The time and space characteristics of magnetomotive force in the cascaded linear induction motor

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1 J. Mod. Transort. (13) 1(3): DOI 1.17/s The time and sace characteristics of magnetomotive force in the cascaded linear induction motor Dajing Zhou Jiaqing Ma Lifeng Zhao Xiao Wan Yong Zhang Yong Zhao Received: 4 July 13 / Revised: July 13 / Acceted: August 13 / Published online: 7 Setember 13 The Author(s) 13. This article is ublished with oen access at Sringerlink.com Abstract To choose a reasonable mode of three-hase winding for the imrovement of the oerating efficiency of cascaded linear induction motor, the time and sace characteristics of magnetomotive force were investigated. The ideal model of the cascaded linear induction motor was built, in which the B and C-hase windings are resectively searated from the A-hase winding by a distance of d and e slots itch and not overlaed. By changing the values of d and e from 1 to, we can obtain different modes of three-hase winding with the different combinations of d and e. Then, the air-ga magnetomotive forces of A-, B-, and C-hase windings were calculated by the magnetomotive force theory. According to the transient suerosition of magnetomotive forces of A-, B-, and C-hase windings, the theoretical and simulated synthetic fundamental magnetomotive forces under different arrangement modes were obtained. The results show that the synthetic magnetomotive force with d = and e = 4is close to forward sinusoidal traveling wave and the synthetic magnetomotive force with d = 4 and e = is close to backward sinusoidal traveling wave, and their amlitudes and wave velocities are aroximately constant and equal. In both cases, the motor could work normally with a D. Zhou (&) J. Ma L. Zhao X. Wan Y. Zhang Y. Zhao Key Laboratory of Magnetic Levitation Technologies and Maglev Trains (Ministry of Education of China), Southwest Jiaotong University, Chengdu 6131, Sichuan, China zdj789@163.com D. Zhou J. Ma L. Zhao X. Wan Y. Zhang Y. Zhao Suerconductivity and New Energy R&D Center, Southwest Jiaotong University, Chengdu 6131, Sichuan, China Y. Zhao School of Materials Science and Engineering, University of New South Wales, Sydney, NSW, Australia high efficiency, but under other 18 arrangement modes (d = 1, e = ; d = 1, e = 3; d = 1, e = 4; ), the synthetic magnetomotive force resents obvious ulse vibration and moves with variable velocity, which means that the motor did not work normally and had high energy loss. Keywords Linear induction motor Three-hase winding Magnetomotive force 1 Introduction Linear motor is a kind of electrical equiment which can directly convert the electrical energy to linear movement. Comared with traditional rotating machine, the drive system of linear motor works without the intermediate gearing, which simlifies the driver system, and makes its linear velocity unlimited and moving rocess without mechanical touch. In addition, the noise level of linear motor is very low. For all these characteristics, linear motor is widely alied to high-seed ground transortation [1 4]. When the ole number of a linear motor is not less than six, the values of negative-sequence current and zerosequence current are much small comared with the ositive sequence current; thus their influence on the cascaded linear motor can be ignored [, 6]. In this case, the asymmetry of three-hase current will not be considered. With three-hase symmetrical current flowing into threehase winding, linear motor will roduce a traveling magnetomotive force (MMF) wave in the air-ga between rimary and secondary windings. Since the energy exchange in mechanic-electronics is achieved by the airga magnetic field of linear motor [7], the time and sace characteristics of the MMF wave directly affect the oerating efficiency and energy consumtion in linear motor. 13

