  # RLC Circuit Analysis

The Java simulation of a classic RLC circuit was great! Now how does it apply to coilguns?

## Role of Resistance

The graph's shape is very sensitive to resistance. At rather high resistance it is the main limiting factor on the current in the inductor, and you see the capacitor merely discharges slowly. At medium resistance you will see one lump of coil current which quickly dies out. At very low resistance it will ring (oscillate) as the energy sloshes from capacitor to inductor and back again.

One goal during construction is to minimize the resistance. Don't be too worried about ringing in a coilgun, because the projectile should absorb most of the coil's energy, leaving only a small amount for ringing. If you use an SCR to switch the coil then it will cut off after the first half cycle leaving the excess energy stored in the capacitor again, helping to prepare it for the next firing.

Another goal is to get the correct timing. If the coil current is too short, then the projectile barely moves before the current is gone. If the current is too long, the projectile is pulled back again after it passes the middle. You must get the right timing for your particular combination of coil and projectile!

## Role of Inductance

The frequency of ring of an LC circuit is in proportion to the square root of the LC product: This is a useful approximation for coilguns. However, the above equation is true for undamped LC circuits, and does not consider resistance or energy absorbed by the projectile, both of which will serve to lower the resonant frequency somewhat.

In a coilgun, inductance L is not constant, but varies with projectile position. Not one of the classic engineering textbook RLC treatments deals with inductance that varies on a sub-cycle basis. We should expect the nominal half sine wave current waveform going into a coil to be heavily phase-distorted by the changing inductance, and by the projectile removing energy from the magnetic field. No, this is more the realm of linear motor design. (Can any of you motorheads out there help me out?)

The projectile moves "in the direction of and in proportion to the change in inductance". Any movement is real work and removes energy from the magnetic field. It looks to the electric circuit like real resistive load.

## What About Negative Coil Current?

You might have noticed that with some combinations of values (notably very low or zero resistance) the coil current oscillates between positive and negative current. Can we use the negative current for some advantage?

In a real solenoid, a negative current gives a magnetic field in the opposite direction. Since magnets only pull (not push) then a negative coil current serves to pull the projectile back into the middle again. So it doesn't matter which polarity you use for the power connection; it will always pull things toward the middle.

The mechanical force is proportional to the magnetic field strength. And the magnetic field strength is proportional to the current (and a few other things such as the number of turns). So as the graph of current over time passes through zero, at that instant the m-field really is zero, and no energy is stored in the coil. In fact, at this instant all the electric energy is stored in the capacitor's electric field.

However the voltage on the inductor depends on the derivative of the current: V = L di/dt. This voltage is about the same as the capacitor's voltage, which will start current flowing again. Inductors tend to resist changes in current because their energy is stored in a magnetic field, which is analogous to capacitors that resist changes in voltage because their energy is stored in an electric field. That's why it's so difficult to get an instantaneous change in coil current, and why they don't switch on or off instantly.

In conclusion, we can't extract useful work out of the negative coil current. However, it does represent energy stored in the capacitor which can be used for the next shot.

## Conclusions and Speculation

When you build a real coilgun, the resistance R should be quite low. You want the net R to be dominated by the work done to move the projectile, not by the series resistance of the capacitor, power transistor, connection wires and inductor.

The RLC equivalent circuit without the projectile should have a quality factor Q fairly high. The projectile's (time-varying) "resistance equivalence of motion" has to be the dominant damping in the equivalent RLC. Since the force to move the projectile varies with projectile position, so does the projectile's damping on the LC.

The field strength of a coil is determined by the ampre-turns, and the product of these two are constant for a given coil geometry. We know that the inductance (proportional to turns squared) is a critical factor in getting a coil "loaded" for the proper duration of time.Timing is extremely important to accelerating the projectile while it's in the right position, and dumping the current before it becomes retarding. It follows that for any multi-coil arrangement the coils are all different from each other. They probably have equal dimensions, but have dramatically different wire size. The first coil would have the smallest wire, and the last coil must have very low inductance so it uses very few turns of very thick wire. Since we can't stretch the projectile during flight, all coils would have the same physical length.

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