A novel method to calculate the thrust of linear induction motor based on instantaneous current value

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1 Journal of Modern Transportation Volume, Number, June 1, Page Journal homepage: jmt.swjtu.edu.cn DOI: 1.17/BF A novel method to calculate the thrust of lear duction motor based on stantaneous current value Jiaqg MA *, Yunpeng ZHU, Jg JIANG Ke Laborator of Magnetic Levitation Technologies and Maglev Tras (Mistr of Education of Cha, Superconductivit and New Energ R&D Center, Southwest Jiaotong Universit, Chengdu 6131, Cha Abstract: Because of the effect, a lear duction motor (LIM runs an asmmetrical state even though the wdg of each phase is smmetric. Based on the basic prciple of the LIM, a new approach was proposed to calculate the thrust of the LIM usg the stantaneous current value. A three-phase LIM model with 1 slots and a sglelaer wdg was designed to validate this method. The experiments show that when the current is small, the calculated results basicall agree with the experiments. The agreement becomes worse with the crease of the current because of the saturation of the primar iron core. The proposed formula is suitable when the iron core of the LIM primar is an unsaturated state. Ke words: lear duction motor (LIM; asmmetrical state; stantaneous current value 1 JMT. All rights reserved. 1. Introduction T he operation mode of conventional rotar duction motors is to transmit the torque generated b a rotator to stressed members through the deliver mechanism. Thus the transmissions can run accordg to the specified mode, such as the rotation of fans and the lear motion of the vehicles. Compared with the conventional rotar duction motor, the lear duction motor (LIM has its herent advantages, i.e., it can run both rotar and translational modes without an other auxiliar devices [1-4]. Yamamamura [1] reviewed the basic theor and analed the structures of several tpes of LIMs details. He stated that the essential difference between the LIM and the rotar duction motor is the open lear air gap due to the entr and exit of the LIM. He also presented the three-dimensional analsis of the LIM to anale its special effects. Long [] presented a universal approach for the LIM s magnetic design. He developed a set of formulas to calculate the ma parameters of LIM such as the pitch, the numbers of pole pairs, the apparent power etc. Ye [3] deduced a formula from Maxwell equations to calculate the thrust of unit wave length of the LIM. The effects of the LIM clude the longitudal and transverse components [5]. Zhang [6] analed the Received Ma 1, 1; revision accepted Jun. 1, 1 * Correspondg author. majiaq_93@qq.com (J.Q. MA 1 JMT. All rights reserved doi: /j.issn.95-87X.1..3 static longitudal effects of the LIM, and presented the formulas to calculate the air gap magnetic field pulsatg coefficient, the magnetic flux densit creasg coefficient, and the longitudal equivalent magnetic leakage length accordg to electromagnetic field theor. Bolton [7] discussed the transverse effects of the LIM with sheet-rotor. He considered that the flux redistribution due to the transverse effect could fluence the performance considerabl for certa motors. Long [8] proposed a general derivg method for the LIM s equivalent circuit from the complex power equivalent relationship the electric circuit and magnetic field analsis with consideration of effect. In summar, substantial efforts [9-15] have been vestigated the theoretical stud of LIM. In order to anale the operation characteristics of an LIM, a general approach is commonl used to describe the current primar wdg as a contuous current sheet. This current sheet that changes contuousl both the temporal and spatial doma is called the travelg wave current laer. This general method, however, is not eas for calculation. In this paper, the stantaneous current value of each primar slot is used to calculate the thrust of LIM. To do this, the stantaneous current each slot has to be calculated first, and the thrust generated b each slot is determed. All thrusts fall need to be summed up to form the fal formula. We designed a three-phase LIM model with 1 slots and a sgle-laer wdg to validate this method.

