# Peak forces of round and rectangular solenoids

1. Jan 9, 2012

### jschwartz6

I am doing some research involving solenoids of varying aspect ratios, and I'm looking for some equations I can use to back it up. I have three different solenoids of the equal n (where n=N/L), with different shapes and different cores/plungers, and I am comparing their forces by testing them with the same current through the wire. Ideally the aspect ratio of the plungers would be my only variable, but there are other variables that I would like to compensate for by using equations.

First, as I understand it, the flux density in a round solenoid is basically
B = μnI.
One of my solenoids (square plunger) is about 9300 turns/m with plunger area 1E-4 m$^{2}$ running 1.5A. I'm using a mild steel core with a rough estimate of μ$_{r}$=50. By the above equation, the field from my solenoids is predicted about 0.88 T. Does that sound logical?

Also I have found a force equation that I don't know whether to believe:
F = B$^{2}$A/(2μ$_{0}$)
(the force exerted on the solenoid plunger). Does anyone have a correction for this? This equation is predicting ~28 N and I'm getting ~0.35 N from that particular solenoid. Having an accurate force equation would be a huge help.

I tested my solenoids on an Instron tester machine. The peak forces for the highest-aspect-ratio solenoid were highest, followed by my medium-aspect-ratio solenoid, and then the one with the square plunger was the least. I'm pretty sure this is because the higher-aspect-ratio solenoids have a larger length of wire per coil. So I think the field for these rectangular-core solenoids might be:
B = μnI + (2)μ$_{0}$I/(2πr)
where the latter term is the field about a straight length of wire, and each wrap of the coil has two lengths (one above and one below the plunger).
Then, for the force of the rectangular-core solenoids, I would add the term NILB to the regular force equation. Does that seem right?

Thanks for the help; sorry for the long post.

2. Jan 9, 2012

### elegysix

you say equal n, n = N/L, is this turn density? (N turns / L length)
The magnetic field in a solenoid depends directly on N, and not so much on L

What exactly do you mean by aspect ratio? n?

I haven't done much in applied magnetics, but I know you can use Biot-Savart law to calculate the magnetic field, B, at a point, due to a wire's configuration. It involves a line integral around the coil's cross section, and you can multiply that by N turns.

the flux is $\Phi$=BA or ∫B$\bullet$dA more precisely (B and dA vectors)

But, have you considered finding the current needed in each coil to produce a particular force?
it might help the experiment by avoiding some nonlinear effects from your secondary magnet - like hysteresis
Plus if the forces are equal, you automatically know that the flux from each coil must be equal. So you can just compare currents to find the most efficient configuration.

3. Jan 10, 2012

### jschwartz6

Yeah I am referring to turns per length.
By aspect ratio I am referring to the cross section of the solenoid, when looking at it from the end. A traditional solenoid is a circle but I have rectangles.

What is the secondary magnet you refer to? With the equations I have, I could indeed back-calculate from a given force to find a current, but that doesn't solve any of my problems, and I know that the equations do not hold for the different shapes of solenoids.

The forces are quite unequal, so the flux seems likewise (because the cross-sectional area of the cores, as well as n, are about the same).

4. Jan 13, 2012

### elegysix

I don't believe turn density is a big factor. I know that in basic physics the B field only depended on the number of turns, not the overall length. so you should be aware of that possibility in any equations with n.

the plunger is your secondary magnet - the force is applied on it. If its another coil with a current running through it, it will act just like a magnet. - secondary coil then.

And yes, if the forces are unequal, the flux will also be unequal. That is why if the forces are the same, the flux' are equal in the region of the plunger. Then you just have to record the current to get your Force vs current data. You can find a best fit curve and then compare it to whatever model you've got so far.

maybe post pictures of the solenoids? might help make it clearer for me