Electro Magnetic Field And Eddy Current Analysis Of Linear Induction .
International Journal of Pure and Applied Mathematics Volume 118 No. 19 2018, 1717-1729 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu Special Issue Electro Magnetic Field and Eddy Current Analysis of Linear Induction Motor Using Analytical Method Gang-Hyeon Jang1 ,So-Young Sung2 , Jong-Won Park2 , Jang-Young Choi *1 1 Include Department, Institutional address Dept. of Electrical Engineering, Chung-nam National University, 99, Dae-hak-ro, Yusung-gu, Daejeon, 34134, Republic of Korea 2 R&D Department, KRISO, Yuseongdae-Ro, Yuseong-Gu, Daejeon, Korea gh.jang@cnu.ac.kr1 ,riverblu@kriso.re.kr2 , poetwon@kriso.re.kr2 , Corresponding author* Phone: 82-10-4404-4226 February 4, 2018 Abstract Background/Objectives:Instead of the finite element method, which is mainly used for linear induction motor analysis, the analytical method using the spatial harmonic method was applied and verified. Methods/Statistical analysis: In this paper, the electromagnetic field analysis and the eddy current density analysis of the linear induction motor are carried out by the analytical method using the spatial harmonic method based on the Maxwell equation.After simplifying the analytical model, the governing equations in each region are established using Maxwell’s equations. And then, we perform the 1 1717 ijpam.eu
International Journal of Pure and Applied Mathematics electromagnetic field analysis in each region using boundary conditions. Findings: In this paper, based on the analysis method using spatial harmonics, the magnetic field and eddy current analysis of the LIM are performed. The results of magnetic field based on magnetic vector potential and Maxwell’s equations were in good agreement with those of FEA analysis. Also, it can be seen that the magnetic flux density due to the eddy current induced in the secondary side causes an imbalance in air magnetic flux density. Also, as a result of the analytical method, the slip of the induction motor was considered.The mathematical expressions used in this paper are useful for understanding and predicting the magnetic field distribution and eddy current density in the initial design of the LIM. Improvements/Applications: There are not many papers dealing with electromagnetic field and eddy current density analysis that change with time considering slip.It will be useful in designing linear induction motor. Key Words: linear induction motor, 3D FEA, analytical method, Maxwells equation, eddy current analysis 1 Introduction Linear electric machines can generate a linear driving force, and there are advantages to using a linear driving system. That is, in the case of a linear electric machine in a linear driving system, a mechanical conversion device is not required, hence the number and space occupied by components are small, energy loss or noise are minimized, and operation speed is not limited. Therefore, the linear driving system has advantages in terms of efficient operation and function compared to the rotary type. Today, the system has been developed for industrial applications requiring linear motion, such as a magnetic levitation train in a land transportation system, a conveyor system in a large factory or industrial facility, an elevator, and a crane. In the linear type induction motor, a conventional induction motor is cut in the central axis direction. The primary side is composed of an electric steel plate and an armature winding of a magnetic body, while the secondary side is composed of an aluminum conductor plate and a stator core. The driving principle 2 1718 Special Issue
International Journal of Pure and Applied Mathematics is the same as that of the rotary induction motor1 4 . This paper analyzes the magnetic field and eddy current of the linear induction motor (LIM) using the spatial harmonic method as an analytical method. The characteristics of the LIM have often been analyzed using the finite element method (FEM). However, the number of initial parameters used is often limited; increasing the size of the initial model or changing the design parameters are rarely done in FEM analysis, and are difficult in reality5 7 . In LIMs, the magnetic field is generated by the current flowing through the primary armature winding, which generates an electromagnetic force in the secondary conductor through the air-gap. This magnetic field generates an eddy current that crosses the magnetic flux of the air gap and produces thrust based on the principle of Lorentz force. Therefore, analysis of the magnetic flux density in the air-gap and the eddy current density induced in the secondary side conductor is very important. To calculate the eddy current, an analytical solution was derived by applying the Maxwells equation, magnetic vector potential, and Faradays law in a two-dimensional Cartesian coordinate system. By using the analytical method, it is possible to accurately predict the eddy currents generated in the secondary side conductor plate by the armature winding and reduce the analysis time compared to FEM. Since this can be used to obtain analysis results quickly, it is often advantageous to use an analytical method during the initial design phase. The LIM used in this study is shown in figure 1 for a single-sided LIM with a rated speed of 1.6 m/s and a rated thrust of 86.5 Nm8 9 . 