Finite Element Analysis and Electromagnetic Field Optimization of Linear Synchronous Motor in High-Speed Maglev Systems

Amid the evolving economic landscape, a heightened demand for rapid transportation modes has arisen. China's research endeavors into high-speed magnetic levitation (maglev) trains have seen substantial progress. Notably, in May 2019, CRRC Sifang unveiled its groundbreaking 600 km/h high-speed maglev train. By 2020, successful trials of a 600 km/h high-speed maglev test vehicle were conducted on the Tongji University maglev test line in Shanghai, and by July 2021, a domestically developed 600 km/h high-speed maglev train was launched in Qingdao [1], [2], [3], [4]. Ensuring the stability of such rapid maglev systems necessitates a thorough exploration into the magnetic fields and electromagnetic interplays during high-velocity operations [5].

Studies on the EMS-type LSM have been conducted by numerous scholars both domestically and abroad. Due to the open slots on the surface of the long stator that hold single-turn three-phase windings, the resultant cogging effect is considered the main cause of fluctuations in traction and levitation forces [6], [7]. Analysis of the vertical force variation under the cogging effect in permanent magnet linear motors was conducted by Xia et al. [8], who used the Maxwell tensor method and finite element analysis to study the effect of pole-arc coefficient and tooth tip width. Wang et al. [9] provided a relatively accurate derivation of the relative magnetic permeability function in the air gap, considering the Dirichlet law and winding characteristics. Liu et al. [10] examined the influence of stator segmentation patterns on the air gap magnetic permeability function, while Zhang and Lu [11] designed a skewed tooth scheme to reduce thrust fluctuation caused by stator cogging effects, optimizing parameters through simulation.

However, most LSM cogging effect studies have concentrated on the analysis of stator cogging effects, with less attention paid to the impact generated by the vehicle's magnetic poles. The characteristics of motor traction and levitation forces are also key performance indicators. The majority of these studies have predominantly utilized finite element analysis in combination with theoretical results from simplified models for comparison [12], [13], [14], [15], [16], [17], [18], [19]. This approach enables the exploration of a variety of factors influencing LSM performance. Three-dimensional motor model simulation methods were employed by other researchers, providing a more nuanced view of LSM behaviors [20]. In a notable first, Maxwell finite element simulation software was used to analyze long stator winding fault conditions [21], contributing a new perspective to the field.

Cao et al. [22] not only analyzed the coupled characteristics of traction and levitation forces under different armature currents and power angles but also examined the effect of aerodynamic lift on these characteristics. It concluded that, under current aerodynamic characteristics, there is a maximum speed for constant-guidance maglev trains. Despite these advancements, there is a noticeable lack of research on the magnetic field and electromagnetic relationships of high-speed maglev trains.

The magnetic field, pivotal for maglev train suspension, plays an indispensable role in assuring the safe and consistent control of high-speed maglev trains. As the crux of the maglev train suspension mechanism, the magnetic field guarantees stable suspension, orchestrates train alignment, dampens external perturbations, and directly influences the system's stability, agility, and safety. In the present study, an analytical magnetic field model of the motor was formulated. Key parameters such as the effective air gap, armature variables, levitation force, and traction force were meticulously computed. With the assistance of ANSYS Electronic Desktop electromagnetic field FEA software, a two-dimensional motor model was employed to discern thrust and levitation dynamics under variant stator currents and disparate linear generator loads. This investigation unveiled salient features encompassing magnetic field distribution, intricate nonlinear associations between current and magnetic force, and saturation characteristics. This exploration furnishes foundational theoretical insights for refining the electromagnetic force in 600 km/h high-speed maglev trains.

To model the magnetic field, an energy-based approach has been utilized wherein the magnetic field is stratified. Within this stratification, an infinitesimally thin layer representing the stator and excitation current, termed the "equivalent current layer," has been introduced. By leveraging the Laplace equation for a curl-free field, the distribution of the magnetic field across each stratified region can be discerned. For the sake of streamlining the analytical process, the following assumptions were incorporated:

•Analysis of the magnetic field and force was confined to a two-dimensional plane.

•Both the stator and rotor were assumed to possess magnetic materials of identical, infinite magnetic permeability.

•Only the fundamental magnetic field for both armature and poles was considered in the analysis.

•The potential effects of magnetic saturation and eddy currents were overlooked during the computation phase.

