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Dr. Shark

Sun Apr 06 2008, 10:54AM

Registered Member #75 Joined: Thu Feb 09 2006, 09:30AM
Location: Montana, USA
Posts: 711

You are right, but since the voltage waveform looks almost linear rather than sinusoidal, I am not too sure about how accurate this is. It would indicate around 4kA. I am not claiming that my LCR simulation here is much better though, since it is based on many assumptions (mostly the capacitors ESR which I cannot measure). Still my initial guess of 5kA seems to be in the right ballpark, which is all I care about. That puts my Rogowski coil at 2mV / kA.

Anyway I had a little think last night which turns out to integrate rather well with what we know already. What I did was to estimate the lifetime of the eddy currents in the projectile, starting with the assumption that the projectile has nearly the same shape as the coil. This way the L and R of the coil can be used to compute tau=L/R, the time constant for the current to decay. Now some educated guessing can be made based on the thickness of the armature, for an induction coil-gun tau is probably shorter than the coil parameters would indicate, whereas for a disk launcher it could be considerably longer, since the disk can be made thick compared to the coil.
For my small coil geometries it turns out tau is about 100us, so the pulse-width I have been working with was more or less spot on.

That's only the first half of the story though. I had another look at the equivalent projectile resistance and used a bit of freshman physics (boy I had a hard time remembering that) and the assumption of a square current pulse to solve for force, acceleration and distance traveled

P = 1/2 i^2 R_p = 1/2 i^2 v L'
F = 1/2 i^2 L'
a= 1/(2m) i^2 L'
s= 1/(4m) i^2 L' t^2

If I plug in the numbers from my gun here, I get something like 1/2mm for the distance, which immediately explains why the amature barely manages to twitch.

This shows the trade-off we are working against:

We want many turns on the coil to get L' up to compete against R
We want few turns so the pulse is as short as tau=L/R
We want a heavy projectile with low resistance for long tau=L/R
We want a light, fast projectile to boost P=1/2 i^2 v L'
We want to wire capacitors in series so C goes down and we get a short pulse even for large L
We want capacitors in parallel to keep parasitic R low

I guess what this leaves us with is what I already said in the first post: Big guns have a big edge, since they have lower R and more v all other things being equal. My implicit goal, to get double digit efficiency from a small gun with a single stage, seems more ambitious (absurd?) than ever.

Dr. Shark

Fri Apr 11 2008, 10:37AM

Registered Member #75 Joined: Thu Feb 09 2006, 09:30AM
Location: Montana, USA
Posts: 711

Just a short update to let you guys know I have not given up yet. I made another set of tests comparing a coilgun and a disk-launcher, but most importantly I have a cool new formula on the theoretical side:

Assuming as usual that we start from rest, have a square current pulse and ignoring resistance in the armature we can use freshman mechanics once more to find the velocity of the projectile from the acceleration, and plug this into the formula for the power to get

v = 1/(2m) i^2 L' t
P = 1/2 i^2 v L' = 1/(4m) i^4 L'^2 t

Now we can compute the ratio of this power to resistive losses

P_r=i^2 R
P_k = 1/2 i^2 v L' = 1/(4m) i^4 L'^2 t

P_k/P_r= 1/(4m R) i^2 L'^2 t

Finally we make the assumption that the efficiency is small (<10%) so the action integral S will be approximated by the resistive component only. E_cap is the energy stored in the capacitor.

S= i^2 t
E_cap= i^2 R t
S = E_cap / R

P_k/P_r= 1/(4m R) L'^2 S = E_cap/(4m) (L'/R) ^2

which is awesome, so I'll repeat:

Form

All the variables like current and time have been eliminated, and we can now see what really matters:
The ratio (L'/R) even makes a squared appearance, but sadly this is more or less a material constant. Using more turns to increase L and L' proportionally increases R, so the best we can do for this factor is to find a clever coil geometry. My experiments indicate that a solenoid coil give an (L'/R)=200uH/m / 20mOhm = 0.01 at best while a pancake manages 500uH/m at 20mOhm, and I doubt that this can be increased much without cryogenic cooling of the coil or something exotic like that.