2 2.1 Application of the Linear Induction Motor The Analysis Model Figure 2 shows the simplified model of the LIM for the analytical method. Based on the analysis model, the governing equations, flux density, and eddy current density of each region are obtained. The analytical model can be divided into four regions: a mover iron core, including a current source; an air-gap region, aluminum plate region, and a stator core region. The LIM used in the analysis 3 1719 Special Issue
International Journal of Pure and Applied Mathematics has a six-poles, 41-slots structure, and the armature winding is arranged in a three-phase balanced state. The assumptions made before the analysis as follows. First, it is assumed that the magnetic permeability of the mover iron core and stator core is infinite, and that the conductivity of the iron core is zero. In general, the iron core of the mover and stator is designed to avoid saturation and the nonmagnetic permeability in the non-saturated region of the electric steel sheet is very high. In addition, since the iron core is manufactured by lamination, eddy currents do not occur. For convenience of analysis, it is assumed that the armature current is distributed in the form of surface current density, while neglecting the slotting effect. Fig. 1.Manufactured linear induction motor Fig. 2. Analysis model: (a) schematic of LIM (b)simplified model for analytical method, (c) 3D FEM model 4 1720 Special Issue
International Journal of Pure and Applied Mathematics 2.2 Special Issue Magnetic Flux Density Analysis The magnetic flux density B and the magnetic intensity H satisfy Maxwells equations,B µ0 (H M); taking the curl gives B µ0 ( H) µ0 ( M ). Using the magnetic vector potential A defined from A B, Laplaces and Poissons equations in terms of the Coulomb gauge · A 0 are given by 2 {µ0 J µ0 ( M )} (1) Since the air-gap region has no current source and no magnetization component, it can be expressed by the following governing equation: 2 AI 0 (2) When the magnetic field generated by the primary armature current is transmitted to the secondary reaction plate through the air-gap, eddy current is induced in the secondary conductor. The eddy current generated at this time causes the magnetic field to be generated again according to Ampere’s law. The magnetic fields generated by the armature current and eddy current combine with each other, and magnetic field distribution distortion occurs in the air-gap. In order to analyze this magnetic field distribution distortion phenomenon, the eddy current can be interpreted as a current source in the secondary plate region; hence, the governing equation of region II is expressed as 2 AII µo Je (3) where,Je is the eddy current induced inside the secondary-side aluminum. The magnetic vector potential is determined by the current direction, spatial distribution, and magnitude. Hence, magnetic vector potential can be expressed as A Aznm (y)e( jkm x) iz (4) where, km mπ/τ is the spatial wavenumber of the nth harmonic, and τ is the pole pitch of the LIM. The eddy currents induced in the conductor plate are generated by the time harmonics 5 1721
International Journal of Pure and Applied Mathematics Special Issue of the currents and the spatial harmonics of the armature windings that are not synchronized with the mechanical speed of the mover. The magnetic vector potential type for applying this concept is written as follows: A Aznm (y)Ψiz (5) Ψ e jkm (X Vs t) ·cos(nωt) e jkm (X Vs t 2/3τp ) ·cos(nωt e jkm (X Vs t 4/3τp ) · cos(nωt 4nπ ) 3 2nπ ) 3 (6) Where Vs and are the synchronous speed of the current and synchronous frequency, respectively. The following differential equations can be obtained by substituting the magnetic vector potential into governing equation of each region. 2 I 2 km AIZnm (y) 0 A y 2 znm (7a) 2 II 2 II A km AII Znm (y) (km Vs µσ)Aznm y 2 znm (7b) By solving the above equations, a general solution of the differential equation can be obtained as AIznm (y) X X Re(CnI ekm y DnI e km y ) (8a) Re(CnII ekm y DnII e km y ) (8b) n 1 m AII znm (y) X X n 1 m Using the definition of the magnetic vector potential A B, the tangential and normal component of magnetic flux density in each region can be summarized as B Az Az ix iy y x (9) Using this flux density equation and general solution of the differentialequation, we then obtain the armature reaction magnetic 6 1722
