Solenoid Force Calculation Questions 1

[Turbinetester]

I am trying to calculate the force a solenoid will develop using the following equation:

F = (N*I)2 μ0 A / (2 g2)

Where:
μ0 = 4π×10-7
F is the force in Newtons
N is the number of turns
I is the current in Amps
A is the area in length units squared
g is the length of the gap between the solenoid and a piece of metal.
Note, any units can be used for A and g so long as they are consistent. For example, in2 and in, or m2 and m, respectively.

My question is what geometry the area (A) and gap (g) represent?

Does (A) represent the cross sectional area of the iron core or the coil? Is it the area of the round face or the rectangular cross section?

Does(g) represent the gap between the outside radius of the iron core and the inside radius of the coil or something else?

Thanks

 

[LiteYear]

Where does the equation come from? All such equations are based on some assumptions. It looks like you might be trying to use it in a scenario where those assumptions don't apply, so the accuracy could well plummet.

If you compare your formula to the energy equation F = (I^2/2) * dL/dx, then your μ0 A / (g2) term is the inductance derivative. That appears to be the formulation of a linear reluctance circuit, so your A and g terms relate to the magnetic path. A is therefore the area of the iron core, since very little flux flows outside the core. If the cross section changes, then it will be some complex average of the different areas, depending on their length and the flux path. The g is the length of the flux path which is not within the core. Have a look at the geometry of your solenoid and measure the flux path length which bridges the two ends of the core. In the case of a typical cylindrical core, this could be a long complex path that bends out from one in around into the other.

 

[Turbinetester]

Thanks LiteYear

The solenoid is basically a simple coil of wire wrapped around a plastic bobbin with a round steel cylinder that gets pulled up inside the hollow bobbin when the coil is energized.

Would the area (A) be across the cylinder diameter (pi*r^2) or (diameter * length)?

I don't follow what you are talking about for the gap (g)

Basically, I have tried different inputs for both (A) and (g) and my numbers are radically different than what I measure the actual solenoid force to be.

 

[LiteYear]

Your solenoid does sound quite non-ideal which could explain the variation from calculation. The geometry you describe means that the majority flux path within the coil will be in the solid cylinder. But at each end, the flux will stray out in an uncontrolled manner, trying to complete the loop back to the cylinder. Something like pictured here:

http://www.cyberphysics.co.uk/topics/magnetsm/Magnetic Fields A2/Solenoid.gif

The cross sectional area of the flux path within the cylinder is easy - since the flux is roughly uniform, it will be pi*r^2. But outside the cylinder it's very hard to predict. So the effective cross-sectional area of the whole path is very difficult to calculate. In fact, the "gap" length is the whole way from one end of the cylinder to the other, but because the flux is not well contained, the formula is not very useful.

Typically, solenoids have an arrangement more like pictured here:

http://www.consult-g2.com/course/chapter11/f11.14.jpg

In that case, the flux is mostly contained partly by the plunger and partly by the return path through the solenoid body. The average cross section of this path is fairly easy to calculate. The gap length is then made up of the small gap between the body and the cylinder wall (marked "plunger wall sliding region" in the diagram) and the changing gap between the centre pin and the plunger. Since the flux is fairly well contained along the entire path (for short plunger extensions), the formula will work quite well.

If you have the appetite for it, you might be better off using a change in energy formula based on F=dW/dx, such as described in this thread:

http://www.eng-tips.com/viewthread.cfm?qid=365686.

If you have access to the numbers for the electrical energy going into the coil, this may work out better for you.