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# Where is the force exerted on a core the highest in a solenoid?

## Where is the force exerted on a core the highest in a solenoid?

(OP)
Hi, I am currently trying to make a solenoid of 2mm length, 2mm outside diameter and .85mm inside diameter with a permanent magnet core of diameter .75mm. Being so small in nature, everything has to be optimized for the greatest force. The solenoid should "push" the core out with the same amount of force that it "pulls" the core back in; what initial position would have the most force for both pushing and pulling the core. Like, should the end of the core start at 1/4 the length of the solenoid, from the tip? Thank You

### RE: Where is the force exerted on a core the highest in a solenoid?

That depends a lot on many parameters. The easy way is to just measure the force.

The "scientific" way is to calculate energy vs position, graph the function and then select the position where the graph is the steepest. That is also where the pull is the strongest.

Gunnar Englund
www.gke.org
--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

### RE: Where is the force exerted on a core the highest in a solenoid?

(OP)
Thank You Shogsgurra
Is there any other way to theoretically get the same result without testing and graphinh? I do not have the right equipment right now...
Kevin

### RE: Where is the force exerted on a core the highest in a solenoid?

There are 3D simulators for electromagnetic simulations. They tend to be expensive and they also need some time to be learned and mastered. I hope that someone that has found a bargain simulator will chime in soon. I would also be interested.

Gunnar Englund
www.gke.org
--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

### RE: Where is the force exerted on a core the highest in a solenoid?

The mathematical relationship is straightforward: F = (I^2/2) * dL/dx

Unfortunately the relationship of L to x is not straightforward, which is where Skogs' simulation suggestion comes in. One possible result from [1] is that dL/dx = -b/(a+bx)^2 for some constants a and b. Obviously this is maximised for x=0, which is the case for maximum insertion of the core in the solenoid. Even if you make that assumption however, you have the problem of trying to rapidly increase I, coincidentally at the moment L is also maximised. I think this is why you'll see a lot of literature on solenoid drivers, which are designed to pump I up as quickly as possible in the presence of a rapidly changing L.

So in summary, greatest force for pushing or pulling is probably possible when the core is fully inserted. Unfortunately that's also the point at which it's hardest to increase I, but you don't get something for nothing.

[1] S.E. Lyshevski. Electromechanical Systems, Electric Machines, and Applied Mechatronics. CRC, 1999.

### RE: Where is the force exerted on a core the highest in a solenoid?

LY and kevinwantstolearn
As LY says, the force follows F = (I^2/2) * dL/dx - which is the same as F = dW/dx, where W is the magnetic energy in the coil/core combination. Since the magnetic energy is the highest when the core is fully inserted in the coil, it follows that dW/dx goes to zero (derivative=0 always means a maximum or a minimum) and hence the force, F, is zero when the core is in that position.

I still think that the simplest way is to plot force (you do not need to measure with absolute accuracy, all you need to do is measure relative force, so even a simple spring-and-scale arrangement will work) vs x and find the steepest derivative.

I realize that the very small dimensions you are working with create a few problems.

Another method would be to use a smart scope with arithmetic possibilities and do the L and I2 calculation. I have used the TiePie USB scopes for such calculations. Very easy to use and available at around USD 1500.

And, finally, a simulator would probably be your best friend. There, I yet have to find something that is easy to master and at a good price.

Gunnar Englund
www.gke.org
--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

### RE: Where is the force exerted on a core the highest in a solenoid?

Sorry, I just realized that you are using a permanent magnet core. That changes a few things. The maximum force may then very well coincide with core at center position.

Gunnar Englund
www.gke.org
--------------------------------------
Half full - Half empty? I don't mind. It's what in it that counts.

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