The E/m term is probably not as interesting. It is quite surprising to see the efficiency depend on the stored energy, but I think this becomes negligible once the intitial velocity v_o is reasonably high. It just tells us that most efficiency is gained at the very end of the launch, when the projectile has already picked up some speed.
The m is again somewhat an artifact, since a very light projectile will have massive resistive losses, which were excluded from the analysis.

EDIT: the following has been superseded, see below
On the experimental front I may have some empirical data about the importance of the eddy current lifetime tau=L/R in the armature. I have build a disk and a solenoid launcher with identical parameters L'=200uH/m and 20mOhm and the disk launcher performs orders of magnitude better. I believe this is due to the fact that the disk projectile has a thickness of 4mm, while the solenoid shoots a piece of copper tube with only 1mm wall thickness. Of course I will have to verify this by testing both launchers with projectiles of different thickness, which I will have to machine first. In the meantime, I think this is quite a plausible explanation.

Dr. Shark

Wed Apr 16 2008, 09:58PM

Registered Member #75 Joined: Thu Feb 09 2006, 09:30AM
Location: Montana, USA
Posts: 711

Another little update for those still interested. I'll start with a few pictures, here is another one of my current setup with the pancake and solenoid coils and their respective projectiles.

I made a new aluminium disk from 1.5mm tread plate, which should have the same resistance as the 1mm copper, so I can make fair comparisons now. First thing I noticed, the disk launcher still works well with the new disk, and second thing I noticed: My measurement of the inductance gradient was wrong, in reality it is more like 500uH/m . This explains why the disk laucher performs so much better: Because the armature is removed from all the turns "simultaneously" rather than turn by turn as is the case with the solenoid, it has more then twice the inductance gradient.

Now what should I make of this? I gathered some supplies for a multi-stage launcher:

But I need to get the first stage of the solenoid launcher working a whole lot better before I can think of adding another stage. Investigating more coil geometries might be helpful, since the first stage only needs a very short, but steep inductance gradient a significantly shorter coil might be a good idea. For later stages the inductance can change more slowly over a longer distance, the first stage really seems to be the most difficult to design.

Why I choose to go with SCRs rather than the IGBTs that I was originally toying with? It turns out that IGBTs have two serious drawbacks compared to SCRs in pulsed power applications: Latch-up and saturation.

Latchup occurs when the parasitic thyristor formed by the PNPN structure of the IGBT gets turned on. Usually the base of the parasitic NPN is shorted to the emitter, but under very high current conditions, the voltage dropped across the die can become high enough to turn it on anyway. If this happens, the IGBT will not turn off until there is no more current through it.

Nothing can be done about this, as it is greatly a parameter of how the IGBT has been manufactured. I don't know how much of a problem it is in practice, but if it turns an expensive IGBT into a cheap SCR, it would certainly be a major annoyance.

Saturation is a lot worse in some respects since it it actually kills the IGBT if it happens, but it has the benefit that it can be controlled to some degree. The saturation current is proportional to the square of the gate voltage ( V_gate - V_threshold to be more exact) so by doubling the drive voltage over the manufacturers recommendation, four times the current can be passed. The DRSSTC crowd has some experience with this, but I don't think this approach has been taken to the limit since high gate capacitances and frequencies make it very hard to charge up the gate to insane voltages in Tesla coil duty.
I have yet to perform experiments on how much static charge the gate can take before dielectric breakdown occurs, but I suspect that this will not be the limiting factor.

Of course even though IGBTs don't come with an "on-state resistance" quoted in the data sheet, there is still a roughly linear relationship between current and forward voltage. This means the conduction losses will scale as the square of the current, leaving the action integral iÂ² t as the ultimate limiting factor.

has more on this.

OZZY

Sun Apr 27 2008, 02:38PM

Registered Member #511 Joined: Sat Feb 10 2007, 11:36AM
Location: Somerset UK
Posts: 55

Hi Dr Shark

I think you are definitely on to something here. Your work has helped my understanding of induction launchers a great deal. I particularly like your list of trade offs for launcher design. Unfortunately it seems to be a universal law of electromechanics that an efficient design will be a compromise between competing factors.

