Study of Induction Eddy Current Distribution Based on FEM Jiangbo Wang1, Junhua Wang2 and Chunsheng Li1 1 2
Institute of Electrical Engineering Chinese Academy of Sciences, Beijing, 100190, China School of Electrical Engineering Hebei University of Technology, Tianjin, 300130, China
Abstract-In some applications, traveling wave induction heating (TWIH) could make the heating distribution more profitable in comparison with transverse flux induction heating (TFIH). In this paper a simplified 3D FEM code is presented for the study of the induction eddy current distribution both in TWIH and TFIH. Results show that there are different eddy current densities in the metal strip when the circle loop lies in the internal or external of the projector.
I.
Draw a concentric circle which shares the same center with C2 and its radius is r0 as show in Fig. 2.
INTRODUCTION
The fundamental principles of the Travelling Wave Induction Heating (TWIH) are known for many years [1-3]. Travelling Wave Induction Heating, as one of the multiphase induction heating systems, has particular features which make them attractive for application to some heating and melting processes in industry. The characteristics of TWIH are the possibility to heat quite uniformly thin strips or regions of a body without moving the inductor above its surface, the feasibility to reduce the vibrations of inductor due to the electro-dynamic forces and the noise provoked by them, and the probability to obtain nearly balanced distributions of power and temperature [4]. Fig. 1 shows a simplified three-dimensional single-coil model of TFIH and TWIH induction heating. It is much simpler than the experimental and practical applications, but it is enough for the preliminary analysis of traveling wave and transverse flux system. ANSYS modeling and simulation are executed based on this model to find the characteristic of eddy current distribution.
Fig. 2. The relationship between eddy current distribution and the projection of the coil geometry
Then feed circular C1 with an alternating current. Because the distance between the coil and the metal strip is far smaller than the area that the coil surrounded, the magnetic flux density approximately remains constant between the projection and the internal coil. According to the Faraday law, the induction electric potential in the circle loop is
e= -
dϕ dt
(1)
where ϕ is the chain magnetic flux induced by the coil current, and t is the time. When exciting current changes, the magnetic flux density changes accordingly. So we get
Φ = ΦM sinωt
and φM is the amplitude of frequency. The resistance of circle loop is
R= ρ
φ
(2) , and
ω
l 2π ar =ρ S drh
is the angular
(3)
where ρ is the resistivity of the metal strip, l is loop length, h is the thickness of metal strip, and S is the cross section area of the circuit. The current density distribution is discussed under two circumstances. Fig. 1. 3D induction heating theoretical model
II.
A. Coil Lies in External of the Projection C2
THEORETICAL ANALYSIS OF EDDY CURRENT DISTRIBUTION
The eddy current is
Suppose that there is a piece of big enough metal strip, parallels to it is a circular C1 with a radius of r1, and the projection of C1 on the strip is circular C2 with a radius of r2.
i=
e ωd h Φ = - r × M cosωt R 2πρ ar
(4)
For metal strip with given thickness, resistivity and
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operational frequency
K1 = -
ω dr h 2πρ
which is a constant. With the increasing α r radius of the circle, φM declines because the projector's internal and external magnetic flux are in the contrary direction. Therefore, when circle loop lies in the external of projector C2, we can get the following conclusion: Conclusions 1: Outside the projection coil, the eddy current density is less intensive from the center along the radius direction.
model. It should be added that eddy current distributions are not the only problems in the use of these systems and for this reason the comparison must be studied further. The electromagnetic problem must be coupled with the thermal and the mechanical ones in order to have a more accurate understanding of the different temperature distributions, mechanical deformations and noises. However, it is difficult to get a specific mathematical equation due to the complication of eddy current distribution and its characteristics [5-9]. So we can only use Finite Element Numerical analysis or other modern methods to analyze this problem.
B. Coil Lies in Internal of the Projection C2 The magnetic flux φ that passes area S can be expressed as
Φ=
∫ Bds S
n
III.
Both analytical and numerical techniques can be used for the
(5) study of these heating systems. Analytical methods are more
Because the metal strip in the magnetic flux density B is continuous, according to the integral median formula there must be a Ba , which satisfies the flowing equation
∫ B ds S
n
= Ba S
(6)
where n is the right law Plane Circle Line direction, and obviously B parallels to n. S is the area that is surrounded by the circuit loop. The magnetic flux φ that passes projection area can be expressed as
Φ = Baπ r22
(7)
Therefore, we get equation (8) through (1), (5), and (7)
e= -
d BaM π r22 sin(ω t ) dt
(8)
where BaM is the amplitude of Ba when Ba has sinusoidal changes. According to equation (4), eddy current in the loop is
i=
e ωd h Φ = - r × M cosωt R 2ρ ar
ANSYS MODELING AND SIMULATION
convenient for the integral parameters determination and analysis, while the numerical techniques are more universal and particularly useful for investigating the induced current and power distributions, taking into account the inductor edge-effects and the slots effects which are usually well pronounced in TWIH systems. The analytical methods which make use of Fourier integral transformation are effective for the simulation of lD, 2D and even 3D multiphase devices, but some simplifications and assumptions must be made. Since in this paper the analysis has been performed by FEM software ANSYS, in which the FEM code is called as an external subroutine. Fig. 3 shows the ANSYS theoretical model. The element type is solid117; air is the 1st material, coil the 2nd material and metal strip the 3rd material. We assume that these parameters do not change as the temperature flux. The value of current in the coil is 60 A, and the frequency is 60 Hz. All physical quantities are analyzed under the frequency-domain.
(9)
For the metal strip with given thickness, resistivity and frequency ωd h K2 = - r 2ρ that is a constant. As the radius α r increases, BaM α r also increases. So, when the circle loop lies in C2, the eddy current distribution is as following. Conclusion 2: Inside the projection coil, the eddy current density is more intensive from the center along the radius direction. The eddy current density in the metal strip is continuous. Hence based on Conclusions 1 and Conclusion 2, we can know: Conclusion 3: The eddy current density attains its maximum value around the coil projection. Outside the projection coil, the eddy current density is less intensive from the center along the radius direction, and inside the projection coil, the change is reversed. All of the conclusions are based on a simple theoretical
Fig. 3. 3D induction heating ANSYS model
After the material definition and the unit dimension, current density is specified to the coil, and VOLT constraint is coupled on the strip, then parallel flux and normal flux boundary condition is added, finally calculation based on FEM is done. For the purpose of simplifying calculation, meshing rate is set to 6, which can greatly curtail the computation time without affecting the solution. Fig. 4 shows the mesh results and current density. The boundary conditions are shown in Fig. 5.
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Fig. 7. The eddy current density distribution in the strip of TWIH systems
Fig. 4. Coil with loading current
IV.
CONCLUSIONS
Through ANSYS simulation, the different eddy current density distribution has been analyzed, and the simulation results certified the theoretical conclusions. The trend of the eddy current distribution in the internal or external of the projection coil is different as the distance from the center becomes larger. REFERENCES
Fig. 5. ANSYS model after loading parameters
Fig. 6 and Fig. 7 show the cloud diagram of magnetic density and the curve of eddy current density respectively. From the results we can see that theoretical analysis of eddy current distribution in section 1 accords with the simulation results. When the distance from the center increases, eddy current intensity weakens in the internal coil projection; on the opposite, it becomes more intensive in the external coil projection.
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Fig. 6. The magnetic flux density distribution in the strip of TWIH systems
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