2 Journal of Modern Transportation 1 (: The formation of travelg current An LIM model is shown Fig. 1, where a, b and c represent the width, height, and length of the primar, respectivel, and l, m and n the width, height, and length of the secondar, respectivel; b is the distance between the first slot and the left edge of the LIM; b k is the pitch of the primar teeth; is the distance of the primar and secondar; and c is the distance of the top side between the primar and the wdg centers. l a c x b b k b m Secondar O Wdg Slot Primar Fig. 1 An LIM model For the convenience of analsis, we assume: (1 The primar and the secondar are parallel, and the centerles of their x- planes are aligned. ( The relative motion between the primar and the secondar takes place onl the direction, and the primar is static. (3 The positive direction of the current is on the negative direction of the x axis. Fig. shows the mode of the wdgs embedded the secondar. When wdgs A, B, and C are a balanced state, the amplitudes of wdgs are equal and their phase difference is 1. Normalied operatg current is shown Fig. 3. To illustrate the formation of the discrete trav A C B a c b c n elg wave current x direction the primar of the LIM, two arbitrar pots t 1 and t are selected with an half of period. Currents I (A At t 1 : t 1 t I A I B I C Time (s Fig. 3 Three phase currents (normalied I I I 1A. (1 A B C At t : 1 1 IA IB IC A. ( The stantaneous values of each slot current are calculated b Eqs. (1 and (, and listed Table 1. The flowg current on 1 slots illustrates the formation of the travelg wave current as shown Fig. 4. The motion direction of the secondar is consistent with that of the travelg wave current. If an two phases are changed, the direction of the wave velocit turns opposite. Slot No. Table 1 Instantaneous current value each slot A Wire t 1 t (ABC t (ACB 1 +A 1 1\ 1\ C 1\ 1 1\ 3 +B 1\ 1\ 1 4 A 1 1\ 1\ 5 +C 1\ 1 1\ 6 B 1\ 1\ 1 7 +A 1 1\ 1\ 8 C 1\ 1 1\ 9 +B 1\ 1\ 1 1 A 1 1\ 1\ 11 +C 1\ 1 1\ 1 B 1\ 1\ 1 Fig. Three-phase sgle-laer wdg LIM

3 78 Jiaqg MA et al. / A novel method to calculate the thrust of lear duction motor based on In a balanced state, the value of the travelg current can be expressed as: bn I(, tk NImAs ( ta (k1 p bn NImA s ( t A (k 1, p k 1,,3,...,1, where k is the serial number of the slots; ImA is RMS current of phase A; p denotes pole pairs of the primar ( this paper; A is itial phase angle of phase A ( this paper; b n is slot pitch; is pole pitch (3 times bn this paper, and N is the turns of each wdg. If phases A and B are exchanged, then bn I(, tk NImAs ( ta (k 1, p k 1,, 3,,1. (4 In the case of asmmetrical factor, Eq. (3 is rewritten as (3 bn I(, tk NImks ( tak (k1, p k 1,,3,,1, (5 are arranged a reasonable order, the travelg magnetic field is produced. In an LIM, when the rotator with closed wires is static, there is a relative motion between the travelg field and the rotator. Then the duction current flows and exerts force on the straight wire. This is the general prciple of the duction motors. The magnetic field is generated b the flow of the current. If other conditions are constant, the strength of the magnetic field is related to the amplitude of the current. The velocit of travelg wave magnetic field is consistent with that of the travelg wave current as shown Fig. 5. Instantaneous current (A t 1 t t (B and C are exchanged Number of slot primar Fig. 4 The formation of the travelg wave current when k=1,4,7,1, Imk is equal to the RMS of phase A; k is A when k=,5,8,11; Imk is equal to the RMS of phase B; and k is B the RMS of phase C and k when k=3,6,9,1; I mk is equal to is C. When Imk ImA, and A, B C Eq. (5 is exactl equal to Eq. (3. The derivation of the wave velocit can be obtaed from Fig. 4. Then, one has O (1 ( (3 ( vs f f. (6 T where v s is the velocit of the travelg current laer; f is the frequenc of power suppl; T is the period of power suppl; and is pole pitch. We can see Fig. 4 that if the currents of B and C are exchanged, the velocit direction of the travelg current laer turns to opposite. Furthermore, if an two phases of A, B, and C are exchanged arbitraril, the direction is evitabl opposite. 