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IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 11, NOVEMBER 2008
Basic Characteristic of Electromagnetic Force in Induction Heating Application of Linear Induction Motor Takahiro Yamada1 and Keisuke Fujisaki2;3 Nittetsu Plant Designing Corporation, Kitakyushu, Fukuoka, Japan Nippon Steel Corporation, Futtsu, Chiba, Japan Graduate School of Environmental Studies, Tohoku University, Sendai, Miyagi, Japan Basic characteristic of electromagnetic force in induction heating application of single-sided linear induction motor (SLIM) which uses electromagnetic AC field is investigated by means of 2-D and 3-D numerical electromagnetic FEM calculation. The electromagnetic force in the normal direction consists of the attractive force caused by magnetization and the repelling force caused by Lorentz force. Although the normal force is canceled by them and becomes zero at crossover frequency, the value is different, which is 2 kHz in 2-D model and is 5 kHz in 3-D one respectively. Since 2-D model expresses only center section of 3-D model and can not express the complex behavior of the eddy current in the whole model, the repelling force caused by Lorentz force is considered to be overestimated when the electromagnetic force value of 2-D model is converted into 3-D one. As a result, the crossover frequency of 3-D model becomes large in comparison with 2-D one. This means that an evaluation by using 3-D model is demanded in order to correctly evaluate the crossover frequency. On the other hand, at low frequency in which the attractive force caused by magnetization becomes dominant, the normal forces of the 2-D and 3-D model are in good agreement. This means that the evaluation by using 2-D model is effective at low frequency. Index Terms—Crossover frequency, electromagnetic force, finite element method (FEM), induction heating, linear induction motor, thin steel plate.
I. INTRODUCTION
HIN steel plate used for many industries such as electric appliance and automobile is representative products in steel making plant. To obtain high quality products, an accurate treatment results in the making process line such as galvanizing, annealing, and color coating, is a very important factor. An induction heating which has advantages such as rapid-heating and local-heating is often used as effective heat control method to realize them. Usually, the treatment in the process line is heated with conveying the thin steel plate in order to realize high productivity by the continuous operation. Then, it is necessary to supply some tension to the thin steel plate in order to prevent slackening of the steel plate by its own weight. Usually, mechanical ways such as pinch roll are used. However, since the thin steel plate contacts to the pinch roll mechanically, it is expected some bad influence to the thin steel plate such as surface crack and exfoliation of coating by the contact friction between the thin steel plate and pinch roll. Therefore, tension supply by non-contact operation is desirable to prevent them. A linear induction motor (LIM) with steel core is practically used because of its non-contact operation and quick response. Also, the eddy current generated by the traveling magnetic field is considered to be used as induction heating equipment. However, in this case, since the thin steel plate is ferromagnetic material, it is expected that the meandering of the thin steel plate caused by the electromagnetic force in the normal direction has an influence on the product quality. Because the normal force
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Digital Object Identifier 10.1109/TMAG.2008.2002786 Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.
is too difficult to handle the linear induction motor and the thin steel plate [1], [2], the reduction of the normal force is an important problem to apply it to such a process. To evaluate the basic characteristics of linear induction motor, two-dimensional (2-D) electromagnetic field calculation by means of finite element method (FEM) is typically researched as simplified method in later work [3]. However, because the real equipment is three-dimensional (3-D) model and especially the eddy current shows complex behavior, the detailed investigation by 3-D electromagnetic field calculation is demanded. The aim of this study is to investigate the basic characteristics of the electromagnetic force in a single-sided linear induction motor (SLIM) from the point of view of the comparison with 2-D FEM calculation and 3-D one. II. ELECTROMAGNETIC FORCE IN SLIM Linear induction motor is classified single-sided type (SLIM) and double-sided type (DLIM). However, since an attractive force in LIM is instability system which sensitively reacts by a little disturbance regardless of SLIM or DLIM [1], the attractive force of LIM is an important problem to solve. Therefore, in order to simplify the problem, a SLIM is investigated in this paper. System construction of SLIM to the thin steel plate and the electromagnetic force operated to the shin steel plate is shown in Fig. 1. Primary part of the LIM is a static electromagnetic coil and secondary part is the thin steel plate. The electromagnetic coil consists of a steel core and several copper coils. The steel core is laminated to depress the induced eddy current in it. Three-phase alternating current is supplied to the copper coils and traveling magnetic field is generated at the steel core. Since traveling magnetic field is alternative and the thin steel plate is electrically conductivity, some eddy current is induced in the
0018-9464/$25.00 Š 2008 IEEE Authorized licensed use limited to: National Taiwan Univ of Science and Technology. Downloaded on May 17, 2009 at 09:19 from IEEE Xplore. Restrictions apply.
