IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 10, OCTOBER 2011
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Dynamic Characteristics of a Linear Induction Motor for Predicting Operating Performance of Magnetic Levitation Vehicles Based on Electromagnetic Field Theory Seok-Myeong Jang1 , Yu-Seop Park1 , So-Young Sung1 , Kyoung-Bok Lee1 , Han-Wook Cho1 , and Dae-Joon You2 Chungnam National University, Daejeon, Korea Cheongyang Provincial College, Chungcheongnam-do, Korea This paper deals with the dynamic characteristics of a linear induction motor (LIM) in terms of acceleration times and jerk conditions. We employed Matlab Simulink for conducting simulations of the dynamic modeling of LIM operated by a space vector pulse width modulation inverter. From the simulation results, the maximum load conditions and minimum acceleration times to guarantee passengers’ safety were determined. Further, the electromagnetic field theory was employed to derive equivalent circuit parameters, and the results were validated by the finite element method. The analysis model was applied to a magnetic levitation vehicle for providing electromagnetic propulsion force, and its dynamic characteristics were analyzed to predict its operating performance; moreover, experimental results were employed to demonstrate the validity. We believe that the proposed prediction technique for the operating characteristics of LIMs can contribute to improving passengers’ safety and riding quality. Index Terms—Electromagnetic fields, finite element methods, magnetic levitation, MATLAB.
I. INTRODUCTION
A
LTHOUGH many researches have been conducted on linear induction motors (LIMs), the dynamic characteristics required for anticipating its operating characteristics have not been sufficiently derived. It is important to obtain good predictions of operating characteristics because when LIMs are applied to transportation systems, they become critical to guarantee passengers’ safety and riding quality. In this paper, we investigate the dynamic characteristics of LIMs applied to magnetic levitation vehicles using the procedure shown in Fig. 1. We first perform electromagnetic field analysis [1] in order to obtain the inductance included in the equivalent circuit parameters. The inductance can be obtained by integrating the flux density, and hence, the flux density is derived using a simplified analysis model. Moreover, since equivalent parameters are essential for the dynamic modeling of LIMs, the procedure to obtain the parameters is extremely significant. Further, its validation can be performed by employing various methods; in this study, we employ two methods that have high reliability and are widely used in the characteristic analysis of electrical machines—Maxwell stress tensor technique and the finite element method (FEM) [2], [3]. Based on the validated analytical procedure, we perform dynamic modeling of a LIM as well as simulations using Matlab Simulink and compare the results with the experimental result [4]–[7]. The analysis model is presented in Fig. 2, and the design specifications are given in Table I. II. DYNAMIC CHARACTERISTIC ANALYSIS OF A LINEAR INDUCTION MOTOR (LIM) A. Electromagnetic Field Analysis In this study, we have employed an equivalent model that translates the structure of teeth and slots to the sheet current, as
Manuscript received February 21, 2011; revised April 25, 2011, May 01, 2011; accepted May 01, 2011. Date of current version September 23, 2011. Corresponding author: Y.-S. Park (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMAG.2011.2153188
Fig. 1. Procedure for dynamic characteristic analysis of LIM.
