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 -=-
222
CHAPTER 8 Eddy-current Problems
1/(36~
x 109),
the ratio ¢¢0/~ drops to about 5
x 10 -16,
and one cannot
seriously object to the neglect of - iO~D. Much higher in the spectrum, in
simulations related to some medical practices such as hyperthermia [GF],
where frequencies are on the order of 10 to 50 MHz, the conductivity of
tissues on the order of 0.1 to 1, and with ¢ ~ 10 to 90 ¢0 [K&], one still
gets a ratio ¢c0 /r~ lower than 0.3, and the low-frequency approximation
may still be acceptable, depending on the intended use. "Low frequency"
is a very relative concept.
f
O
V
FIGURE 8.3. A case where capacitive effects may not be negligible (d << L).
But there are circumstances in which the electric field outside
conductors is far larger than inside, and these are as many special cases.
Consider Fig. 8.3, for instance, where L is the length of the loop and d
the width of the gap. A simple computation (based on the relation
V - diE I) shows that the ratio of ¢O~E
in the gap
to the current density
in
the conductor
is on the order of ¢ coL/dG and thus may cease to be
negligible when the ratio L/d gets large. This simply amounts to
saying that the
capacitance
C of this gap, which is in ¢/d, cannot be
ignored in the computation when its product by the
resistance
R, which
is in L/o, reaches the order of c0 -1 (recall that RC, whose dimension is
that of a time interval, is precisely the time constant of a circuit of resistance
R and capacitance C). One can assert in general that dropping iCOD
from the equations amounts to neglecting
capacitive effects.
This is a
legitimate approximation when the energy of the electromagnetic field is
mainly stored in the "magnetic" compartment, as opposed to the "electric"
one, in the language of Chapter 1.
One then has, instead of (7), rot H
-- J.
Consequently, div j = 0, and if
the supports of
Jg
and G are disjoint, one must assume div
Jg -- 0,
after (11 ).
To sum up, we are interested in the family of problems that Fig. 8.4
depicts: a bounded conductor, connected, a given harmonic current,
8.1 THE MODEL IN H 223
with bounded support, not encountering the conductor. (The last
hypothesis is sensible in view of Fig. 8.1, but is not valid in all conceivable
situations.) The objective is to determine the field H, from which the
eddy currents of density j = rot H in the conductor will be derived.
S /ff~
/
\. C //
/
supp( J g )
FIGURE 8.4. The theoretical situation. Note the convention about the normal
unitary field on S, here taken as outgoing with respect to the "outer domain" O =
E 3 -C.
From the mathematical point of view now, let thus C be a bounded
domain of space, S its surface (Fig. 8.4), and jg a given complex-valued
field, such that supp(j g) r~ C = ~, and div
Jg =
0. Again, we denote by O
the domain which complements C u S, that is to say, the topological
interior of E 3 - C (take notice that O contains supp(jg)). Domains C
and O have S as common boundary, and the field of normals to S is
taken as outgoing with respect to O. Conductivity o and permeability
being given, with supp(o) = C, and o 1 >_ o(x) > o 0 > 0 on C, where ~0
and c~ 1 are two constants, as well as ~t(x) _> G on C and ~t(x) = ~t 0 in
O, one looks for H
E ILarot(E3 ),
complex-valued, such that
(12) ic0~t
H 4-
rot E = 0, J
= Jg 4- (~Er
rot
H -- J.
8.1.3
The problem in., in the harmonic regime
Let us set, as we did up to now, IH
= IL2rot(E3)
(complex), and
IH g=
{H E IH" rot H
-- Jg
in O},
IH°= {H ~ IH" rot H
=
0 in O}.
We shall look for H in IH g. One has
IH g =
H g + IH °, with, as in
magnetostatics (cf. the h i of Chapter 7),
H g=
rot A g where
224 CHAPTER 8 Eddy-current Problems
1 f~ jd(y)
Ag(x)= ~~3 Ix-yl
dy.
It all goes as if the source of the field was H g, that is, the magnetic field
that wouldsettle in the presence of the inductor alone in space. The
difference H = H- H g between effective field and source field is called
reaction field.
Let us seek a weak formulation. From the first Eq. (12), and using
the curl integration by parts formula, one has
0 = SE3 (i03~t H + rot E). H' =
i03 ~E 3 ~1, H. H' q- SE3 E.
rot
H' V H'E
IH °.
As
rot H'= 0
outside
C (this is the key point), one may eliminate E by
using the other two equations (12): for E = C~-10 - Jg) = r~-lJ in C, and thus
E = O-a rot H. We finally arrive at the following prescription:
find
H E IH g
such that
(13)
SE3i03~tH.H'+~Cr~-lrotH.rOtH'=0 VH'~IH °.
Proposition 8.1.
If
H g E IL2rot(E3 ),
problem
(13)
has a unique solution
H,
and
the mapping
H g---) H
is continuous from
IL2(E3 )
into
IH.
N
Proof.
Let us look for H in the form H
+ H g.
After multiplication of both
sides by 1 - i, the problem (13) takes the form a(H, H') = L(H'), where L
is continuous on IH, and
a(H, H') -- YEs CO I.L H. H' q- SC (~-1
rot H. rot
H'
+
i (SE3
03 , H. H' -- fC (~-1
rot H. rot H').
As one sees, Re[a(H, H*) > C (S~
I H12 q- S C
Irot HI 2) for some positive
constant C. This is the property of
coercivity
on II-I ° under which
Lax-Milgram's lemma of A.4.3 applies, in the complex case, hence the
result.
Remark 8.1. Problem (13) amounts to looking for the point of stationarity
("critical point") of the complex quantity
Z (H) =
i03 SE B
, H 2 q- SC (~-1
(rot
H) 2,
when H spans IH g. (There is a tight relationship between z and what is
called the
impedance
of the system.) So it's not a variational problem in
the strict sense, and the minimization approach of former chapters is no
longer available. By contrast, this emphasizes the importance of
Lax-Milgram's lemma. 0
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.