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IEEF T R A N S A C T I O N S O N MAGNETICS. VOL 32, N O 3. M A Y 1996
A 3D Impedance Calculation for an Induction Heating System for Materials with Poor Conductivity S. M. Mimoune, J. Fouladgar, A. Chentoufand G. Develey GE44 / LRTI / C R n , B.P. 406 - 44602 Saint Nazaire Cedex, France Absrrad-A study o f three-dimensional magneto-thermal finite element phenomena is presented. The (AV-A) formulation usingthe Impedance Bdundary Condition @BC) is applied to multiply connected regions. From the exterior inductor voltage and the current density distribution, the impedance and the total power o f the induction heating devices are calculated.
I. INTRODUCTION
(inductor)
\
Provided that the magnetic properties of the are linear, the f ~ l l3d (A,v-A) formulation with the pedty term in the harmonic case is 121 : curt (v curl A) - grad (vdiv A) +(o+joE) (jwA +grad
v> = 0 (1)
div[-(o+ jm)(joA +gradV)] = 0
In the induction heating devices, the theoretical determination of inductor-load characteristics (impedance) presents some difficulties in mean and high frequencies. The electromagnetic source in this case is the generator voltage, U, applied to the inductor. This leads to a non homogenous concentration of the current in the skin depth, due to the proximity effects resultant from the interaction between inductor coils and heating load [ 11. In our case the device consists of a transformer in which the secondary is replaced by a single turn torus shaped vessel containing a material with poor conductivity, such as salted liquids, acids or plasma (Fig. 1). The system consists of a conducting region with non zero conductivity ( transformer primary and secondary ) surrounded by a non conducting region containing the air and a magnetic material ( transformer core ). It presents a three dimensional and multiply connected aspect, due to the torus shape of the primary and the secondary. In this case (A,V-A) finite element formulation may be used (21. In our previous work, a current driven model was studied [3]. In this paper a voltage driven model is presented. To avoid the costly volume meshing in the skin depth, the (?Be) is used at the surface of the primary [41,[51. transformer primary
11. MAGNETIC FORMULATION
where o is the electric conductivity, v is the magnetic reluctivity and E is the permitivity. The displacement current is negligible inside the conductors but it is introduced in non conducting region for the calculation of the secondw electrical potential. The weak formulation with V = jov leads to :
S,
( v cu rt N, .curl A
+ v d i v ~d, i v ~
+ jo(cT+ jo&)Ni.(A+gradv))dQ - jNi.(vcudA x n)dF
rlBc
I
- jNi.ndivAfl =0 rl, jo(o + jos) gradNi .(A +gradv)dn
n
+
Ijo(cr+jws)N;(-A-gradv’).ndF = O r1BC
the boundary conditions are :
transformer secondary
/
A = 0, V = 0
at the infinite :
(3)
and on the surface of inductor FE, where KBC is used : 1
n x H = --(n Z
x
1
E ) x n = --Es Z
(4)
where Es is the tangential component of the electric field. In the (A,v-A) formulation, (4)leads to : Fig. 1. Schematic view of the device Manuscript received July 10, 1995. S . M. Mimoune, A Chentouf, fax (33) 40.17.26.18 phone: (33) 40.17.26.36; J. Fouladgar, e-mail f o u l a d g ~ ~ l r t i . n t t s n . u n i v - phone n ~ ~ . (33) ~ , 40.17.26.38; G. Develey e-mail develey@,lrti.rrasn.univ-~~.~, fax: (33) 40.17.26.18.
vnxcurfA=V(nx[A+gradv’I)xa A
In the air side of inductor, o = 0
00 18-9464/96$05 .OO 0 1996 IEEE
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(5)
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The impedance of the installation at the generator side is then calculated by : Taking the scalar product of (6) with the normal vector, U, gives : n.curlH= josE, (7)
U = (R +jwL).I
(13)
1- j IJ.dl 2
(14)
= IF applying the identity of the vector analysis,
*lX
div(Hx n) = n.curlH-H.curln
(8)
to (7), with regarding curl n = 0 and the continuity of tangential component of H, (6) leads to
J =-oE=-o[jo(A+gradv)+-i,]U 2xr
(15)
where I, J, R,L, IN are respectively, the total current, the current density, the resistance, the inductance and the contour of the inductor. 111. THERMAL FORMULATION
where n is the normal vector pointing inward the inductor, A is the complex thickness, Z is the surface impedance and 6 is the skin depth.
For the computation of the temperature profile in the secondary, one should solve the coupled magneto-thermal equations. The thermal equation is : pCp v.gradT= div(KgradT) +Qm -QMy
J ~J .~ * 20 J, = -joo(A +gradv) QTH =-
As (2), (3) and (5) are not sufficient for uniqueness of the solution, the normal component of the vector potential, n.A must be zero and then the second surface term in ( 2 ) disappears [ 5 ] . With these formulations the resulting system is symmetric. As the inductor has a circular form the gradv‘ has an important azimuthal component, we divide it then into :
U jwgradv’ = -i, 2nr
+ jwgradv
( 1 1)
where 6 is the azimuthal vector. This decomposition improve the convergence of the algorithm. The weak formulation (2) can be written as follows :
5,
where p, Cp, K, Qm, QRAYandJs are respectively, the mass density, specific heat at constant pressure, thermal conductivity, local electromagnetic power dissipation, volumetric radiation heat losses and current density. The secondary contains a plasma of argon with o,p, Cp, K, Qm and QRAy dependent strongly on temperature [SI. The coupling algorithm of the electromagnetic and thermal equations is presented in Fig. 2. It consists of an iterative algorithm based on the bloc diagonal Gauss-Seidel method. It permits to separate the symmetric and linear magnetic equation from the non symmetric and non linear thermal one [3].