shown in Fig. 3 [1]. The inner characteristics of the equivalent model are expressed in terms of permeability and conductivity. In a 2-D model, some reasonable assumptions can be made for simplicity. First, we assume that all regions extend infinitely in the x-direction, whereas the primary extends infinitely in the y-direction; the primary moves in the x-direction. Secondly, the physical constants of the regions are homogeneous, isotropic, and linear, and variations in the z-direction are ignored. Thirdly, all the currents flow only in the z-direction. Moreover, the secondary consists of a back-iron and an aluminum-conducting plate (Fig. 2) with permeability identical to that of air. On the other hand, the permeability of back-iron is considered as infinity. In practice, the back-iron is usually not laminated, and hence its conductivity is directly adopted. In addition, it is assumed that current does not exist in region 1 and 4 due to high permeance. Moreover, since the primary is typically laminated, its conductivity can be ignored. From Maxwell’s equations, the governing equation of the analysis model in Fig. 3 is derived as (1); further, from the above can be expressed assumptions, the magnetic vector potential as (2) Therefore, (1) can be rewritten as (3)
r A = @A @t 0 V 2 (r 2 A) ; V A = A(y)e @ A +@ A @x @y
= velocity
i = = ; = pole pitch @A : = j!A + V @x
(1) (2) (3)
The velocity of the mover is expressed by (4) in terms of slip and synchronous speed. Since (3) can be rewritten as (5), and if (6) is defined, the final governing equation can be obtained as (7)
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(8) (9) (10) (11) To derive the unknown coefficients in the and components of the flux density, the boundary conditions are considered as (12)–(14). First, this analysis assumes that all the field compo. The continuity is represented in terms nents appear at of the normal component of the flux density and magnetic field . However, the boundary between the primary and at air gap has discontinuous characteristics in the case of both sheet current and current density distributions, as shown in (14)
Fig. 2. Analysis model applied to magnetic levitation vehicles. TABLE I DESIGN SPECIFICATION OF ANALYSIS MODEL
(12) (13) (14) Finally, to obtain the unknown coefficients of the flux density, and . (15)–(17) are written as calculation matrices stands for Matrix (shown at the bottom of the page): Here,
M2 = j C1 C2 D2 C3 D3 C4 D4 D5 jT M3 = j 0 0 0 0 0 0 0 Jn jT ; M1 1 M2 = M3 :
Fig. 3. Equivalent model for electromagnetic field analysis.
Vx = (1 0 s)Vs 2 0 2 Az + @@yA2z = (j!Az 0 Vx Az ) jsVs 2 1 + = 2 @ 2 Az 0 2 Az = 0: @y2
(4)
Since the magnetomotive force (MMF) distribution of a distributed winding is not sinusoidal, the actual slot-embedded winding should be replaced by a train of pulses of the appropriate height. These pulses repeat periodically over every pair of poles, and they can be Fourier-analyzed to obtain the predominant harmonic component. The sheet current of one phase is given by (18)–(19)
(5)
Jn =
(6)
Jm =
(7)
of When (7) is calculated, the magnetic vector potential each region is derived, and the flux density of each region can be finally obtained as (8)–(11) by using
(16) (17)
1
n=1;odd Jd
(Jm 0! iz )ej (!t0 x) ;
4 sin(n p =2) n NI ! Jd = ; = ; ! = 2f: ! p
(18) (19)
To validate the analytical method, we employed the Maxwell stress tensor technique using the derived flux density [2]. Here, the electromagnetic propulsion force and levitating force can be obtained by (20) and (21), respectively; they vary with the slip. FEM has also been employed to validated the analytical
(15)
JANG et al.: DYNAMIC CHARACTERISTICS OF A LINEAR INDUCTION MOTOR
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The inductance is defined as the ratio of the linking magnetic flux to the current producing the flux (23). Therefore, substituting the flux density equations into (23) leads to (24)
Lph = N I N L1 = I N L2 = I
s
B 1 ds = N I By3 ds;
s
A 1 dl
By4 ds; L3 = N I
Bx4 ds:
(23)
(24)
The mutual inductance is obtained from the surface integral of the normal flux density of region 3 shown in (25), whereas the primary leakage inductance can be derived from (26)
Lm = L1 Ll = (L2 0 L1 ) + L3 :
Fig. 4. Force characteristic comparison versus slip.
(25) (26)
B. Dynamic Characteristic Considering Operating Condition In the equivalent circuit, if it is assumed that the neutral point is not connected to the outer device and the instant sum of each phase voltage is 0, the d-q axis voltage in a certain speed can be expressed as (27)–(28). Here, and indicate primary and represent the resissecondary, respectively, whereas and tance and differential operator, respectively, in terms of time. In addition, the flux linkages of primary and secondary can be calculated as (29)–(30), respectively
Fig. 5. Equivalent circuit of LIM.
