhibit the outside field from penetrating.

...

The more i think about it, the more it comes down to the B-Field being Low-pass filtered due to eddy currents. i mean.. thats what eddy currents do.. right?

Yes! Which, along with the eddy losses, is why i've always thought coilgun projectiles should be laminated in some fashion.

That doesn't help with your modelling though -sorry.
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