220
CHAPTER 8 Eddy-current Problems
8.1 THE MODEL IN H
The model under study in the present chapter is further characterized by
a few simplifications, the most noteworthy being the neglect, for reasons
we now indicate, of the term -i~D in (7).
8.1.1 A typical problem
Figure 8.1 ([Na], pp. 209-247) shows a case study, typical of computations
one may have to do when designing induction heating systems, among
other examples: An induction coil, fed with alternative current, induces
currents (called "eddy currents" or, in many national traditions, "Foucault
currents") in an aluminum plate, and one wants to compute them.
J / Y .... j~
I
FIGURE 8.1. The real situation ("Problem 7" of the TEAM Workshop [Na]): compute
eddy currents induced in the "passive conductor" C by an inductive coil, or
"active conductor", which carries low-frequency alternating current. (The coil has
many more loops than represented here, and occupies volume I of Fig. 8.2
below.) The problem is genuinely three-dimensional (no meaningful 2D modelling).
Although the pieces are in minimal number and of simple shape, and the constitutive
laws all linear, it's only during the 1980s that computations of similar complexity
became commonplace.
Computing the field inside the coil, while taking its fine structure
into account, is a pratical impossiblity, but is also unnecessary, so one
can replace the situation of Fig. 8.1 by the following, idealized one, where
the inducting current density is supposed to be given in some region I,
of the same shape as the coil (Fig. 8.2).
This equivalent distribution of currents in I is easily computed,
hence the source current
Jg
of (11), with I as its support. (One takes as
given a mean current density, with small scale spatial variations averaged
out.) If {E, H} is the solution of (7-11), H is then a correct approximation
to the actual magnetic field (inside I, as well), because the same currents
8.1 THE MODEL IN H 221
are at stake (up to small variations near I) in both situations. However,
the field E, as given by the same equations, has not much to do with the
actual electric field, since in particular the way the coil is linked to the
power supply is not considered. (There is, for instance, a high electric
field between the connections, in the immediate vicinity of point P of
Fig. 8.1, a fact which of course cannot be discovered by solving the problem
of Fig. 8.2.)
r~= I I j
I
\
FIGURE
8.2. The modelled imaginary situation: Subregion I (for "inductor") is
the support of a known alternative current, above a conductive plate C.
These considerations explain why emphasis will lie, in what follows,
on the magnetic field H, and not on E (which we shall rapidly eliminate
from the equations).
8.1.2 Dropping the displacement-currents term
Let us now introduce the main simplification, often described as the
"low-frequency approximation", which consists in neglecting the term of
"Maxwell displacement currents", that is ico D, in (7). By rewriting (7),
(9), and (10) in the form rot H
--
Jg + (~E 4- ico ~E, one sees that this term is
negligible in the conductor inasmuch as the ratio ¢co/r~ can be considered
small. In the air, where o = 0, everything goes as if these displacement
currents were added to the source-current jg, and the approximation is
justified if the ratio
llicoDli/liJgll
is small (11 II being some convenient
norm). In many cases, the induced currents j = o E is of the same order
of magnitude as the source current jg and the electric field is of the same
order of magnitude outside and inside the conductor. If so, the ratio of
icoD to
Jg
is also on the order of ~co/c~.
The magnitude of the ratio ~co / c~ is
thus often a good indicator of the validity of the tow-frequency approximation.
In the case of induction heating at industrial frequencies, for instance co
= 100 ~, the magnitude of c~ being about 5
X 10 6,
and ¢ = ¢0 -=-

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