Question

I need an equation to calculate how much time a coil has to and can remain active in a coilgun to reach a specific velocity. In coilguns the coils must be disabled once the projectile is 1/2 inside the coil or the projectile will bounce.

Fixed variables are as follow:
-Mass of projectile
-Initial velocity of projectile assuming vacuum
-Final velocity of projectile assuming vacuum
-Projectile and coil length (Projectile length is equal to coil length
-Coil attraction distance in millimeters
-Coil amperage draw
-Voltage
-Efficiency

Answer 1

Suppose we have a coilgun system which satisfies the small induced voltage assumption, how do we determine the approximate muzzle energy from the force curve? It's actually quite straight forward, all that is needed is a simple integration. The only tricky part here is deciding what function to use. Since we are only concerned with the left half of the curve (we're turning the current off at the midpoint) let's dump the right-hand side and get a curve fit from a spreadsheet programme.

Fig 3.1

A 3rd order polynomial fits this part of the curve very well so integrating this equation gives us

The energy units are in mJ because the distances are expressed in mm. So what does this mean in terms of projectile speed? Firstly we need to know the mass of the projectile, in the simulation above the projectile dimensions give rise to a mass of about 10g so the velocity can be worked out as follows

Clearly, this isn't going to create any sonic booms. What you need to appreciate is that very large current densities will be needed to deliver a high-velocity projectile from a single-stage coilgun. In this example the current was set to 50Amm^-2, this is small compared to what is needed for a really fast projectile. Current densities of around 1000Amm^-2 will produce 'respectable' velocities.

There is a very neat little formula which describes the force on the plunger (projectile) of a solenoid, it goes like this:

Eqn 4.1

where N is the number of turns, I is current, and ^df/_dx is the rate of change of flux linkage with plunger displacement. N and I are straightforward, but the flux linkage is quite difficult to determine since it is dependent on the geometry of the coil and the plunger material. Perhaps the best thing to take away from examining this formula is that the force can be increased by either increasing the number of turns, increasing the current, or increasing the change in flux linkage

You need to know the attractive force at each position as the projectile enters the coil. The force will be near zero when the steel projectile is outside the coil. The force should increase proportionally to the amount of steel inside the coil. It tends to pull the steel to the midpoint of the coil length; therefore we'll call the force positive as the steel is introduced to the magnetic field, and negative after the steel goes past the midpoint. Looking ahead, in a real coil gun, we need to turn off the coil before the projectile reaches the midpoint.

So here are some ideas. I've come up with one way to measure these forces with a minimum of laboratory equipment. All you need is a ruler and an adjustable dc power supply. Suppose the projectile is a flat-head screw (so it can stand up by itself). Stand the screw with the tip pointing up, and position the coil at certain measured distances above it. At each position, slowly increase the applied voltage until the screw is lifted off the table.

This method yields acceleration directly. You don't need to know the mass of the projectile or the force in grams / ounces / pounds/tons / newtons. You don't have to weigh it because "one gee" force is our calibration standard. How convenient that your friendly local gravity field provides a free calibrator for you!

You recall that f = ma (force equals mass times acceleration). Instead of measuring tricky little quantities of force and mass in tiny amounts, we'll get the acceleration figure directly. This is what we really want, anyway, to fill into the basic equations of motion:

velocity as function of time: v = at

distance as function of time: d = at²/2.

My method goes like this. First, measure the length of the screw. Then follow these steps:

1. Position the coil above the screw.
2. Measure the height from the table to the coil. (We want the depth of the screw into the coil, but it's hard to measure directly because we can't easily reach inside the coil.) I strapped a micrometre firmly onto the side of the coil to measure the height above the table.
3. Hold the coil steady.
4. Now gradually increase the supply voltage, and make note of the exact voltage at which the screw is barely lifted from the table. This force is (by definition) precisely one gee.
5. Now repeat this measurement for each position as the coil progressively encloses more of the screw.

At each height of the coil above the table, it shows the voltage necessary to barely lift the projectile from the table. This voltage (actually current) will later be increased to the limit of my power supply to hurl the projectile at high speeds. Go here to see the calculated possible forces.

The particulars for these particular measurements in particular are:

• two identical coils were placed closely end-to-end
• each coil is 40.35mm long, total of 80.7mm
• both coils use 228 feet of wire in 790 turns, 5.05 ohms
• two coils are temporarily wired in series
• the wire is 24-gauge coated magnet wire
• the projectile is 31.6mm drywall screw (about 1 1/4" long)

Coil and Screw

Vmax	= 12.5 v
Lscrew	= 31.6 mm
Amax	= 2.5 A

Measurements

Height (mm)	Voltage Force @ 1G
38	>12.5 (coil cannot lift screw)
37	12.5
36	12
35	9.8
34	8.1
33	6.7
32	5.7
31	5.2
30	4.6
29	4.2
28	4.05
27	3.6
26	3.3
25	2.9
24	2.8
23	2.5
22	2.3
21	2.35
20	2.2
19	2
18	2
17	2.05
16	2
15	1.95
14	1.92
13	1.85
12	1.87
11	1.9
10	1.9
9	1.9
8	1.9
7	1.84
6	1.85
5	2
4	2.1
3	2.12
2	2.35
1	2.5
0	2.5
-1	2.8
-2	3.1
-3	4.5