YAMADA AND FUJISAKI: ELECTROMAGNETIC FORCE IN INDUCTION HEATING APPLICATION OF LINEAR INDUCTION MOTOR
Fig. 1. Electromagnetic force in thin steel plate. (a) Attractive force; (b) repelling force.
thin steel plate. As a result, the thin steel plate is heated by the joule heat generated through the eddy current. When the traveling magnetic field operates on the thin steel plate as an external field, the thin steel plate is magnetized. As a result, an electromagnetic force as attractive force mode operates between the magnetized thin steel plate and linear induction motor. On the other hand, the eddy current and the traveling magnetic field makes an electromagnetic force as Lorentz force to the thin steel plate in the thrust direction as well as in the normal direction. Therefore, the electromagnetic force in the normal direction consists of the attractive force caused by magnetization and the repelling force caused by Lorentz force.
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Fig. 2. Specification of calculation model. (a) X-Z plane; (b) Y-Z plane.
Fig. 3.
B –H curve of thin steel plate (SS400).
III. FEM CALCULATION To evaluate the basic characteristics of the LIM, an electromagnetic field analysis is calculated by way of finite element method (FEM). Fundamental equations in electromagnetic field are formulated from Maxwell’s equations. The equations method are expressed as folin eddy current field with lows [4]: (1) (2) (3) Here,
is vector potential [Wb/m], is current density , is inverse of permeability [m/H], is scalar potential [V/m], is electric conductivity [S/m], is eddy . current density Because the thin steel plate is ferromagnetic material and has the nonlinear magnetic characteristic, it is solved by the Newton-Laphson method. For the time iteration, the step-bystep method with the backward difference method is used. The specification of the calculation model is shown in Fig. 2. Since the 3-D model is symmetry shape on the width center of the thin steel plate, the half model is calculated. Thickness of the thin steel plate is 0.1 mm which is almost a minimum thickness in the steel making plant. The pole pitch of the LIM is 60 mm and pole number is 2. Three-phase alternating current such as U, V, W-phases is supplied to the LIM to make traveling magnetic flux. The specific boundary condition that vector potential becomes to be zero is used for all boundary condition.
TABLE I MATERIAL CONSTANTS
Material constants are shown in Table I. The material of the thin steel plate is SS400 and the – curve is shown in Fig. 3 [5]. The frequency changes from 10 Hz to 20 kHz. AC current . density in the LIM is constant as 5.0 The calculation mesh of the 2-D and 3-D model are shown in Fig. 4. In case of 2-D model, the X-Z plane model shown in Fig. 4 is calculated. In the FEM calculation, a FEM package named “JMAG” was used. The nonlinear – curve of the SS400 indicates the maximum specific relative permeability at 1.0 T and is about 1800. When the frequency is 20 kHz, the skin depth of the thin steel plate is 0.046 mm. The surface division in the thin steel plate is 0.01 mm which satisfies one-third of the skin depth. IV. CALCULATION RESULT Some methods are introduced as the calculation method of the global force in a rigid body [6], 1[7]. However, since the electromagnetic force is evaluated as the global force acts to the whole steel plate, Maxwell’s stress method is used [8], [9]. This method calculates the global force by the integration of the is expressed surface force density. Maxwell’s stress tensor as follows:
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 11, NOVEMBER 2008
Fig. 6. Mechanism of the crossover frequency. Fig. 4. Bird’s eye view of the calculation mesh.
Fig. 5. Frequency characteristic of the electromagnetic force in the normal direction.
