Modeling of Induction Heating in Oxide Czochralski

Modeling of Induction Heating in Oxide Czochralski SystemssAdvantages and Problems Mohammad Hossein Tavakoli* Physics Department, Bu-Ali Sina UniVersity, Hamedan 65174, Iran

CRYSTAL GROWTH & DESIGN 2008 VOL. 8, NO. 2 483–488

ReceiVed April 19, 2007; ReVised Manuscript ReceiVed October 29, 2007

ABSTRACT: Two mathematical models of induction heating for oxide Czochralski crystal growth systems are reviewed, and additional results of electromagnetic field and volumetric heat generation have been computed for both models using a finite element method (ENTWIFE package). In the first model, the eddy current in the RF coil (i.e., the self-inductance effect) has been neglected while for the second model, it is taken into account. For the calculations, the electrical current input and total voltage of induction coil are set to be 1000 A and 200 ν, respectively, with a frequency of 10 kHz; the heat generation in all metallic parts, that is, crucible, afterheater, RF coil, and chamber, has been calculated for a real Czochralski setup. It has been found that by including the self-inductance effect in the RF coil the results change dramatically. Also, it was shown that the spatial distribution of the heat generation in the crucible is more realistic in the second model than in the first model. Concerning the distribution of heat generation in all metallic parts (including the RF coil), the results for the first model are not reasonable, but they are acceptable for the second model.

1. Introduction Today, the method of radiofrequency induction heating plays an important role in the production of high-quality single crystals. This method is applied to generate the required thermal power in several crystal growth techniques such as the Czochralski (CZ) method, floating zone, and vapor growth (modified Lely method). Induction heating is a convenient method applying direct dissipation of electrical energy to melt a certain material. The induction heating equipment is relatively small in size and easy to use. Because the power goes directly into the heated metal, the process is clean, fast, repeatable, relatively efficient, and allows automatic control. In the oxide CZ system (Figure 1), this induction power is produced by an induction heater coil (copper) that surrounds the metal crucible (usually iridium or platinum). A highfrequency electric current (∼10 kHz) in this coil induces an eddy current distribution in the metallic parts, and the final product is the rate of energy dissipation in the cruciblesJoulean heating (I2R)sin the form of temporal and spatial volumetric heating. This internal heat generation is partly transferred into the melt via the inner wall and bottom of the crucible and partly lost to the surroundings via the insulation and gas flow. Therefore, it is clear that any modeling and simulation of the fluid flow and heat transport in the inductively heated crystal growth systems such as the CZ method cannot be achieved without understanding the nature and physics of this kind of heat generation. As a consequence, the modeling and calculation of the induction heating process play an important role for these numerical simulations. Up to now, there have been two mathematical models that have been applied for the calculation of induction heating in crystal growth systems. The first model was introduced by Gresho1 and has been applied by several researchers for the CZ method (Derby,2 Tsukada,3 Chen,4 and Tavakoli5–7). In this model, the coupling between driving electrical current and induced eddy current in the RF coil itself has been * To whom correspondence should be addressed. Tel: +98 811 8210424. Fax: +98 811 8280440. E-mail: [email protected].

Figure 1. Schematic diagram of the inductively heated CZ furnace.

ignored. The second model has been applied for modeling of the SiC growth during the modified Lely method (Klein8), in which the induced eddy current in the RF coil has been taken into account. Every model has some special advantages and disadvantages that I will explain hereafter. The aim of this article is to review both models of induction heating and compare the corresponding results for an oxide CZ system using a FEM numerical approach.

10.1021/cg070378+ CCC: $40.75  2008 American Chemical Society Published on Web 01/23/2008

484 Crystal Growth & Design, Vol. 8, No. 2, 2008

2. Mathematical Model For calculating an electromagnetic field, it is necessary to solve Maxwell’s equations (Leatherman9 and Rudnev10). The following assumptions are made in my numerical calculations: (i) The system is axisymmetric. (ii) All media are linear, isotropic, and stationary. (iii) All materials are nonmagnetic and have no net charge. (iv) The displacement current is neglected. (v) The distribution of electrical current (also voltage) in the coil is uniform. Under these assumptions, Maxwell’s equations in differential form and in the “mks” units (meter-kilogram-second-coulomb) can be written as: ∇ · E ) 0 (from Gauss’s law)

(1)