3. The formation of travelg wave the magnetic field A straight wire carrg current duces magnetic field around it [9-1]. If a number of wires, each of which has a suitable variable phase angle and amplitude of current, Fig. 5 Formation of travelg wave the magnetic field Fig.5 shows the magnetic field generated b the flowg current of primar slots at time t 1 shown Fig. 3. It should be emphasied that the magnetic field of a travelg wave is contuous both the temporal and spatial doma. The velocit of travelg magnetic field is stated as v v. sm s (7 In contrast to the magnetic field of the travelg wave, the secondar with an electric conductor sheet has a motion, and its velocit is given b v ( v e v e v e ( v v, (8 r rx x r r sm where v is the relative velocit of the secondar when takg travelg wave magnetic field as reference; v is the r ab-

4 Journal of Modern Transportation 1 (: solute velocit of the secondar when takg the primar as reference; e, e, and e are unit vectors of orthogonal coordate sstem, respectivel; and v, v, and v are the x rx r r correspondg components of each axis respectivel. Because the motion onl occurs direction, v rx and v are equal to ero. Thus, the slip is given as r s v vsm where Then v v sm sm, is the component of vsm the direction. (9 v ( v v sv. (1 r sm sm To calculate the magnetic field on each slot, polar coordate is used at first and then replaced with the orthogonal coordate. Accordg to the magnetic disribution as shown Fig. 6, the magnetic field is given b I B e (11 ( r,, r where B (, r is magnetic flux densit polar coordate; is air permeabilit; I is the current of straight wire; r is the distance of the wire and some pot around the wire; and e is the unit vector direction polar coordate. v r Conductive sheet Current (out of the paper Fig. 6 Distribution of the magnetic field around the straight wire with current (the direction of the current is out of the paper The relationship of the polar and the orthogonal coordate is e e s e cos. (1 Thus, B( x,, ( Bxex Be Be K I( tk, ( e e ( ri( t, k ( es e cos. (13 r r 4. The process of calculatg thrust Fig.7 shows the force at 4 pots the magnetic field (marked (1, (, (3, and (4 Fig. 5. As shown Fig. 7, we take the 4th slot as an stance to formulate the thrust. The ductive electromotive force is generated at a ver small portion on the secondar conductive sheet. The area of this portion is dd and the length is l. Thus the ductive electromotive force is described as: v r B d l vr Bl, (14 l where dl le x, and B is magnetic flux densit the direction. Thus the ductive currenc is d ( ll dd lvrb d I /, (15 RR d s ( ll where is the resistivit of the secondar material, and l. Hence the force generated b this small participation is d f (dfxex dfe d fe didlb d li ( Be Be. (16 In (16, the force the direction is the normal force and that the direction is the thrust. Then the force df is obtaed b tegratg Eqs. (13, (15 and (16: l K I ( tk, vr df li d ( Be Be e e ( ( lvr ( Be BBe dd. (17 ( l l d B e d df 4l B B e (a Left of the 4th slot B e B e df 7l (c Left of the 7th slot B e Fig. 7 The force generated b each slot embedded with wire B B e df 4R df 4R (b Right of the 4th slot B e B e (d Right of the 7th slot

5 8 Jiaqg MA et al. / A novel method to calculate the thrust of lear duction motor based on Fig. 7(a dicates the force on the left part of the 4th slot of the secondar sheet. This force is generated b the movg magnetic field, and this field is caused b the stantaneous current flowg on the 4th slot. The forces caused b other slots as shown Fig. 5 are similar to this case. Then the force generated b the kth slot is l K I v ( tk, r ( kt, f e dd. e (18 ( ( The total force is 1 l K vr F( t f( kt, k 1 1 I k 1 l K vre 1 cn m I (, tk dd. c k 1 ( (, tk e e ( ( dd Sce the motion occurs onl direction as shown Fig. 1, even when all slots are considered, the position of the force is changed onl direction. Hence equation (19 is modified as (19 l 1 Kvre cn m F( t I(, tk dd. c k 1 ( [ b ( x1 bn ] ( If the slip is taken to consideration, Eq. ( is rewritten as: l Ksvsme F( t 1 I k 1 ( cn [ ( 1 ] m b x bn (, tk c [ b( x1 bn ] dd. (1 From Eq. (1, the relationships between the thrust and the current, the thrust and the slip, as well as the thrust and the gap can easil be obtaed. Wdg A Wdg B Wdg C PF34 5. Comparison of calculated results and measured data To validate Eq. (1, we observed the operation of a three-phase LIM with the sgle-laer wdg as shown Fig. 8. The maglev tra is levitated above