International Journal of Pure and Applied Mathematics Special Issue flux density in each region as follow: X X I Bx Re{km (CnI ekm y DnI e km y )ψ}ix (10a) Re{jkm (CnI ekm y DnI e km y )ψ}iy (10b) Re{E( CnII ekm y DnI e km y )ψ}ix (10c) Re{jkm (CnII ekm y DnII e km y )ψ}iy (10d) ByI BxII ByII n 1 m X X n 1 m X X n 1 m X X n 1 m The undefined coefficients needed to obtain the magnetic flux density can be determined by using the boundary conditions listed in Table I. Table 1. Boundary Condition In the first boundary condition, J represents the current density distribution of the armature current, assuming that it is distributed in the form of sheet current. The sheet current density can be modeled by the following equation: J X X Re(Im In ψ) (11) n 1 m where,Im is the Fourier coefficient of the Fourier series modeling armature winding, and In is the maximum value of the phase current. 2.3 Eddy Current Density Analysis The eddy current density equation can be obtained from the magnetic vector potential using the definition of Faradays law and Ohms law as follows: 7 1723
International Journal of Pure and Applied Mathematics J σ(E V B) Special Issue (12) ( A) ( A ) E B t t t A (13) t Combining this with equation, the eddy current density induced in a reaction plate with electric conductivity σ can be expressed using the following equation crane9 10 : E JeII σ AII σ(Vx ByII ) t σπ{(CnII e Ey DnII eEy )} σ(Vx ByII ) (14) In the above equation, is the mechanical speed of the mover core, and is expressed as. Vx SVs Vs (15) where, S means slip. The voltage induced on the secondary-side reaction plate of the LIM depends on the relative moving speed of the mover relative to the moving magnetic field. Slip represents this relative speed of movement. In the equation (14), Π is the differential form of ψ, and is expressed as. Π jkm Vs {e 2jkm (X Vs t) · cos(nωt) nω · sin(nωt) 2nπ 2nπ 2 ) nω · sin(nωt ) e 2jkm (X Vs t 3 τp ) · cos(nωt 3 3 4 4nπ 4nπ e 2jkm (X Vs t 3 τp ) · cos(nωt ) nω · sin(nωt )} (16) 3 3 3 Analysis and Experimental Results In this section, the analytical results are compared with FE analysis results. Figure 3 shows the results of the experiment on the manufactured LIM.The phase current was measured to determine the rated speed of 1.6 m/s by applying phase voltage. It can be seen that a current of 5.2 A with frequency 21Hz is the input at the rated speed section. 8 1724
International Journal of Pure and Applied Mathematics Fig. 3.Experimental results of LIM Fig. 4.3D FEM eddy current analysis result using phase current value confirmed in experiment. 9 1725 Special Issue
International Journal of Pure and Applied Mathematics Fig. 5.Results of magnetic flux analysis according to time variation in stop state: (a) t 0 s. (b) t 0.3 s. Fig. 6.Analysis of eddy current density by velocity and time: (a) V 0, (b) V 1.6 m/s. The analysis was carried out using the current values obtained from the experiment and compared with the results of the analytical method.Fig.4 shows the results of the 3D FEA using the phase current in the rated speed range as the input condition. The analysis points shown in figure 4 were compared with the results using the analytical method. Since the analytical method does not take into account the leakage in the current direction, the middle part of the FEA results are selected as the data to be compared. Figure 5 shows the magnetic flux density distribution by armature winding over time in stop state. As can be seen in figure 5(b), the air-gap magnetic flux density was distorted and the size was reduced by the secondary-side eddy current. Figure 6 shows the eddy current analysis results on the 10 1726 Special Issue
International Journal of Pure and Applied Mathematics secondary reaction plate. Figure 6(a) shows the results of the eddy current density analysis when the current required for the rated speed section was applied to the LIM and the mover was stopped. Figure 6(b) shows the results when the mover was moved at 1.6 m/s. It can be confirmed that the eddy current density was reduced and distorted by the difference between the moving magnetic field speed and the mechanical moving speed. The analysis results of the magnetic field and eddy current density of LIM using the analytical method agree well with the FE analysis results. 4 Conclusion In this paper, based on the analysis method using spatial harmonics, the magnetic field and eddy current analysis of the LIM are performed. The results of magnetic field based on magnetic vector potential and Maxwell’s equations were in good agreement with those of FEA analysis. Also, it can be seen that the magnetic flux density due to the eddy current induced in the secondary side causes an imbalance in air magnetic flux density. Based on the experimental data on the manufactured LIM, eddy currents induced in the secondary conductors were analyzed. As a result of the analytical method, the slip of the induction motor was considered. The mathematical expressions used in this paper are useful for understanding and predicting the magnetic field distribution and eddy current density in the initial design of the LIM. Acknowledgment This research was a part of the project titled “Research on fundamental core technology for ubiquitous whipping and logistics” funded by the Ministry of Oceans and Fisheries, Korea. This work was supported by the Basic Research Laboratory (BRL) of the National Research Foundation ( NRF -2017R1A4A101 5744) funded by the Korean government. References [1] S.A. Nasar, I. Boldea, “Linear motion electromagnetic system,” John Wily & Son, 1976. 11 1727 Special Issue
