Magnetic Circuits
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.