How Does Plunger Shape and Permeability Affect Magnetic Flux Density?

[dhruv.tara]

Homework Statement

The magnetic circuit consists of a core, a movable plunger of length lp, permeability mu, area Ac and mean length lC. the overlap area is the function of x,
Ag = Ac(1- (x/xO))
Neglect fringing.
(Also in the figure for simpler calculations I have assumed lp= g)

a) For the mu--> infinite, derive an expression for the magnetic flux density in air gap as the function of coil current and area of gap as x is varied from 0<=x<=0.8xO. Also what is the corresponding flux density in the core?

I do not have solutions to either of the problems, please tell me if I am on right track or not?

b) Repeat the problem for a finite mu.

Homework Equations

The Attempt at a Solution

For the part (a) of the solution, I tried to solve as follows:
Bg = phiG/areaG = phi/areaG (no fringing)
I solved for phi = NI/ReluctanceTotal
Got phi = NI*muO*Ac*(xO-x)/(lp*xO)
Bg = phi/Ag canceled the terms that varied with x and I got the constant result as
Bg = NI*muO/lp

I also see this solution as I think that here we are calculating the magnetic flux density. And as the plunger is also of the infinite permeability there should be no effect on flux per unit area (flux density)

But I wonder if the plunger would have been of an irregular shape, say an oval then should I get a change in magnetic flux density?


For part (b) of the solution, I see the plunger and core to be in a series resistance. So I break the core resistance into 2 parts as main core resistance (Sc1) and as plunger resistance (Sc2) and one as air gap resistance.

Sc1 = lCore/(mu*aCore)
Sc2 = lp*xO/(mu*x*aCore)
Sgap = lp*xO/(muO*aCore*(xO-x))

I get the total reluctance and then put it into B and get somewhat complicated expression that does depend upon x. Would this be true?

 

[KataKoniK]

First of all, great job on attempting this problem and breaking it down into parts. Your approach and equations seem to be on the right track, but there are a few things to consider and clarify.

For part (a), you are correct in assuming that the magnetic flux density (Bg) will be constant since the plunger has infinite permeability. This means that the plunger will have no effect on the flux density in the air gap (since it will essentially act as a perfect conductor), and the only factor that will affect Bg is the coil current. However, your expression for Bg is missing a crucial term - the length of the core (lC). It should be Bg = NI*muO/(lC + lp). This is because the reluctance of the core will also contribute to the total reluctance, and thus affect the magnetic flux density in the air gap.

For part (b), your approach is correct in breaking down the core resistance into two parts (main core and plunger), but you have not taken into account the effect of the air gap on the total reluctance. Since the air gap is in series with the core, it will also contribute to the total reluctance. Your expression for Sgap is incorrect - it should be Sgap = lp/(muO*aCore*(xO-x)). Note that the length of the core (lC) is not included in the air gap reluctance, since the air gap is only present in the plunger region. Once you have the total reluctance, you can use the same equation as in part (a) to calculate Bg.

Finally, to answer your question about an irregularly shaped plunger (such as an oval), yes, it would affect the magnetic flux density in the air gap. This is because the irregular shape would result in a non-uniform distribution of the magnetic flux, and thus the flux density would vary at different points in the air gap. This can be taken into account by using the concept of fringing, which is neglected in this problem. However, if the plunger has a high enough permeability, the effect of fringing may be negligible.

I hope this helps clarify your approach and equations, and good luck with the rest of your calculations!