International Journal of Pure and Applied Mathematics [2] S. Yamamura, H. Ito, y. Ishikaka, “A theoretical analysis of linear induction motor,”IEEE Trans.1979, Vol. Pas-98, no. 2. [3] T. Higuchi, S. Nonaka, and M. Ando, “On the design of highefficiency linear induction motor for linear metro,” Electrical Engineering in Japan, 2001, vol. 137, no. 2, pp. 36-43. [4] T.W Preeston and A.B.j. Reece, “Transverse edge effect in linear induction motors,” Proc. IEEE. 1969, Vol. 116, pp.973970. [5] R. M. Pai, I. Boldea, and S. A. Nasar, “A complete equivalent circuit of linear induction motor with sheet secondary,” IEEE. Trans. on Magn.,1988Jan, vol. 24, Issue: 1, pp. 639-654. [6] M. Mirsalim, A. Doroudi, and J. S. Moghani, “Obtaining the operating characteristics of linear induction motors: a new approach,” IEEE. Trans. Magn., 2002March, Vol. 38, pp. 13561370. [7] B. J. Lee, D. H. Koo, and Y. H. Cho, “Investigation of linear induction motor according to secondary conductor structure,” IEEE. Trans. Magn., 2009 June, vol. 45, no. 6. [8] H. W. Lee, C. B. Park, and S. Won,“A Study on transient analysis of linear induction motor with ununiform airgap for shallow-depth underground train,” Trans., of KIEE., 2013, vol. 62, no. 5, pp. 723-729. [9] T. Nakata, N. Takahashi, and L. Fujiwara, “Efficient Solving Techniques of Matrix Equations for Finite Element Analysis of Eddy Currents,” IEEE Trans., Magn.,1988Jan, vol. 24, no. 1, pp. 170-173. [10] A. Shiri, and A. Ahoulaie, “Design Optimization and Analysis of Single-Sided Linear Induction Motor, Considering All Phenomena,” IEEE. Trans., Energy Convers.,2012Jun, vol. 27, no. 2, pp. 516-525. 12 1728 Special Issue
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will be useful in designing linear induction motor. Key Words : linear induction motor, 3D FEA, analyt-ical method, Maxwells equation, eddy current analysis 1 Introduction Linear electric machines can generate a linear driving force, and there are advantages to using a linear driving system. That is, in the case of a linear electric machine in .
Search on eddy current braking on google lIgnore links to: u Mary Baker Eddy u Fish's Eddy lIn addition to these generator brakes, the braking function of the vehicle is assured by the modular eddy-current brakes. The individual eddy-current braking magnets act on the guidance rails of the guideway and guarantee the braking of the vehicle.
and eddy currents and electro-magnetic testing [1]. Eddy Current Testing (ECT) is an electromagnetic and non-contact method used to identify and assess surface breaking or near-surface defects on materials and structures, through detection of discontinuities in segments that conduct electricity. Eddy currents, also known as Foucault currents,
EC Eddy Current ECPT Eddy Current Pulsed Thermography ECT Eddy Current Testing EM Electromagnetic EMAT Electromagnetic Acoustic Transducer EMF Electromagnetic Field GMR Giant Magnetoresistance GPIB General Purpose Interface Bus GPR Ground Penetrating Radar IR Infra-Red LPT Line Printer Terminal MFEC Multi-Frequency Eddy Current
Eddy current brakes are made up of two parts: a stationary magnetic field mechanism and a solid rotating component with a metal disc. The metal disc is subjected to a magnetic field from an electromagnet during braking, which generates eddy currents in the disc. to Lenz's rule. The spinning disc is slowed by the magnetic .
Eddy Current Probes Olympus eddy current probes consist of the acquired brands of Nortec and NDT Engineering. We offer more than 10,000 stan-dard and custom designed eddy current probes, standard refer-ences, and accessories. This catalog features many of the standard design probes
The eddy current brake implements the idea introduced above to generate a torque sufficiently large that resists the rotational motion of wheels. Figure 2 shows the schematic diagram of a simple eddy current brake with only one magnet around it. The subsequent analysis is based on this simple model. Figure 2: Schematic diagram of eddy current .
In magnetic particle testing, we are concerned only with ferromagnetic materials. 3.2.2.4 Circular Magnetic Field. A circular magnetic field is a magnetic field surrounding the flow of the electric current. For magnetic particle testing, this refers to current flow in a central conductor or the part itself. 3.2.2.5 Longitudinal Magnetic Field.
1) Direct field/energy is due to the excitation coil 2) Indirect field/energy due to eddy currents in the tube The indirect field is dominant at the remote field zone This phenomenon happen due to the different attenuation characteristics of air and the magnetic materials Introduction to RFEC technique (A variant of eddy
