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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