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Registered Member #511
Joined: Sat Feb 10 2007, 11:36AM
Location: Somerset UK
Posts: 55
Hello everyone
I intend to use this thread to discuss my thoughts on coilgun theory and practice. I am not a mathematician or a scientist but I will try to back up my opinions with equations and experimental results where I can. I welcome any comments or discussion on this subject.
Saturation The first topic I will tackle is saturation of the projectile. I have found a lot of contradictory information on the web about how saturation affects performance so I decided to investigate it myself. The general rule of thumb is that below the saturation point Force is proportional to current squared and above the saturation point Force is proportional to current. But how does this affect the efficiency and when does the projectile saturate?
To answer this question I constructed a FEMM model of a coilgun similar to mine and used it to get values of force at different currents. If you assume constant acceleration which it isn`t and constant current which it isn`t, then you can calculate the efficiency. From the force you can calculate acceleration, velocity and kinetic energy. The energy lost as heat in the coil is I^2*R*t where t is the time it takes the projectile to fill the coil, the initial velocity is zero. I put all this into an Excel spreadsheet and ploted a graph. Graph
What dose this graph tell us? First of all the force does appear to start off with a curve that suggests a squared relationship but after 20A it becomes a straight line. Conclusions: 1, the projectile saturated at about 20A. 2, at high values of current it is safe to assume force is proportional to current so saturation does not place a limit on coilgun performance. The efficiency curve is much more interesting. I did not expect the maximum efficiency to be at the saturation point . The value of 30% is not significant, it is a product of the coil parameters I used in the FEMM model. Conclusion, operating below the saturation point is a very bad idea because force and efficiency will be low. Above the saturation point the efficiency drops off but not very fast so operating in this region is a good idea. The coilgun this simulation is based on had a peak current of 800A where the efficiency will be much lower .
So now we know that an efficient coilgun will have low acceleration and need a very long barrel, and a high power coilgun will have poor efficiency and need a very big capacitor.
Now for some maths A voltage is applied to a coil by some sort of power supply, the coil contains a moving steel projectile.
V = Ri + L di/dt + i dL/dt
V is applied voltage, R is resistance of coil, L is inductance of coil & i is current.
The first term is back emf due to the resistance. The second term is back emf due to the inductance and the third term is back emf due to the chainging inductance caused by projectile motion.
This can be re-written as
V = Ri + L di/dt + iv dL/dx
where v is velocity and dL/dx is the inductance gradient caused by the projectile. If we multiply both sides by the current we get
Vi = Ri^2 + iL di/dt + i^2 v dL/dx
Now the LHS is Power delivered to the coil, the first term is power dissipated as heat, the second term is power going into the magnetic field and the third term is power going into the projectile. Up to this point these equations are quite general but now we need to make some assumptions. As usual we assume constant current and constant acceleration, the projectile covers distance S in time t. This gives us.
Vit = Ri^2 t + i^2 vt dL/dx
The inductance term has vanished because current is now constant.Velocity times time is distance so this can be re-written as.
Vit = Ri^2 t + i^2 S dL/dx
Now the LHS is energy taken from the supply, the first term is energy dissipated as heat and the second term is kenetic energy of the projectile. This does not take into account the energy of the magnetic field E = Li^2 /2 it is assumed that this energy is taken from the supply and then returned to it.
Energy is force times distance so therefore the force on the projectile is.
F = i^2 dL/dx
This looks like a nice simple result but don`t be fooled, dL/dx is not a constant. The inductance gradient will vary with projectile position, so you get the force/displacement graphs that are shown on some websites. At very low current the gradient profile is constant but when the projectile saturates it will start to drop, this accounts for the shape of the force/current graph above.
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Reluctance coilgun efficiency
. The value of 30% is not significant, it is a product of the coil parameters I used in the FEMM model. Conclusion, operating below the saturation point is a very bad idea because force and efficiency will be low. Above the saturation point the efficiency drops off but not very fast so operating in this region is a good idea. The coilgun this simulation is based on had a peak current of 800A where the efficiency will be much lower 
.