Given a three-phase winding with $N$ turns per phase, a phase difference of $120^{\circ}$ between each successive phase, and a designated current density $J_x$ (with $x$ adopting values of $a, b, c$), alongside parameters like stator slot width $b_s$, slot pitch $a_s$, and pole pitch $\tau_s$, the Fourier expansion of the current density can be articulated as:

Subsequently, the cumulative three-phase currents were computed, and the fundamental wave was isolated, resulting in:

Building on the premise wherein the stator pole pitch mirrors the vehicle excitation pole pitch, the Fourier expansion of the excitation current density is described as:

It's noteworthy to highlight that while varying stator and rotor pole pitches might be harnessed in real-world LSM applications with the objective of minimizing electromagnetic force oscillations, the theoretical computations stemming from equal and unequal pole pitches demonstrate negligible divergence, particularly in the fundamental wave outcome of the Fourier transformation. As such, in this discourse, attention is solely dedicated to the calculations associated with equal pole pitch.

To model the tooth-slot structure, which comprises both the air gap and iron core mediums, media with differing magnetic permeabilities in the $x$ and $y$ directions were employed. Herein, magnetic permeabilities in the $\mathrm{x}$ and $\mathrm{y}$ directions are represented as $\mu_x$ and $\mu_y$, respectively. The magnetic permeability of air is denoted as $\mu_0$, while that of the stator iron core is expressed as $\mu_{F e}$. Drawing upon magnetic circuit principles, the following relationships were derived:

As shown in Figure 1, the layered magnetic field model of LSM is presented.

In this model, a layered calculation method was applied. The magnetic fields generated exclusively by the stator current layer in regions I and III were determined, and the magnetic fields attributed to the vehicle excitation current layer were subsequently superimposed. This approach enabled the quantification of magnetic fields in each region, facilitating the computation of the electromagnetic forces therein.

For instance, when focusing on the excitation current layer, the Laplace equation in air-gap region I, influenced only by the vehicle's magnetic pole, is represented as:

Taking into account solely the fundamental wave, solutions were derived from the established boundary conditions. The results are:

The procedures for calculating the composite magnetic field in other regions were not elaborated further within this section. The formula derived for electromagnetic force computation via the magnetic field energy method is:

Upon evaluation, when the power angle is set at 90°, the expressions for the traction and levitation forces of the linear motor were identified as [23], [24]:

These equations highlight that the traction force is directly influenced by the product of the stator and excitation currents, while the levitation force is dependent on the square of the excitation current. Both forces were observed to amplify as the air gap, represented as $g_0$, diminishes.

The electromagnetic tool, ANSYS Electronics Desktop – the latest rendition of ANSYS Maxwell – was selected for its proven efficacy in low-frequency phenomena and device analyses. Notably, its compatibility with other ANSYS electromagnetic tools facilitates seamless integration within a multi-physics analytical platform. For the context of this research, the structural design of the high-speed maglev train linear synchronous motor (LSM) from Tongji University's Maglev Center was employed, drawing inspiration from the German TR08 maglev train and incorporating several structural optimizations. A visual representation of this model is provided in Figure 2.

To ensure a consistent terminological framework throughout this study, the following definitions were adopted:

•The X-axis is identified with the direction of the train's motion, with motor parameters along this axis being classified as 'length'.

•The Y-axis corresponds to the train's levitation direction, with parameters in this direction designated as 'height'.

•Lastly, the Z-axis signifies the train's guidance direction, with the motor's size in this axis termed as 'width'.

A summary of the primary parameters is tabulated in Table 1, Table 2 and Table 3.

A two-dimensional transient field simulation method was adopted for all models. In order to examine the traction magnetic field's impact on electromagnetic force under decoupling conditions, a power angle of 90° had to be maintained in the model. This condition was met by modulating the initial phase angle of alternating current to ensure the direct-axis current (id) of the stator was zero. As such, the magnetic pole axis at the initial transient simulation position was aligned with the stator's magnetic pole axis. Parameterized simulations were subsequently conducted, with the A-phase alternating current initial phase being the primary variable. The phase angle maximizing the traction force was identified and used as the initial phase angle. The desired power angle of 90° was achieved with an A-phase current phase angle of 210 degrees, B-phase angle of 90 degrees, and a C-phase angle of -30 degrees.

Subsequently, a two-dimensional CAD model, representing these parameters, is showcased in Figure 3.