(4) , , are the magnetic flux density in the x, y, Here, z component respectively [T]. Then the electromagnetic force supplied to thin steel plate is expressed as follows: (5) Here, is a normal unit vector of the thin steel plate surface. Frequency characteristic of the electromagnetic force in the normal direction is shown in Fig. 5. The electromagnetic force calculated from the 2-D model is converted to the value of the steel plate width in the 3-D model. The normal force of the both models decreases as the frequency increases. The crossover frequency in which the normal force changes from attractive force to repelling force is 2 kHz in 2-D model, though the one is 5 kHz in 3-D model. The crossover frequency is different between 2-D model and 3-D model. The mechanism of the crossover frequency is explained as follows by means of Fig. 6. Under current constant condition,
since the magnetic field derived from the motor is assumed to caused the magnebe constant, the attractive force tization is also considered to be constant at any frequency. The becomes large according to the higher repelling force frequency. As a result, the attractive force and reare considered to be canceled at the pelling force crossover frequency. Therefore, the effect of the normal force is considered to be removed by operating at the crossover frequency. When evaluating the efficiency of the whole system, since the frequency is one of the important parameter which has an influence on the joule heat of steel core, power source capacity, and heating characteristic of steel plate, it is very important to correctly decide the frequency which is closely related with them. Therefore, to make clear the difference of the crossover frequency in the 2-D and the 3-D model is important. In Fig. 5, although the normal forces of the 2-D and the 3-D model are in good agreement at the low frequency, the crossover frequency of the 3-D model is larger than the 2-D one. From the mechanism of crossover frequency as shown in Fig. 6, the difference of the normal force at the high frequency such as crossover frequency is considered to be effect of the repelling force caused by the Lorentz force. Lorentz force depends on the distribution and the intensity of the eddy current. The eddy current density vector is shown in Fig. 7. Then, although the 2-D model is calculated by using 2-D plane model, the thin steel plate of the 2-D model is showed with some thickness in the width direction in order to simplify the comparison between 2-D and 3-D result. The calculated area of the 2-D model is located at the width center of the 3-D model (enclosed area by dotted line) as shown in Fig. 7(b), the eddy current distribution in the both models are almost same at this area. On the other hand, because the 3-D model has edge part of the steel plate, the eddy current returns around the edge part. The difference of the distribution of the eddy current is considered to have influence on the Lorentz force. The time averaged contour of the Lorentz force in the normal is shown in Fig. 8. The 2-D model and the width direction center of the 3-D model are almost same distribution and intensity. Although the distribution of the 3-D model shows high intensity at the width center of the steel plate, it becomes small gradually in the edge direction. This shows that the Lorentz force of the 2-D model converted by the steel plate width in the 3-D model becomes overestimated value.
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YAMADA AND FUJISAKI: ELECTROMAGNETIC FORCE IN INDUCTION HEATING APPLICATION OF LINEAR INDUCTION MOTOR
Fig. 7. Eddy current density vector at frequency 5 kHz (! t = 0 ). (a) Twodimensional model. (b) Three-dimensional model.
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repelling force caused by Lorentz force. Although the normal force is canceled by them and becomes zero at crossover frequency, the value is different, which is 2 kHz in 2-D model and is 5 kHz in 3-D one respectively. Since the 2-D model expresses only center section of the 3-D model and can not express the complex behavior of the eddy current in the whole model, the repelling force caused by Lorentz force is considered to be overestimated when the electromagnetic force value of the 2-D model is converted into the 3-D one. As a result, the crossover frequency of the 3-D model becomes large in comparison with the 2-D one. This means that an evaluation by using 3-D model is demanded in order to correctly evaluate the crossover frequency. On the other hand, at low frequency in which the attractive force caused by magnetization becomes dominant, the normal forces of the 2-D and the 3-D model are in good agreement. This means that the evaluation by using 2-D model is effective at low frequency. Future research should focus on evaluating the distribution of the electromagnetic force by using nodal force method. REFERENCES
Fig. 8. Time averaged contour of the Lorentz Force in the normal force (F ). (a) Two-dimensional model. (b) Three-dimensional model.
Also, in this study, although the laminated steel core is used as the steel core because of the basic investigation, the use of ferrite core should be considered at high frequency such as crossover frequency in the design of the real equipment. V. CONCLUSION Basic characteristic of electromagnetic force in induction heating application of single-sided linear induction motor (SLIM) which uses electromagnetic AC field is investigated by means of 2-D and 3-D numerical electromagnetic FEM calculation. The electromagnetic force in the normal direction consists of the attractive force caused by magnetization and the
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Manuscript received March 03, 2008. Current version published December 17, 2008. Corresponding author: T. Yamada (e-mail: T-Yamada@npdx.co.jp).
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