On the mathematical side I do have a problem, in your analysis you state that armature resistance is ignored, I do not think this is a safe assumption. Heating of the armature will be a significant loss in the system, if you intend to calculate the efficiency then you should take it in to account.

Have you read the thesis paper by Braam Daniels posted on Barry`s website? He describes the system using a set of three simultaneous differential equations. These equations appear to take all the important variables into account but I don`t think they will yield to an analytical approach, he used numerical methods.

The action intergral of the current pulse will be related to the intergral of force over time for the armature. Force x time is impulse so the action of the system will relate to the change in momentum not change in kinetic energy.

I have been doing some FEMM modeling of a disc launcher. The force is huge near the coil but drops away with distance, for a 50mm OD coil the force is practically gone when the armature 12mm away. This leads me to the conclusion that a disc launcher will need a current pulse with a short rise time so that the peak current occurs when the disc is still close to the coil.

OZZY

Dr. Shark

Sun Apr 27 2008, 08:01PM

Registered Member #75 Joined: Thu Feb 09 2006, 09:30AM
Location: Montana, USA
Posts: 711

OZZY wrote ...

Hi Dr Shark

I think you are definitely on to something here. Your work has helped my understanding of induction launchers a great deal. I particularly like your list of trade offs for launcher design. Unfortunately it seems to be a universal law of electromechanics that an efficient design will be a compromise between competing factors.

Thank you for your kind words, I had almost resigned to the fact that I created this thread just for myself. It's good to see some interest after all.

wrote ...

On the mathematical side I do have a problem, in your analysis you state that armature resistance is ignored, I do not think this is a safe assumption. Heating of the armature will be a significant loss in the system, if you intend to calculate the efficiency then you should take it in to account.

I agree with that, basically all my formula gives you is an upper bound on the efficiency. I mostly like the formula for it's mathematical simplicity, it's something that even a person without a science degree can put a few numbers in, and see whether their design could work at all.

For my own launcher, the formula easily predicts an efficiency of 0.1, but what it get is more like 10^-4

wrote ...

Have you read the thesis paper by Braam Daniels posted on Barry`s website? He describes the system using a set of three simultaneous differential equations. These equations appear to take all the important variables into account but I don`t think they will yield to an analytical approach, he used numerical methods.

Yes I am aware of that paper, but for some reason I only skimmed through it very quickly. I'll have another look, it is here

for anyone else that is interested.

wrote ...

The action intergral of the current pulse will be related to the integral of force over time for the armature. Force x time is impulse so the action of the system will relate to the change in momentum not change in kinetic energy.

I was not able to put it in words as elegantly as you, but this was what I identified as the key benefit of multistage designs: The higher the initial velocity, the more kinetic energy you get for a constant amount of momentum.

OZZY wrote ...

I have been doing some FEMM modeling of a disc launcher. The force is huge near the coil but drops away with distance, for a 50mm OD coil the force is practically gone when the armature 12mm away. This leads me to the conclusion that a disc launcher will need a current pulse with a short rise time so that the peak current occurs when the disc is still close to the coil.

OZZY

I am trying to stay away from FEM models because it is something I don't understand very well, and I'd rather write down some formulas that I can intuitively assign meaning to.
This way I am making my own approximations along the way rather than relying on some hard coded approximations in the software; if something goes wrong I have only myself to blame.
Also there is some very good FEM code from WaveRider here

, so why reinvent the wheel? I have not looked at his program myself though, since my iMac and I are not able to compile C programs.

Coilgun4

Progress on my own model has been slow, but I'll put what I have up for criticism. Basically what I am trying to do is start with an ordinary RLC circuit, for which Kirchoff's law tells us:

V(resistor) + V(capacitor) + V(inductor) = 0
R i + 1/C q + L i' = 0
R q' + 1/C q + L q'' = 0

Where q, the charge on the capacitor, is a function of time and q' indicates time derivatives. Now I am trying to put an extra term in there to model the back EMF induced by the moving armature. I did this using Lenz's Law to find the voltage and associated current induced in the armature, and Lenz's Law again to find the induced EMF back in the coil. I am using the superposition principle to treat the magnetic field in the circuit as a sum of the primary coils field and the opposing field produced by the armature.