(1) Finite Element Model Geometry and Material Configurations

A simplified motor, comprising 4 magnetic poles and a 10mm air gap, served as the basis for the finite element simulation, as depicted in Figure 4. The long stator core and excitation magnetic pole core were recognized as being composed of 0.5 mm/piece silicon steel laminations. Therefore, the nonlinear material silicon steel sheet (DW470-50) was utilized. Building upon the properties of DW465-50 in the ANSYS material library, the lamination type was set as "Lamination," featuring a stacking factor of 0.97 and stacking direction v3, aligning with the z-axis in Cartesian coordinates. This stacking factor represents the ratio between the magnetic conduction length and the total length, considering that silicon steel sheets typically comprise magnetic cold-rolled steel and non-magnetic insulating material. The conductivity of these sheets was determined to be 2000000 S/m, encompassing eddy current losses in the core loss calculation.

(2) Excitation Sources and Parameter Settings for Primary and Secondary Windings

Electricity for the stator three-phase current of the high-speed maglev train's linear motor was derived from terrestrial power stations, with the effective current fluctuating based on the train's speed. Current frequency was derived from the equation $f=v / 2 \tau_r$. For a transient field simulation at 100m/s, the default effective current value was set at 1200A, with a frequency of 100/0.516Hz. The winding sorting method was assigned as A-Z-B-X-C-Y, and the current type was defined as "Stranded". The alternating currents loaded onto the three-phase winding were specified as follows:

(3) Secondary Movement Parameters, Motion Boundaries, and Solution Configurations

A rectangular region, termed Region A, was initially employed to encase the secondary moving magnetic pole, permitting Region A to execute unified motion within the stipulated motion range during transient calculations. The boundary of Region A was marginally extended, slightly exceeding 2 mm past the secondary winding boundary. Subsequently, a secondary motion area (Region B), spanning 1500 mm, was introduced, with its movement aligned with the positive X-direction of the Cartesian coordinate system. Parameters, such as the solution's initial position and maximum position, were configured as mentioned. The "Region" boundary condition was defined as "VectorPotential". Transient analyses factored in a maglev train speed of 360 km/h, with designated time intervals. The origin of the Cartesian coordinate system for the motor model was aligned with the stator's far-left position, with transient simulations commencing from this origin. A total of 61 transient analysis points were plotted, enveloping a cumulative distance equivalent to two stator pole pitches in 5.16 ms. The model depth was fixed at 180mm, corresponding to the core's width, and mesh refinement was conducted, as illustrated in the finite element model grid division (Figure 5) for the two-dimensional linear motor.

In a departure from traditional wheel-track transportation systems, magnetic levitation trains harness electromagnetic forces for vehicle levitation and propulsion. Specifically, in Electromagnetic Suspension (EMS) systems, the levitation of the vehicle is achieved through the excitation of electromagnets positioned beneath the tracks equipped with a three-phase armature stator. The interaction between this stator core and the vehicular levitation electromagnets yields attractive forces that keep the vehicle elevated.

Within the electromagnetic stator, the three-phase current produces a moving magnetic field. This field engages in an interactive dance with the vehicle's magnetic poles, creating forward momentum. Post-processing procedures on the finite element model yielded visual interpretations in the form of magnetic induction intensity cloud diagrams and magnetic force line distribution cloud diagrams, which are illustrated in Figure 6 and Figure 7. These diagrams effectively capture spatial variations and intensity gradients of the magnetic field. Such a design is intuitive: regions of high intensity are typically rendered in warmer hues such as red, whereas lower intensities might be depicted in cooler shades like blue.

An examination of Figure 7 reveals that magnetic force lines follow a looped trajectory. Beginning their journey in the stator, they traverse the air gap to eventually connect with the secondary excitation magnetic poles, with minor leakages noted within this gap. A distinct red zone in the stator is discernible, indicating the three-phase winding's magnetic pole distribution. It is notable that during transit, a consistent misalignment exists between these and the excitation magnetic poles. This misalignment is crucial, playing a pivotal role in generating forward traction. Concurrently, the highest magnetic induction intensity observed within the motor was approximately 1.8T, indicating a persistent unsaturated operational state.