∇ · B ) 0 (from Gauss’s law)

(2)

∇ × B ) µJ (from Ampere’s law)

(3)

∇ × E ) -

∂B ∂t

(from Faraday’s law)

J ) σE

(4)

B ) ∇ × A

(6)

and assuming axi-symmetric condition, we can transform eqs 1–5 into a simple scalar equation (Gresho1)

( )

( )

∂ 1 ∂ψB ∂ 1 ∂ψB + ) - µJφ ∂r r ∂r ∂z r ∂z

(7)

in which ψB is the magnetic stream function defined by ψB(r, z, t) ≡ rAφ(r, z, t), where Aφ is the azimuthal component of A and (r, φ, z) are the cylindrical coordinates. The boundary conditions are ψB ) 0 for

{

r ) 0 (r, z) f ∞

(8)

In fact, the wall of a metallic chamber strongly reduces the electromagnetic fields, and so, we define out of the chamber as the limit (r, z) f ∞. 2.1. Model a (Neglecting the Eddy Current in the Induction Coil). Assuming that the self-inductance effect (i.e., eddy current) in the coil is negligible and Jcoil φ ) Jd ) J0 cos ωt as the driving current in it and Jconductor ) Je ) σcrEφ ) φ

( )

σcr ∂ψB r ∂t

in the conductors as the eddy current, we can find a solution of the form ψB(r, z, t) ) C(r, z) cos ωt + S(r, z) sin ωt

(9)

where C(r, z) is the in-phase component and S(r, z) is the outof-phase component of the solution. Now, the coupled set of elliptic PDEs for C(r, z) and S(r, z) is

(10)

0

∂ 1 ∂S ∂ 1 ∂S + ) ∂r r ∂r ∂z r ∂z

( )

( )

in the coil µcrσcrω C in the conductors r 0 elsewhere

(11) where σcr and µcr denote the electrical conductivity and magnetic permeability of conductors, that is, crucible, afterheater, and chamber, respectively, and µco is the magnetic permeability of the induction coil. The quantity δcr ) (2/µcrσcrω)1/2 has the dimension of a length and is called the skin depth. It is a measure of the field penetration depth into the conductors. After solving eqs 10 and 11 for C(r, z) and S(r, z), the energy dissipation rate can be computed via P(r, z, t) )

(5)

where E is the electric field intensity, B is the magnetic flux density, J is the free charge current density, µ is the magnetic permeability, and σ is the electrical conductivity of the medium, which is a nonzero value only in metallic parts. Introducing the vector potential A as

{ {

-µcoJ0 in the coil µcrσcrω S in the conductors r 0 elsewhere

∂ 1 ∂C ∂ 1 ∂C + ) ∂r r ∂r ∂z r ∂z

( )

Tavakoli

( )

J2φ σ ∂ψB ) 2 σ r ∂t

2

σω2 (S cos ωt - C sin ωt)2 r2 σω2 ) 2 (C2 + S2 - CS sin 2ωt) r Thus, the power is generated in all metallic parts (including the RF coil) as a function of 2ω. The period for this time dependence is τ ) 2π/ω ) 10-4 s for a 10 kHz induction system. Because the time-harmonic function is so short, representation of the heat generation by the time averaged quantity is more useful, which we average over one period to obtain the volumetric heat generation rate, ω 2π⁄ω q(r, z) ) P(r, z, t) dt 2π 0 σω2 2 ) (C + S2) (12) 2r2 This equation is valid for all metallic parts. 2.2. Model b (Considering the Eddy Current in the Induction Coil). If we include the self-inductance effect in the induction coil as an eddy current represented by Je ) -(σco/ r)(∂ψB/∂t), then the total current in this part will be σco ∂ψB Jcoil ) Jd + Je ) J0 cos ωt φ r ∂t σcoω σcoω S) cos ωt + C sin ωt ) (J0 r r where σco is the electrical conductivity of the RF coil. Consequently, the new set of coupled elliptic PDEs for C(r, z) and S(r, z) is ∂ 1 ∂C ∂ 1 ∂C + ) ∂r r ∂r ∂z r ∂z σcoω S in the coil -µco J0 r µcrσcrω (13) in the conductors S r 0 elsewhere σcoω C in the coil -µco r ∂ 1 ∂S ∂ 1 ∂S µcrσcrω + ) ∂r r ∂r ∂z r ∂z C in the conductors r 0 elsewhere (14) )

∫

{

( )

( )

( )

( )

)

(

{

Modeling of Induction Heating in Oxide Czochralski Systems

Crystal Growth & Design, Vol. 8, No. 2, 2008 485

Table 1. Values of Electrical Conductivity (mho/cm) Used in Our Calculationsa symbol

value

σco σcr σch

5.9 × 105 1.72 × 104 4.0 × 104

ref 6 2 1

a The subscripts co, cr, and ch denote coil (copper), crucible (iridium), and chamber (stainless steel), respectively.