( v curl N f . curl A + v d i v ~d i v ~ I
+Jo(o+jo~)N .(A+gradv))dn ~
+
I I- IBC
N i . ~ ( n x [ A + g r a d v ] ) x n U= A
1j w ( o +jm)grad N .(A +grad v) dR + 1 -gradNi.(nx[A+gradv))xndT A i
R
V
= 0
r- iE€
Fig. 2. Coupling algorithm
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Iv. RESULTS 1
I
A . Magnetic Formulation The magnetic formulation is applied at two axisymmetric designs. The 1/8 of the geometry is taken into account for both examples. The first one represented in Fig. 3 is a circular inductor of one turn [6], fed by an arbitrary voltage U with a frequency of 100 kHz and an electric conductivity equal to 58.106 S/m. The second one shown in Fig. 4 is a load with an electric conductivity equal to 1.106 S/m and an inductor with an electric conductivity equal to 58. lo6 S/m at a frequency of 10 kâ‚Ź-k 171. Fig. 6 and Fig. 7 present the variation of the real and imaginary electric field along the ABCD and OF paths of the Fig. 5 .
f
0.w
ah
0.02
0.01
ABCD path (m)
Fig. 6. Real and imaginary componentsof electric field along ABCD path example 2.
i � e t u c ~
F i g 3. Example 1 : circular inductor (unit in mm). Fig. 7. Real and imaginary components of electric field along OF path example 2. k
The oscillation shown in Fig. 7 is due to the difference of two large values of the same order. With these oscillations the total current in (14) can not be calculated with accuracy. An alternative method is to calculate the mean value of J on the surface of the inductor and evaluate then the total current by :
Fig. 4. Example 2 : load + inductor (unit in mm).
TABLE I COMPARISON
BEWEENCALCULATED AND MEASUREDIMPEDANCE
-
~~
Exmole 1 161
Resistance (mQ) Inductance (pH) Example 2 [71 Resistance (mQ)
Inductance (pH)
-~
calculated (3d)
measured
7.24 1.21
5.6
calculated (3d) 0.823 0.0716
calculated (aX;) 0.925 0.081
Fig. 5 . 1/ 4 of 3d geometry of the inductor in example 2.
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1.349
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The results obtained by this method are not very far from the experimental ones as is shown in Table I.
T (1 0‘ “K) 1.12
B. Magneto-Thermal Formulation 8.84
.
/
For the magneto-thermal equation the study was made with an induction heating system at 10 kHz (Fig. 1). The transformer primary is supplied by a voltage of 200 V. Fig. 8 and Fig. 9 represent respectively a regular distribution of the current density Js and the temperature profile in the secondary.
V. CONCLUSION At mean frequencies, the (A,V-A) formulation is suitable for materials with poor conductivity. Associated with the (IBC) in the high conducting regions, it can be used for the problems with voltage driven coils. Some difficulties appear in the calculation of current distribution at the surface of the inductor. Some studies are carried out to improve the model by using the modified (IBC), the edge elements or first order elements.
Distance 6om the inner to the outer radius at the middle level of the tore section (lO-’m)
Fig. 9. Temperature profile in the cross section ofthe secondary REFERENCES
[ 11 J. Fouladgar and A Chentouf, “The Calculation of the Impedance of an
[2]
[3] [4]
[5]
[6] [7]
[8] Fig. 8. Current density field in the secondary
Induction Plasma Installation by a Hybrid Finite-Element BounQryElement Method,” IEEE Transachon on Magnehcs, vol. 29, pp. 24792481, November 1993. 0. Biro and IC Preis, “On the Use of the Magnetic Vector Potential in the Finite Element Analysis of Three-Dimensional Eddy Currents,” IEEE Transachon on Magnetics, vol. 25, pp. 3145-3 195, July 1989. S. M. Mimoune, J. Fouladgar and G. Develey, “Modeling of 3D Electromagnetic and Heat Transfer Phenomena for Materials with Poor Conductivity.” IEEE Transaction on Magnetics, September, 1995, in press. M. Gyimesi, D. Lavers. T. Pawlak and D. Ostergaard, “Impedance Boundary Condition for Multiply Connected Domains with Exterior Circuit Conditions” IEEE Transaction on Magnehcs, vol. 30, pp. 30563059, Szptember 1994. M. Gyimesi, D. Lavers, T. Pawlak and D. Ostergaard, “Vector Potential Formulation for the Impedance Boundary Condition,” Presented in Budqxst on the III. Hungarian-Japan seminar on electromagnetic jelds, July, 1994. D. Delage and R Emst “Prididion de la r6partition du courant dans un inducteur a symetrie de rivolution destini au chauffage par induction MF et HF,”Revue Generale de 1‘Electricit&,pp. 225-230, Avril, 1984. A ChentouS “Contribution a la modilisation electrique, magnetique et thermique d’un applicateur de plasma indudif haute fiiquence,” These de 1 ‘Universite de Nantes. ler Decembre, 1994, France. M. I. Boules, P. Fauchais and E. Pfender, Thermalflasma. FundamentaZs andApplications, Vol. 1, Plenum Press, New York, pp. 385-448,1994.
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