the permanent magnetic guidewa with a height of 1 mm and can onl move one direction because of the characters of the high temperature superconductg materials. The secondar of the LIM is pasted on the tra, and the primar and the force meter are fixed on the ground. When the LIM is operation, the forcemeter shows the value of the thrust. The PF34 is an strument to measure the three-phase parameters of LIM such as currents, voltages, power factors, etc. We can get the different thrusts b changg the wdg currents which are measured b PF34. Even though the impedance of each of the wdg is balanced, the runng states of phases A, B, and C are not completel smmetrical due to the effect of LIM. We assume that the phases A, B and C are smmetrical and Eq. (1 s=1. Thrust (N Maglev Secondar of LIM Primar of LIM Forcemeter Fig. 8 The measurement of the thrust and currents Calculated Measured Current of phase A (A Fig. 9 Comparison of two results

6 Journal of Modern Transportation 1 (: The current of phase A is used to calculate the thrust and the calculated results are compared with the measured results shown on the forcemeter. As shown Fig. 9, the current for the calculated results ranges from to 3 A, and the measured data corresponds to the currents at 3.6, 4.9, 6., 6.8, 7.8, 11.4, 16.3, and 1.5 A, respectivel. When the current is small, two results basicall agree with each other. The agreement degraded with the crease of the current because of the saturation of the primar iron core. Eq. (1 is suitable when the iron core of the LIM primar is an unsaturated state. Because the thrust is contuous the time doma, the secondar of the LIM is forced at an time pot. The calculated thrust with the slip of 1 and current of A is shown Fig. 1. We can see that the fluctuation is so small that the thrust variation can be ignored. Thrust (N Time (s Fig. 1 The relationship between the thrust and the time 6. Conclusions Accordg to the LIM s prciple, when the motor is the runng state, the travelg wave magnetic field is formed the gap between the primar and the secondar. Accordg to this prciple, the stantaneous current value is used to deduce the thrust of the LIM with copper secondar. The calculated results agree with the experimental data, demonstratg that this approach is feasible and applicable. The calculation formula clearl shows the relationships between the thrust and the current, the thrust and the slip, as well as the thrust and the air gap, etc. Acknowledgements This work was supported b the National Natural Science Foundation of Cha (Nos , 56778, and , the National Basic Research Program (973 program, No. 7CB61696, the Australian Research Council (Grant No. DP55987 and DP881739, and the PCSIRT of the Mistr of Education of Cha (No. IRT751. References [1] S. Yamamamura, Theor of Lear Induction Motor, Toko: Universit of Toko Press, [] X.L. Long, Theor and Magnetic Design Method of Lear Induction Motor, Beijg: Cha Science Publishg, 6 ( Chese. [3] Y.Y. Ye, The Prciple and Application of Lear Mache, Beijg: Cha Mache Press, ( Chese. [4] A.I. Bolde, S.A. Nasar, Lear Motion Electromagnetic sstems, New York: John Wile and Sons Inc. Press, [5] E.R. Laithwaite, S.A. Nasar, Lear motion electrical maches, In: Proc. of the IEEE, 197, 58(4: [6] G.Y. Zhang, Research on the static longitudal effect of lear electric mache, Transactions of Cha Electrotechnical Societ, 1999, 14(5: 18-1 ( Chese. [7] H. Bolton, Transverse edge effect sheet-rotor duction motors, PIEE, 1969, 116(5: [8] X.L. Long, A general method derivg the equivalent circuit of lear duction motor, Transactions of Cha Electrotechnical Societ, 1993, 8(4: 55-6 ( Chese. [9] J.D. Kraus, Electromagnetics, New York: McGraw-Hill, [1] E.M. Purcell, Electricit and Magnetism, New York: McGraw-Hill, [11] W. Xu, J.G. Zhu, Y.C. Zhang, et al., Equivalent circuits for sgle-sided lear duction motors, IEEE Transactions on Industr Applications, 1, 46(6: [1] C.S. Chen, Equation for Mathematical Phsics, Nanjg: Southeast Universit Press, ( Chese. [13] R. Hellger, P. Mnich, Lear motor-powered transportation: histor, present status, and future outlook, Proc. IEEE, 9, 97(11: [14] S.J. Chapman, Electric Macher Fundamentals (4th Edition, New York: McGraw-Hill, 5. [15] Y.Q. Tang, Electromagnetic Field of Electric Mache, Beijg: Science Press, 1998 ( Chese. (Editor: Yao ZHOU