Table 2. Operating Parameters Used for Calculations description (units)

symbol

value

crucible inner radius (mm) crucible thickness (mm) crucible inner height (mm) afterheater inner height (mm) afterheater hole (mm) coil inner radius (mm) coil thickness (mm) coil wall thickness (mm) height of coil turns (mm) distance between coil turns (mm) distance between two coils (mm) current frequency of RF coil (kHz) electrical current in RF coil (A)

rc lc hc haf raf rco lco sco hco dco Dco f Id

49 2 98 100 10 85 10 1 20 3 55 10 1000

By solving eqs 13 and 14 for C(r, z) and S(r, z), the rate of heat generation in the RF coil can be computed via Pco(r, z, t) ) σcoω2 2

r

[ ( C2 +

(Jd + Je)2 ) σco

) (

J0r -S σcoω

2

+ C

)

]

J0r - S sin 2ωt σcoω

and also the time average of this heat generation is qco(r, z) )

σcoω2

[ ( C2 +

)]

J0r - S σcoω

2

(15) 2r 2.3. Numerical Method. The set of fundamental equations with boundary conditions have been discretized in standard Galerkin formulation of the finite element method using the ENTWIFE package.10 A graded mesh of second-order finite elements was carefully designed to obtain a sufficiently accurate approximation of the solution. The discrete equations are solved using a Newton–Raphson method. Because the equations are linear, only one iteration is necessary. 2.4. Calculation Conditions. Values of electrical conductivity employed for our calculations are presented in Table 1, and operating parameters are listed in Table 2. The induction coil has two parts with the 6 and 1 hollow rectangular-shaped copper turns, respectively. In a real growth system, this coil is usually cooled very efficiently by water flowing inside the coil turns. Therefore, it is realistic to assume that the coil is always at room temperature. For the calculations, I have assumed in the coil an electrical current of 1000 A and total voltage of 200 V with a frequency of 10 kHz. For the magnetic permeability (µ), I assume that it is everywhere the constant value of free space µ ) µrµ0 = µ0 (i.e., µr = 1 where µr is the relative magnetic permeability). The results based on this set of parameters will be presented now. 2

3. Results and Discussion I computed the electromagnetic field and heat generation based on both models a and b in an oxide CZ furnace including crucible, active afterheater, induction coil, and chamber corre-

Figure 2. Components of the magnetic stream function (ψB) calculated from model a. The right side shows the in-phase component C (centered on the coil with Cmax ) 1.2 × 10-4 weber), and the left side shows the out-of-phase component S (located on the crucible wall and close to the bottom with Smax ) 8.6 × 10-6 weber).

sponding to an often used growth situation in the CZ dielectric laboratory at the Institute for Crystal Growth (IKZ) (Berlin, Germany). 3.1. Model a (Neglecting the Eddy Current in the Induction Coil). Figure 2 shows the contours of the in-phase component (right side) and the out-of-phase component (left side) of the magnetic stream function for the CZ system. The maximum value of the in-phase component (C) is located in the middle part of the main coil, and its value rapidly decreases toward the crucible wall, while the maximum value of the outof-phase component (S) is located on the crucible wall and close to the bottom corner. Although Cmax is about 13 times greater than Smax, the magnitudes of C and S0 are similar within the crucible and afterheater (also chamber), where the required heat generation occurs. It means that within the conductors both components (C and S) contribute to the heat generation about equally. The volumetric heat generation rate (q) in the crucible and afterheater wall is shown in Figure 3. The maximum value of crucible energy deposition in the crucible and afterheater is qmax ) 3 146 W/cm and is located at the crucible bottom edge. Figure 4 shows the profile of the generated heat in the outer surface of the crucible and afterheater side wall. It indicates that the heat is mostly generated in the crucible wall (especially in the lower part) and less in the afterheater. The total energy deposition crucible rate in the crucible and afterheater is Qtotal ) 5.73 kW. Within the induction coil, the maximum value of heat coil generated is qmax ) 2.1 × 104 W/cm3 and at the middle turns coil of the main coil, and the total heat generation is Qtotal ) 340 kW! Obviously, this value of heat generation in the RF coil is not correct, because the total input power for the induction coil is usually ∼10 kW and this heat generated is about 30 times more. This unrealistic result arises from the spatial distribution of the magnetic stream function components. In the coil, the magnitude of component C is ∼10-4 weber (the maximum of component C), while component S is ∼10-6 weber and so