2 MMF in the cascaded linear induction motor 19 The motor can work efficiently with the waveform of the MMF close to a sinusoidal traveling wave [8]. The finite element method (FEM) is an effective and accurate method to investigate the linear motor characteristics. Lu et al. [9] analyzed the features of air-ga magnetic field in large air-ga linear induction motor. Selcuk and Kurum [1] built a simlified FEM model of an actual short rimary linear induction motor and solved it for airga magnetic field distribution. Lu et al. [11] analyzed the two-dimensional transient air-ga magnetic field of long rimary induction motor with the hel of FEM. Li et al. [1] studied the characteristics of temerature field for tubular linear motor with the FEM. In this aer, we will solve the single-hase winding MMF and three-hase winding-synthesized MMFs in single-side linear induction motor (SLIM) by the classical theory of MMF [13] and adot the FEM to validate the theoretical results. The work conditions and efficiency of SLIM are then otimized by analyzing MMF characteristics with different three-hase winding arrangement. Theoretical analysis and simulation model for SLIM.1 The MMF of A-hase winding The established theoretical model of SLIM is shown in Fig. 1, in which the leak flux in this model is ignored. In this model, it is assumed that rimary and secondary are infinite long, rimary iron yoke is not in magnetic saturation, the magnetoconductivity of rimary and secondary is infinite, and magnetic induction intensity only contains the comonent in the y axis direction. It is also assumed that the current flows along the z axis direction, the equivalent air-ga d between rimary and secondary is distributed evenly along the x axis direction, the magnetic otential difference of three-hase winding is distributed evenly along the air-ga, the number of ole airs is and ole itch is three slots itch long, and the three-hase wingding is a bi-layered full-itch winding. Without loss of generality, the sinusoidal current as shown in Eq. (1) is assumed flowing into A-hase wingding, i.e., i A ¼ ffiffi IA cos xt: ð1þ With reference to Fig. 1, according to the Amere s circuital Law, A-hase winding MMF in air-ga can be exressed as [14] f A ¼ H y d ¼ N i A f A ¼ H y d ¼ N i A when m h m; ðaþ when m h 3 m; ðbþ where m = 1,, 3,, electrical angle h = x/s, the number of turns in series winding N = N c, and N c is the number of turns in a single coil. Since the quantity f A (h) reresents a eriodic square wave along the air-ga, it can be reresented by the Fourier series as f A ðhþ ¼f A1 cos h þ f A3 cos 3h þ f A cos h þ þ f An cos nh; ð3þ where f An is f An ¼ 4 T Z T f A ðhþ cosðnx hþdh ¼ 1 n 4 Ni A sin n ; ð4þ n = 1,, 3, Substituting Eqs. (1) and (4) into Eq. (3), we have the instantaneous value of A-hase wingding MMF: f A ðh; tþ ¼ 4 ffiffi NI A cos h 1 3 cos 3h þ 1 cos h þ 1 n sinðn Þ cos nh cos xt: ðþ From Eq. (), we can find that A-hase winding MMF after the Fourier series transformation can be decomosed Fig. 1 The theoretical model of SLIM and A-hase winding MMF distribution diagram J. Mod. Transort. (13) 1(3):

3 196 D. Zhou et al. into fundamental wave and a series of higher harmonics, and the amlitudes of fundamental wave and higher harmonics ulse over time with the current frequency. Because fundamental wave determines the energy conversion of linear motor and its main erformance, it is most imortant and fundamental to analyze the fundamental MMF [1].. The MMFs of three-hase winding As shown in Fig., when the B and C-hase windings are resectively d and e slots itch away from the A-hase winding, we can obtain the MMF of B-hase winding and C-hase winding resectively by moving the MMF of A- hase winding d/3 and e/3 along the ositive direction of h axis, resectively. Let three-hase current flow into three-hase winding, through above analysis we can obtain the exression of the fundamental MMF for A, B, C hase windings, resectively, F A ðh; tþ ¼ 4 ffiffi NI A cosðhþ cosðxtþ; ð6aþ F B ðh; t; dþ ¼ 4 ffiffi NI A cos h 1 3 d cos xt 3 ; ð6bþ F C ðh; t; eþ ¼ 4 ffiffi NI A cos h 1 3 e cos xt þ 3 : ð6cþ Adding the instantaneous values of fundamental MMF with A, B, C hase windings, we have the fundamental MMFs: Fig. Three-hase winding distribution diagram Fðh; t; d; eþ ¼F A ðh; tþþf B ðh; t; dþþf C ðh; t; eþ: ð7þ According to the theoretical model of SLIM, the values of d and e can be taken from 1,, 3, 4 and in a cycle of MMF, so that the number of arrangement modes of threehase winding is with the combination between d and e. The results of fundamental MMF S with different arrangements of three-hase windings will be analyzed later..3 The simulation model for SLIM The simulation model of SLIM is established by Ansoft Maxwell as shown in Fig. 3. Table 1 resents the secific arameters related to this model. Different simulation data of MMFs will be obtained by changing the relative ositions of three-hase winding. 