(27) (28) (29) (30) The electromagnetic propulsion force of LIM is calculated as (31), and with the assumption that is zero, the flux linkage of the primary and secondary can be expressed as (32)–(33) Fig. 6. Dynamic modeling of LIM.
procedure [3]; the 2D FEM is applied only to several slip conand ). Only ditions ( these conditions are considered primarily because if all the cases were analyzed by FEM, the analysis times would be very long. Furthermore, we considered that the seven slip conditions are sufficient to validate the performed analytical procedure. Fig. 4 shows the results of the analytical and FEM methods in the form and refer to the elecof lines and dots, respectively. Here, tromagnetic propulsion force and levitating force, respectively 2
Fx = 0 Fy = 0 1
2
0
2
l =2
0l
0l
0
1 Re(Hal x H 3 )dxdz al y 2 1 Re(Hal y H 3 0 Hal x H 3 )dxdz: al y al x 2
=2 l =2
=2
(20) (21)
The LIM can be simply expressed as an equivalent circuit as shown in Fig. 5, and the circuit parameters are calculated by the flux density obtained previously in this paper. As a solution to obtain the parameters, the magnetic flux passing through the plane of the close loop is defined as (22)
8 = B 1 ds = (r 2 A) 1 ds = A 1 dl: s
s
c
(22)
Fd = 3 (dp iqp 0 qp idp ) 2 t dp = (Vdp 0 rp idp )dt; qp = 0
ds =
t 0
(0rs ids 0 !s qs)dt; qs =
(31) t 0 0
(Vqp 0 rp iqp )dt t
(32)
(0rs iqs + !s ds )dt: (33)
As shown in Fig. 6, by using the equations illustrated above and the derived equivalent circuit parameters, the dynamic modeling of LIM was performed, and it is operated by Space Vector Pulse Width Modulation (SVPWM) inverter [4]–[6]. Further, the LIM of the propulsion system is operated at a constant voltage/frequency ratio [7]. For the dynamic characteristic analysis, the dynamic simulations considering load conditions were first performed as shown in Fig. 7(a), and the primary weight was varied for lighter and heavier load conditions. In the figure, the load is varied by 20 (ton) steps from 10 (ton) to 70 (ton). As the load becomes heavier, the time to follow the reference is delayed. In particular, when the load condition is 70 (ton), the speed becomes approximately 12 (km/h) in 7 (s), which cannot follow the speed reference. Here, 10 (ton) 50 (ton) are possible load conditions; we determined that for better speed characteristics, the maximum load condition is 50 (ton). On the other hand, Fig. 7(b) shows the case for the acceleration time and the simulation results. The acceleration time is varied from 3 (s) to 7 (s), and the speed follows the reference
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 47, NO. 10, OCTOBER 2011
properly from 5 (s) to 7 (s). However, when the magnetic levitation vehicle is accelerated in 3 (s), vibrations occurs and the speed characteristic does not follow the reference. As a result, the minimum acceleration time is determined as 5 (s) for stability. In addition, during the operation of LIMs, the rail conditions or external disturbances can affect their speed characteristics. In practice, the curve region included in the rail causes an outer disturbance of 300 (N) to a vehicle system; in this paper, the external disturbance has been considered as a jerk. When the external disturbance is applied to the mechanical equation loop in Fig. 6, the speed characteristics by dynamic simulations in terms of the jerk are presented in Fig. 8(a), whereas the experimental results are shown in Fig. 8(b). For the simulations, the external disturbances at two times values has identical amount with different time conditions, thereby indicating the continuous nature of the speed characteristics. Further both the results from simulations and experiments corresponded well. III. CONCLUSION
Fig. 7. Speed characteristic with operating conditions: (a) load variation and (b) acceleration time.
In this paper, dynamic modeling and speed characteristic analysis have been performed for predicting the operating characteristics of magnetic levitation vehicles with LIMs. To derive the equivalent circuit parameters, electromagnetic field analysis was applied, and the results were validated by FEM. In addition, from the simulation results, the maximum load conditions and minimum acceleration times to guarantee the passengers’ safety were determined. In particular, the simulation and experimental results were shown to be similar, and the prediction procedure was validated. We believe that our proposed operating procedure can make significant contributions to achieving safety and reliability in vehicles and be the basis of related researches. ACKNOWLEDGMENT This work was supported in part by KESRI (No. 20101020300080), which is funded by MKE (Ministry of Knowledge Economy), and in part by the Human Resources Development of the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government Ministry of Knowledge Economy (No. 20104010100600). REFERENCES
Fig. 8. Speed characteristic under jerk condition of external disturbances of 300(N): (a) simulation result and (b) experimental result.
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