486 Crystal Growth & Design, Vol. 8, No. 2, 2008 coil

q

=

σcoω2 2

C2

2r ∼104 W/cm3 therefore the magnitude of power generated in the coil is too high. In other words, this distribution of components C and S in the RF coil (and also in the whole system) is not correct. Therefore, this is a wrong result of model a. More details of heat generated in the different parts of the system are shown in Table 3. 3.2. Model b (Considering the Eddy Current in the Induction Coil). Figure 5 shows the contours of the in-phase component (right side) and the out-of-phase component (left

Figure 3. Contours of the volumetric power distribution (q) in the side wall of the crucible and afterheater, computed from model a. The maximum value of energy deposition is qmax ) 146 W/cm3 at the bottom edge.

Figure 4. Profile of the heat generated along the outer surface of the crucible and afterheater side wall for model a. The vertical axis is the crucible and afterheater wall, and for clarity, the RF coil is shown.

Tavakoli Table 3. Detail Information about the Heat Generated in the Different Parts of the CZ System, Calculated from Model a and Id ) 1000 A part

heat generated (watt)

percentage (%)

crucible bottom crucible wall afterheater wall afterheater top cover chamber coil total

353 4468 870 40 60 340000 346000

0.1 1.3 0.3 0.01 0.02 98.3 100

side) of the magnetic stream function computed from the second model. The maximum of the in-phase component (C) is located at the lowest edges of the main coil and also the upper edges of the second coil, while the minimum is located on the crucible wall and close to the bottom. Also noteworthy is that the gradient close to the maximum points is too high and the gradient close to the minimum point (crucible wall) is considerably inside and outside the crucible. In the other parts of the system, this component is nearly constant. For the out-of-phase component (S), the maximum is located at the outer surface of the main and second coil and Smax is 12 times greater than Cmax (absolute value). The spatial distribution of this component (S) is similar to the in-phase component (C) in model a, but here, this component (S) is stronger in the upper coil. The volumetric heat generation rate (q) in the crucible and afterheater wall is shown in Figure 6. The maximum value of crucible this energy deposition is qmax ) 3 × 10-3 W/cm3 and can be found at the bottom edge. Figure 7I shows the profile of the generated heat at the outer surface of the crucible and afterheater wall. It indicates again that the heat generation mostly occurs in the crucible wall (similar to model a), and it is nearly uniform except near the bottom, and in addition, heating in the afterheater wall is considerable. The total heat generation in the crucible crucible and afterheater is Qtotal ) 0.12 kW! Of course, this value should not be correct. Corresponding to the induction coil, the

Figure 5. Components of the magnetic stream function (ψB) calculated from model b. The right side shows the in-phase component (C) with Cmax ) 1.4 × 10-8 weber on the lowest and top edges of the coil and Cmax ) -3.5 × 10-8 weber on the crucible wall. The left side shows the out-of-phase component (S) with Smax ) 4.5 × 10-7 weber on the main and upper coil.

Modeling of Induction Heating in Oxide Czochralski Systems

Crystal Growth & Design, Vol. 8, No. 2, 2008 487 Table 4. Detail Information about the Heat Generated in the Different Parts of the CZ System, Calculated from Model b and Id ) 1000 A part

heat generated (watt)

percentage (%)

crucible bottom crucible wall afterheater wall afterheater top cover chamber coil total

0.007 0.075 0.034 0.003 0.001 0.014 0.135

5 56 26 2 1 10 100

Table 5. Detail Information about the Heat Generated in the Different Parts of the CZ System, Calculated from Model b and eq 16, That Is, Id ) 1.5 × 105 A

