Magnetic circuits problems and solutions
The first topic of this course is actually a continuation of magnetostatics from EE 381 last semester. This topic is magnetic circuit analysis and it’s a lumped-element method for solving certain types of magnetic field problems. Magnetic circuit analysis is not always applicable, but when it is it can greatly simplify the solution to magnetostatic problems. This process is similar to the simplification of electric circuits using R, L, and C lumped elements to represent the effects of actual devices with physical dimensions. To illustrate magnetic circuit analysis, consider the toroid:
This problem is similar to toroid problems you have encountered earlier in EE 381. There is one difference though; the turns of wire exist only over a portion of the toroid.
However, with m >> m _{0} , the B field will largely stay within the toroid. Here we will ignore all such flux leakage. Then, by Ampère’s law
j H.dl=NI
By symmetry,
We will assume that this toroid has a small cross section such that Bj is nearly uniform over the cross section. Therefore
Now, the magnetic flux is
where A = w^{2} is the cross-sectional area. Therefore, with (2) in (3)
We now develop the equivalent magnetic circuit for this toroid using this last equation by defining
where
. V_{m} = (NI=x)is the source. Called an mmf or “magnetomotive force,”
. The reluctance of the core (units of H-1).
where l = 2pa is the mean length. The equivalent magnetic circuit for the toroid can be drawn as
There is a direct analogy between electrical and magnetic circuits.
Analogous quantities in electrical and magnetic circuit analysis
Example N1.1: Determine the magnetic flux through the air gap in the geometry shown below. The structure is assumed to have a square cross section of area 10^{-6} m^{2}, a core with m_{r} = 1,000, and dimensions l_{1} = 1 cm, l_{3} = 3 cm, and l_{4} = 2 cm.
So how do we solve such a problem? Can we use Ampère’s law
No, we cannot because there isn’t sufficient geometrical symmetry for us to solve for H using Ampère’s law. The assumptions inherent in magnetic circuits allow us to find an approximate solution, however. A drawback to magnetic circuit analysis is that generally we can’t check our solutions with a simple analytical formula because there isn’t one. Usually our only recourse to check the accuracy of our magnetic circuit solutions is to use a computational magnetostatics tool, which can compute H everywhere in space.
A distinguishing characteristic of this problem is the air gap. We’ll assume
.The length of the gap is small with respect to the cross section, thus
.No “flux fringing” effects. Assumptions:
1. No “flux leakage”: B remains entirely within the magnetic material and the small air gap.
2. No “flux fringing”: B remains vertical in the air gap in this problem:
With no “flux fringing,” then the air gap can be modeled as another reluctance in series with R_{2} and shown in the equivalent magnetic circuit above. Compute the four reluctances using
The magnetic flux through the source coil is the mmf divided by the total reluctance seen by the source:
Using “flux division” (analogous to current division), then
A practical application of this problem could be to find B in the air gap, for example. This can be determined from y m _{2}and the no flux fringing and no flux leakage assumptions