Simple equations for coilgun

1. Assessment of current.

We often need to suggest a value of current in a coil of gauss-gun i, given deposited energy is E. This can be performed by following simple equation. Other parameters are coil length l, projectile velocity v and capacitor voltage U. Then:

i ≈ (E·v)/(U·l)

The formula is precise when its components are constant. Thus, it works best for middle and last coils of multistage accelerator fed by common capacitor, where velocity and voltage change negligbly.

2. Point of maximum current.

It is sometimes necessary to determine a moment of time when curent in a coil achieves its maximal value. This moment is described by the next complicated equation (as it is shown here):

where

is capacitive constant of the circiut,

inductive constant,

damping factor,

R, C, L - resistance, capacitance and inductance, respectively,

arth - hyperbolic arctangent ( arth (x) = ½ln {(1+x)/(1-x)} ).

Fortunately, this formula can be substantially simplified  for strongly overdamped systems, where k<<1. This case realizes in high-capacity circuits (e.g. multistage coilguns driven by common capacitor) or in single stage gauss guns with thin coils (i.e. large length-to-diameter and length-to-thickness ratio).

Alas, decomposing the equation above to Taylor series and neglecting the second-order terms, we have

just:

3. Assessment of damping resistor.

If a closeable switch circuit is utilized, it is necessary to properly choose a value of damping resistor in R-D-chian which limits the inductive voltage spike.

 Transistor-equipped circuit with R-D-damping chain.

It can be made as following. Maximum current obtained in RLC-circuit is no more than imax=U0/RL (where U0 is initial cap voltage, RL - active resistance of the chain including wire, switch resistance, parasitic elements etc). Then the voltage spike on the switch makes no more than Uk =U0+imaxRD=U0+U0·RD/RL.

Thus, the damping resistance is RD = (U- U0)·RL/U0.

Uk here is maximum voltage rating for the switch according to its datasheet.

Category: Coilgun calculations | Added by: Eugen (21.06.2017)
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