Figure 6. Contours of the volumetric heat distribution (q) in the side wall of the crucible and afterheater calculated from model b and Id ) 1000 A. The maximum value of heat generation is qmax ) 3 × 10-3 W/cm3 at the bottom edge.

part

heat generated (watt)

percentage (%)

crucible bottom crucible wall afterheater wall afterheater top cover chamber coil total

150 1670 770 60 30 520 3200

5 52 24 2 1 10 100

To obtain a better value for heat generation in this model, the researchers have used Klein8 J0 )

σcoVcoil 2πRcoN

(16)

for the driving current in the RF coil instead of directly using the electrical current input, where Vcoil is the total voltage of the coil, Rco is the mean value of the coil radius, and N is the number of coil turns. For a normal voltage in a CZ system, that is, 200 V, the value of the electrical current computed from eq 16 is about I0 ) 1.5 × 105 A, and it is clear that this value cannot be correct (the magnitude of the electrical current in the coil is in the range of ∼1000 A). Of course, by using this equation for J0, I can suppose it as a trick to find a realistic quantitative result of heat generation. For this value of electrical current input, the total heat generation in the crucible and crucible afterheater is Qtotal ) 2.65 kW and it is in the range of that which I expect. The details of this heat generation are presented in Table 5. It should be mentioned that by using eq 16, the spatial distribution of the electromagnetic field and the heat generation do not change.

4. Conclusions and Outlook Figure 7. Profiles of the heat generated along the outer surface of the crucible and afterheater side wall for model b and different values of skin depth δcr. The vertical axis is the crucible and afterheater wall, and for clarity, the RF coil is shown. By increasing the δcr, the heat generation at the crucible side wall increases as compared to the bottom 0 0 corner. (I) δcr ) δcr ) 3.8 mm, (II) δcr ) 1.4δcr ) 5.3 mm, and (III) 0 δcr ) 1.8δcr ) 6.8 mm. coil maximum value of heat generated is qmax ) 0.009 W/cm3 at the middle turns of the main coil and its total heat generation coil is Qtotal ) 0.014 kW. It means that about 10% of the power is produced in the RF coil and 90% in the conductors. It could be a right result, and we expect such percentages of heat generation for different metallic parts of the CZ system. The origin of this good result is that the maximum of the absolute value of the in-phase component (C) is located on the crucible wall (-3.5 × 10-8 weber) and not on the coil (contrary to model a) and also in most parts of the coil this component is too weak (∼10-10 weber). Table 4 shows more details of the heat generated in the different parts of furnace.

I have presented and demonstrated some results of two mathematical models of induction heating for an oxide CZ crystal growth system by using a finite element method. The results of both models show that most of the heat generation in the crucible and afterheater is produced in the lower part of the crucible wall and close to the bottom. Therefore, this part of the crucible should be the hottest part of it. In fact, this result has also been found in other calculations including temperature and fluid flow by using these volumetric heat generation, as a source term (not presented here). However, from observation in IKZ with this orientation of coil-crucible, the middle part of the crucible side wall is the hottest part and the bottom is relatively cold. In fact, in this configuration, there is a solid part of molten material just above the crucible bottom in the melt. To make this solid part melting, it is necessary to add more turns below the coil or to use a bottom heater in the system. The reason for this intense heating at the bottom corner is the edge effect. In an edge, the electromagnetic field penetrates