3 Results and analysis 3.1 The MMFs with d = 1 and e = Referring to Fig., we move B-hase winding and C-hase winding from the osition of A-hase winding 1 slot itch and slots itch, resectively. Then, substitute d = 1 and e = into Eq. (7) and the exression of fundamental MMFs is calculated by trigonometric formula [16]: Fðh; t; 1; Þ ¼ ffiffi NI A½ cosðxt þ hþþcosðxt hþ þ cosðxt þ h Þþcos xt h 1 3 þ cos xt h þ 4 3 ð8þ The MMF is comosed of three forward traveling waves and two back traveling waves referring to Eq. (8), and the results are shown in Fig. 4. As can be seen in Fig. 4a, when the satial MMF waveform is at the transient time of xt =, /,,3/, resectively, and with a scaling factor b ¼ ffiffiffi NIA =, the MMF ushes to the left in sace and exists obvious vibration. The movement with variable velocity is shown in Fig. 4c, d. These features will reduce Fig. 3 The simulation model of SLIM 13 J. Mod. Transort. (13) 1(3):

4 MMF in the cascaded linear induction motor 197 the efficiency of linear motor thrust in the horizontal direction, and at the same time also bring high energy loss. According to the simulation model for SLIM, when arranging three-hase winding to be d = 1 and e =, the simulation transient waveform of the MMF is shown in Fig. 4b, which further validates the characteristics of MMF with d = 1 and e =. 3. The MMFs with d =, e = 4, and d = 4, e = In order to kee the amlitude and wave velocity of MMF invariant over time and avoid the ulse vibration and Table 1 SLIM arameters for simulation model Parameter Value Parameter Value Virtual current I A 8 A Air-ga d. mm Power frequency f Hz Slots er ole er hase q 1 Pole itch s 36 mm Turns er coil N c 6 Pole airs 8 movement with variable velocity of MMF, we solved the MMFs under different arrangements of three-hase winding. The result indicates that the MMFs in the condition of d =, e = 4ord = 4, e = are a traveling wave with its amlitude and velocity being constant as shown in Fig. a, c, which are exressed, resectively, by Eqs. (9a) and (9b): Fðh; t; ; 4Þ ¼ 6 ffiffiffi NI A cosðxt hþ; ð9aþ Fðh; t; 4; Þ ¼ 6 ffiffiffi NI A cosðxt þ hþ; ð9bþ where h ¼ k x ¼ s x. From Fig. a, c, we can see that F(h, t,, 4) is a forward traveling wave and F(h, t, 4, ) is a backward traveling wave. Their amlitude and wave velocity can be exressed as F m ðh; t; ; 4Þ ¼F m ðh; t; 4; Þ ¼ 6 ffiffi NI A; ð1aþ mðh; t; ; 4Þ ¼mðh; t; 4; Þ ¼ x k ¼ s x : ð1bþ (a) F /β ω t= ωt=π/ ωt=3π/ (b) H y /(A/km) ω t= / ω t=3π/ (c) 3. (d).. 1. F m /β. v /v max π /4 π / 3π /4 π π /4 3π / 7π /4 ω t /(rad) π. π /4 π / 3π /4 π π /4 3π / 7π /4 ω t /(rad) π Fig. 4 The MMFs with d = 1 and e = and its amlitude and velocity characteristics. a theoretical fundamental MMFs. b simulated MMFs. c amlitude of theoretical fundamental MMF. d velocity of theoretical fundamental MMF J. Mod. Transort. (13) 1(3):

5 198 D. Zhou et al. (a) F /β ω t= / ω t=3π/ (b) ) H y /(A/km 8 ω t= / ωt=π ω t=3π/ (c) 4 3 ω t= / ω t=3π/ (d) 8 ωt= / 6 ω t=3π/ F /β 1 H y /(A/km) Fig. The MMFs with d =, e = 4 and d = 4, e =. a theoretical fundamental MMFs with d = and e = 4. b simulated MMFs with d = and e = 4. c theoretical fundamental MMFs with d = 4 and e =. d simulated MMFs with d = 4 and e = Based on the simulation model for SLIM, we also changed the arrangement mode of three-hase winding to d =, e = 4 and d = 4, e =, resectively, and obtained the MMFs simulated by Ansoft Maxwell. The results are shown in Fig. b, d. Since the simulation model for SLIM is not fully ideal and its MMFs contain other higher harmonics, the MMFs are close to sinusoidal traveling wave with constant amlitude and velocity. 