End Effect and Electromagnetic Force Characteristics of Two Adjacent Linear Induction Motors

The medium and low speed maglev is driven by linear induction motors (LIMs), which are composed of finite-length primary winding and infinite-length secondary induction rail. This configuration offers several advantages, including structural simplicity, low construction costs, and ease of maintenance. However, the structural characteristics of a discontinuous primary and a continuous secondary plate result in mutual interference of secondary eddy currents at adjacent positions. This interference alters the air-gap magnetic field of the motor, which directly impacts both the traction force and the normal force of the motor. To elucidate the influence characteristics and underlying mechanisms of adjacent motor configurations on electromagnetic forces, enhance motor traction performance, and address issues related to the normal force, this study focused on two adjacent and series-connected LIMs. Theoretical and simulation models were developed to investigate the effects of two key factors, including the motor spacing and slip frequency, on the traction and normal forces of the two motors.

To investigate the influence of adjacent LIM arrangements on electromagnetic forces in medium and low speed maglev and gain a deeper understanding of the magnetic field interference between two LIMs, as well as its intrinsic effects on electromagnetic forces, firstly, vector magnetic potential equations were established for each region of the adjacent motors based on Maxwell’s equations, and the two-dimensional vector magnetic potential for each region was solved by using boundary conditions. Secondly, the expressions for the air-gap magnetic field, traction force, and normal force of the two LIMs were derived based on the air-gap vector magnetic potential and the distribution of primary currents. These expressions were used to analyze the characteristics and underlying mechanisms of how magnetic field interference between adjacent motors affects electromagnetic forces. Thirdly, a finite element analysis (FEA) model of two adjacent motors was developed by using Maxwell software to calculate the electromagnetic forces under different motor spacings. The simulation results were compared with theoretical predictions to validate the accuracy of the theoretical calculation method. Finally, the established theoretical model was utilized to explore the effects of two critical factors, including motor spacing and slip frequency, on the electromagnetic forces of the adjacent motors. By analyzing the air-gap magnetic field under various operating conditions, the study revealed the intrinsic mechanism by which the electromagnetic force of LIM2 was influenced by motor spacing.

The variation trends of the traction force between the two motors, as calculated through both theoretical and finite element simulation methods, exhibit a high degree of consistency, thereby validating the accuracy of the theoretical model. In the boundary conditions of the theoretical model, the magnetic permeability of the iron core and back iron is assumed to be infinite, and the primary current is approximated as a thin layer. Consequently, the theoretical calculations yield slightly larger fluctuations in the traction force of LIM2 with respect to motor spacing compared to the simulation results. From the expression of the air-gap magnetic field of the motor, the backward traveling wave caused by the end effect at the exit of LIM1 is influenced by factors such as the primary current, slip frequency, and motor spacing of LIM2. Similarly, the forward traveling wave caused by the end effect at the entry of LIM2 is affected by the parameters of LIM1 and the spacing between the two motors. Consequently, the air-gap magnetic field of the linear motor is altered due to the influence of the adjacent motor. The impact of adjacent motor arrangements on the longitudinal air-gap magnetic field is relatively minor, whereas the effect on the vertical air-gap magnetic field is more pronounced. This influence leads to corresponding variations in the motor’s traction force and normal force. As the motor is less affected by the backward traveling wave induced by the exit-end effect and more significantly affected by the forward traveling wave induced by the entry-end effect, LIM1 is less affected by LIM2, whereas LIM2 is more significantly influenced by LIM1. The computational results indicate that the traction force of LIM1 is nearly unaffected by motor spacing and exhibits minimal variation under different slip frequencies. For various slip frequencies, the operating speed and input current of LIM1 remain consistent, resulting in negligible differences. In contrast, for LIM2, as the slip frequency is smaller, its traction force is more significantly influenced by motor spacing. As the spacing increases, the amplitude of traction force fluctuations for LIM2 decreases across all slip frequencies. When the slip frequency is 8 Hz, and the spacing is 1.5 times the pole pitch, the traction force of LIM2 reaches 8.4 kN, which is 1.83 times that of LIM1 (4.6 kN), representing an 83% increase. Conversely, when the spacing is reduced to 0.6 times the pole pitch, the traction force of LIM2 drops to 0.6 kN, which is merely 13% of LIM1’s traction force, signifying an 87% reduction. These results underscore the importance of spacing design in optimizing the traction performance of LIM2. Furthermore, as the slip frequency decreases, the normal force of LIM1 increases. However, the normal force of LIM1 is almost unaffected by the motor spacing at different slip frequencies. In contrast, the normal force of LIM2 exhibits fluctuations with varying spacing under different slip frequencies. Smaller spacing and slip frequencies lead to a greater impact on the normal force of LIM2. A well-designed motor spacing can effectively reduce the normal force of LIM2, thereby enhancing the stability of maglev vehicles. When the slip frequency is 8 Hz, and the motor spacing is 2.1 times the pole pitch, the normal force of LIM2 is 1 kN, which is 6.6 kN lower than that of LIM1 measured at 7.6 kN. This reduction translates to a weight decrease of 660 kg per motor for the levitation system. However, when the motor spacing is reduced to 1.2 times the pole pitch, the normal force of LIM2 rises dramatically to 28.2 kN, which is 3.7 times greater than that of LIM1, representing a 270% increase. This additional force imposes approximately two tons of extra load on the levitation system per motor.

The electromagnetic force of a rear-positioned LIM is significantly influenced by the presence of a front-positioned LIM, as well as the spacing between the motors. However, the optimal spacing for achieving the desired traction force and normal force varies under different operating conditions and system requirements. Therefore, in practical engineering applications, both the traction demands and levitation capabilities of the vehicle should be considered for the design of the motor spacing. The findings can provide theoretical guidance and technical support for the application of LIMs in medium and low speed maglev trains. However, due to the complexity of LIM theoretical derivations and the numerous parameters involved, certain idealized assumptions and simplifications are necessary during the mathematical analysis. The quantitative results presented should be interpreted as reference values, and the results of the study are based on two-dimensional electromagnetic field theory. Future research may explore the electromagnetic interaction between adjacent motors by using three-dimensional field models to achieve more precise findings.