The primary current rise creates an EMF in the armature as

EMF= d Phi / dt = sqrt(k) L i' = i_sec R_sec

The coupling k features as a root, because this is only half of the "round trip" coupling. Now in contrast to my posts above I am treating the L as constant, because the inductance only _appears_ to change when it is coupled to the armature. I am also substituting V=iR in the armature, so this is where the losses make their appearance. This can easily be rearranged for the secondary current

i_sec = sqrt(k) L i' / R_sec

which we can use with Lenz's Law again to get the EMF inducted in the primary again:

EMP_pri= d Phi_sec / dt = sqrt(k) L_sec i_sec' = k L_pri L_sec i'' / R_sec

If this term is added to the above expression of Kirchoff's law, we get something like

R q' + 1/C q + L q'' = k/R L^2 q'''

Where I have assumed that L_pri = L_sec which should be a reasonable approximation for the geometry under consideration.

Not only is this a third order differential equation, but also it is a PDE, since the coupling between the coil and the armature, k, is also a function of time. In fact the whole term is a bit of a bitch since in order to express k(x) as k(t), we need to open the whole Pandora's box of Newtons mechanics again. Starting again from the energy in the field, but this time using i rather than L as the dynamical variable, we can try to get an expression for the armature speed:

(to be continued)

I gladly take hints on how to solve this, but I'll keep editing this post as I make progress... Or should that read "IF I make progress"?

Steve Conner

Thu May 08 2008, 09:22AM

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

Hi,

Dr. Shark asked me to comment on his discussion of IGBTs vs. SCRs, so here I go.

Latchup: This was only a problem with first-generation IGBTs and has been fixed.

Saturation: True to an extent. No matter how much you overdrive the gate, the IGBT will come out of saturation at some arbitrarily high current. Many IGBTs are rated to withstand this condition, passing hundreds of amps with hundreds of volts dropped, for a few hundred microseconds, long enough for a protection circuit to shut them down.

But this is still not as good as a SCR, which can never come out of saturation, because it's self-exciting. This is why SCRs can be protected by fuses and breakers, while IGBTs need active protection circuits. It also explains why SCRs last longer in the clumsy hands of noobs, and I suspect this is the reason why they are preferred for EM gunnery.

Overdriving IGBT gates is not that difficult, even in high frequency operation, but it only gets you so far before reliability suffers. 15V is about the limit for applications that need industrial-grade reliability. I've never dared to go above 28V. The dielectric punctures instantly at around 50V. Overdriving is never done in industry anyway, because it negates the short-circuit withstand capability that I discussed in "Saturation" above.

Voltage drop: In theory, the voltage drop across an IGBT, just like a SCR or diode, is a logarithmic function of the current. They are all conductivity-modulated devices and can be modelled by the Ebers-Moll equation or whatever.

In practice, all real devices have ohmic components in the current path, that make the function "Log with a hint of linear" at low currents, transitioning to "Linear with a hint of log" at high currents.

Hence the action integral is a little dubious in theory, but everyone just uses it anyway. I even used it myself in my OLTC and DRSSTC modelling work. Presumably the argument is that the ohmic parts of the device will fail first.

WaveRider

Thu May 08 2008, 07:46PM

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

Just a couple of comments...

It's always easier to solve a system of first-order DEs than higher order coupled equations. In fact, you can do the complete simulation in about 5-6 lines of Matlab/Octave code this way. Try to break up the system into a set of 1st order DEs. This is readily solved using classical Crank-Nicolson or implicit time stepping. If you want to get more sophisticated, try an adaptive 4th order Runge-Kutta method on it...

Be careful about the definition of "action integral". What you are really talking about is "impulse integral". In variational mechanics, "action integral" often refers to the integral of the Lagrangian in time (which is extremised to find a stationary solution).

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