1 Introduction - Finite Element Method Magnetics

2 Overview
The goal of this section is to give the user a brief description of the problems that FEMM solves.
This information is not really crucial if you are not particularly interested in the approach that
FEMM takes to formulating the problems. You can skip most of Overview, but take a look at
Section 2.2. This section contains some important pointers about assigning enough boundary
conditions to get a solvable problem.
Some familiarity with magnetism and Maxwell’s equations is assumed, since a review of this
material is beyond the scope of this manual. However, the author has found several references that
have proved useful in understanding the derivation and solution of Maxwell’s equations in various
situations. A very good introductory-level text is Plonus’s Applied electromagnetics [1]. A good
intermediate-level review of Maxwell’s equations, as well as a useful analogy of magnetism to
similar problems in other disciplines is contained in Hoole’s Computer-aided analysis and design
of electromagnetic devices [2]. For an advanced treatment, the reader has no recourse but to refer
to Jackson’s Classical electrodynamics [3].
2.1 Relevant Maxwell’s Equations
For the low-frequency problems addressed by FEMM, only a subset of Maxwell’s equations are required.
By definition, low-frequency problems are those problems in which displacement currents
can be ignored. Displacement currents are typically relevant only at radio frequencies.
2.1.1 Magnetostatic Problems
Magnetostatic problems are problems in which the fields are time-invariant. In this case, the field
intensity (H) and flux density (B) must obey:
∇ × H = J (1)
∇ · B = 0 (2)
subject to a constitutive relationship between B and H for each material:
B = µH (3)
If a material is nonlinear (e.g. saturating iron or alnico magnets), the permeability, µ is actually a
function of B:
µ= B
H(B)
(4)
FEMM goes about finding a field that satisfies (1)-(3) via a magnetic vector potential approach.
Flux density is written in terms of the vector potential, A, as:
B = ∇ × A (5)
Now, this definition of B always satisfies (2). Then, (1) can be rewritten as:
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∇ × ∇ × A = J (6)
µ(B)
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