4 Conclusion Based on the classical MMF theory of rotating machine, we established the theoretical model and simulation model for SLIM, in which three-hase winding has arrangement modes in a cycle of MMF, and the arrangement mode determines the time and sace characteristics of MMFs. Through the calculation with electromagnetic theory and finite element software simulation, we solved and discussed the MMFs under arrangement modes of three-hase winding, resectively. On the basis of the above analysis results, the conclusions are drawn as follows: (1) The MMFs with d = and e = 4 are close to sinusoidal wave, and it travels toward the ositive direction with the constant amlitude and velocity. In this case, the motor can work normally with a high efficiency. () The MMFs with d = 4 and e = are close to sinusoidal wave, and travel toward the negative direction with the constant amlitude and velocity. Also the motor can work normally with a high efficiency. (3) In other 18 conditions, the MMFs show obvious ulse vibration and movement with variable velocity, such as the MMFs under d = 1, e =. This means that the motor does not work normally, and instead it has high energy loss. Acknowledgments This work was suorted by the National Magnetic Confinement Fusion Science Program 11GB111, 13 J. Mod. Transort. (13) 1(3):

6 MMF in the cascaded linear induction motor 199 Program of International S&T Cooeration S13ZR9, the financial suort of the National Natural Science Foundation of China (No. 1711), the Fundamental Research Funds for the Central Universities (SWJTU11ZT16, SWJTU11ZT31), the Science Foundation of Sichuan Province 11JY31, 11JY13. Oen Access This article is distributed under the terms of the Creative Commons Attribution License which ermits any use, distribution, and reroduction in any medium, rovided the original author(s) and the source are credited. References 1. Ye YY () Princile and alication of linear motor. China Machine Press, Beijin (in Chinese). Wang TC (1971) Linear induction motor for high-seed ground transortation. IEEE Trans Ind Gen Al 7(): Deng JM, Chen TF, Tang JX et al (13) Otimum sli frequency control of Maglev single-sided linear induction motors to maximum dynamic thrust. Proc CSEE 33(1):13 13 (in Chinese) 4. Wang K, Shi LM, He JW et al (9) A decouling control of normal-and-thrust forces in single-sided linear induction motor. Proc CSEE 9(6):1 14 (in Chinese). Lu JY, Ma WM, Sun ZL et al (9) Research on static longitudinal end effect of linear induction motor with multi-segment rimary. Proc CSEE 9(33):9 11 (in Chinese) 6. Sun ZL, Ma WM, Lu JY et al (1) Research of static longitudinal end effect and imedance matrix for long rimary doublesided linear induction motors. Proc CSEE 3(18):7 77 (in Chinese) 7. Long LX (6) Theory of linear induction motor and its electromagnetic design method. Science Press, Beijing (in Chinese) 8. Fan SD (1996) Determination for magnetic density wave in the ga of the ermanent magnet DC motor. Small Sec Electr Mach 3: (in Chinese) 9. Lu QF, Fang YT, Ye YY () A study on force characteristic of large air ga linear induction motor. Proc CSEE (1): (in Chinese) 1. Selcuk AH, Kurum H (8) Investigation of end effects in linear induction motors by using the finite-element method. IEEE Trans Magn 44(7): Lu JY, Ma WM, Xu J (8) Modeling and simulation of high seed long rimary double-sided linear induction motor. Proc CSEE 8(7):89 94 (in Chinese) 1. Li LY, Huang XZ, Kou BQ et al (13) Numerical calculation of temerature field for tubular linear motor based on finite element method. Trans China Electrotech Soc 8(): (in Chinese) 13. Lio TA (4) Introduction to AC machine design. University of Wisconsin Press, Chicago 14. Liu CY, Wang H, Zhang ZJ et al (11) Research on thrust characteristics in ermanent magnet linear synchronous motor based on analysis of nonlinear inductance. Proc CSEE 31(3):69 76 (in Chinese) 1. Xie MT, Zhang GY (4) Electromechanics. University of Chongqing Press, Chongqing (in Chinese) 16. Zhou SL (3) The analysis of stator magnetic otential in the three-hase asynchronous motor. J Tongling Coll ():73 74 (in Chinese) J. Mod. Transort. (13) 1(3):