488 Crystal Growth & Design, Vol. 8, No. 2, 2008

into the conductor via both sides of it and an interaction between these penetrated fields yields a high heat generation in that area rather than a single side wall. Although I expect this effect at the bottom edge of the crucible, it seems that it is too high to accept. In fact, the skin (or penetration) depth of the crucible δcr ) (2/µrµ0σcrω)1/2 depends on µr (relative magnetic permeability of crucible), σcr (electrical conductivity of crucible), and ω (frequency of input power). The relative magnetic permeability is a complex function of several parameters such as frequency, magnetic field intensity, and temperature.8,9 In my calculations, the skin depth of the crucible is δcr ) 3.8 mm (on the order of the crucible wall thickness lc) and so this local high maximum of heat production occurs in the crucible edge. Some extra calculations for different values of δcr (i.e., µr and/or σcr) show a different distribution of the electromagnetic field components and also heat generation in the crucible. Figure 7 shows three profiles of heat generation at the outer surface of crucible and afterheater wall for model b and different values of δcr. This figure indicates that by increasing δcr, the heat generation at the crucible side wall increases as compared to the bottom corner. In this case, the crucible side wall is heated very effectively rather than at the bottom. It seems this kind of spatial energy generation in the crucible is more realistic as compared to previous results. In addition, the following conclusions were obtained as well: (i) By including the self-inductance effect in the RF coil as an eddy current [Je ) -(σco/r)(∂ψB/∂t)], the results change dramatically, and consequently, this effect is very important. (ii) In the first model, the magnitude of heat generation in the crucible and afterheater calculated using the electrical current input (∼1000 A) seems to be correct, but for the second model, it is not reasonable. To avoid this problem in the second model, using eq 16 is very useful and so the results could be realistic. (iii) The magnitude and percentage of heat generation in the RF coil for the first model are not correct, while in the second model, they are acceptable. (iv) The spatial distribution of the heat generation in the crucible side wall is more uniform (i.e., more realistic) in model b. Finally, it has been indicated that model b (i.e., with the selfinductance effect in the RF coil) can produce more realistic results than model a. In addition, it seems that the modeling of induction heating for the crystal growth systems is not completed yet and further studies are necessary. Although the basic electromagnetic phenomena of induction heating are quite simple and discussed in several textbooks,8,9 the induction heating itself is a complex combination of electromagnetic and heat transfer phenomena. Heat transfer and electromagnetic behavior are

Tavakoli

tightly interrelated because the physical properties of heat-treated materials depend strongly on both magnetic field intensity and temperature. Also, a nonuniform current distribution can be caused in the RF coil and conductors by several electromagnetic phenomena, including (i) self-inductance effect, (ii) skin effect (the current density will decrease from the surface of the conductor toward its center), (iii) proximity effect (the conductors have their own magnetic fields, which will interact with nearby fields, and as a result, the current and power density distributions will be distorted), (iv) ring effect (concentration of electrical current on the inside surface of the induction coil), and (v) electromagnetic end and edge effects (the distortion of electromagnetic filed in its end and edge areas). These effects play an important role in understanding the induction heating mechanism. Thus, it is required to do further studies on the “coil physics” to find better quantitatively and qualitatively results and to correctly describe the phenomena. Acknowledgment. I thank H. Wilke, P. Uecker, P. Reiche, and M. Ziem in the IKZ CZ-dielectric group for their helpful discussions.

References (1) Gresho, P. M.; Derby, J. J. A finite element model for induction heating of a metal crucible. J. Cryst. Growth 1987, 85, 40–48. (2) Derby, J. J.; Atherton, L. J.; Gresho, P. M. An integrated process model for the growth of oxide crystals by the Czochralski method. J. Cryst. Growth 1989, 97, 792–826. (3) Tsukada, T.; Hozawa, M.; Imaishi, N. Global analysis of heat transfer in CZ crystal growth of oxide. J. Chem. Eng. Jpn. 1994, 27, 25–31. (4) Chen, Q. S.; Gao, P.; Hu, W. R. Effects of induction heating on temperature distribution and growth rate in large-size SiC growth system. J. Cryst. Growth 2004, 266, 320–326. (5) Tavakoli, M. H.; Wilke, H. Numerical study of induction heating and heat transfer in a real Czochralski system. J. Cryst. Growth 2005, 275, e85–e89. (6) Tavakoli, M. H.; Wilke, H. Numerical study of heat transport and fluid flow of melt and gas during the seeding process of sapphire czochralski crystal growth. Cryst. Growth Des. 2007, 7, 644–651. (7) Tavakoli, M. H.; Wilke, H. Numerical investigation of heat transport and fluid flow during the seeding process of oxide Czochralski crystal growthsPart 1: Non-rotating seed. Cryst. Res. Technol. 2007, 42, 544– 557. (8) Klein, O.; Philip, P. Transient numerical investigation of induction heating during sublimation growth of silicon carbide single crystals. J. Cryst. Growth 2003, 247, 219–235. (9) Leatherman, A. F.; Stutz, D. E. Induction Heating AdVances; National Aeronautics and Space Administration: 1969. (10) Rudnev, V.; Loveles, D.; Cook, R.; Black, A. M. Handbook of Induction Heating; New York, NY, 2003. (11) Cliffe, K. A. SERCO Ltd., http://www.sercoassurance.com/entwife/.

CG070378+