In a granular observation at a specific temporal juncture, magnetic induction intensities Bx and By at the air gap's epicenter along the motor's longitudinal axis are elucidated in Figure 8 and Figure 9. The magnetic induction intensity predominantly gravitated in the Y direction (corresponding to levitation force), peaking at around 0.79T. A fluctuating, yet comparably diminutive, magnetic induction component was also identified in the X direction (pertaining to the traction force). A series of discernible troughs in the magnetic induction intensity distribution at regions corresponding to the stator tooth slots suggests an increase in magnetic reluctance at these sites. This, in turn, attenuates the levitation force.

Within long-stator linear synchronous motors, a distinction is observed wherein the length of the secondary or the vehicle's magnetic poles remains finite. Conversely, the primary length, represented by the three-phase stator windings serving as the track, significantly surpasses that of the secondary. As a result, distinct terminations – a beginning and an end – are inherent to each full vehicle magnetic pole. Such demarcations culminate in a distinctive longitudinal end effect, emblematic of long-stator linear motors.

Of note, both the vehicle's magnetic poles and the stator have been constructed using 0.5mm silicon steel laminations. It is suggested that due to this composition, eddy currents birthed by the end effect are essentially negligible. Furthermore, the impact exerted by these eddy currents on the air gap magnetic field appears to be inconsequential. As elucidated in Figure 10, magnetic leakage at the terminal of the motor is relatively minimal.

To fully comprehend the implications of the longitudinal end effect, it is crucial to delve deeper into its genesis. The structural difference in length between the primary and secondary components plays a pivotal role. Yet, the negligible effect of eddy currents, despite the pronounced difference in lengths, speaks volumes about the efficacy of the materials employed. Further investigations could potentially illuminate strategies to further minimize even this minor magnetic leakage, optimizing LSM motor performance.

The relationship between the electromagnetic forces and the excitation and armature currents remains a pivotal area of investigation in linear motor applications. For this discourse, a detailed examination of the traction force and levitation force over time, with set parameters of excitation current at 20A and armature current at 1200A, has been undertaken.

Figure 11 depicts the time-based waveform alterations of both forces. A variance in the traction force, for the 4-pole linear motor, is identified to oscillate between 0.035 and 1.1009 kN. In contrast, the levitation force exhibits fluctuations between 10.26 and 13.43 kN, thus demonstrating a magnitude approximately ten times that of the traction force. It was discerned that the air gap magnetic field is dominated by the 5th and 7th harmonic components. Interactions between these harmonics and the three-phase armature current induce traction force fluctuations, occurring at a frequency six times the fundamental. A formula detailing the computation of traction force fluctuations due to slotting effects, as delineated in reference [24], is:

where, $p$ symbolizes the number of pole pairs, with Ns representing the stator slots. With an established model where $p=1$ and $N_s=6$, the harmonic components of traction force are computed to be six times the foundational frequency. Validation of this was found in Figure 11, showcasing traction force harmonic values of 6, 12 , and 18- consistent with prior theoretical analyses.

In practical applications, such as maglev trains, symmetrically positioned excitation poles are found on either side. A 180° phase difference in the traction forces on both ends, if achieved by controlled armature current phase, might attenuate the slotting effect, subsequently minimizing traction force deviations.

(a) Influence of Excitation Current

With a regulated motor power angle at 90° and a consistent armature current of 1200A, variances in the excitation current were documented between 15A and 30A. Figure 12 and Figure 13 exhibit these traction and levitation force fluctuation curves. Numerical disparities in both forces, subject to excitation current alterations, are tabulated in Table 4. An enhancement in traction and levitation forces was observed with increasing excitation current. No evidence of negative curvatures in the plots was identified, suggesting the absence of core saturation near 30A excitation current. From Maxwell's stress tensor derivations, levitation force calculations, contingent on the magnetic field, are given as:

Comparative analyses between theoretical predictions and simulated outcomes revealed significant coherence.

(b) Implications of Armature Current

Maintaining a motor power angle of 90°, an excitation current of 20A, and varying only the armature current in the range of 600A to 1800A, resultant changes were documented in Figure 14 and Figure 15. Here, it was observed that while traction thrust proportionately augmented with increments in three-phase winding current, levitation force increments remained more subdued. The numerical contrasts between levitation and traction forces, influenced by armature current, are presented in Table 5. The disparity in the influence of the levitation force between excitation and armature currents exceeded a tenfold magnitude. These observations imply that, by ensuring a 90° power angle and an invariant air gap magnetic field, modulation of the armature current could effectively decouple and control traction force without significant levitation force perturbation.

Conclusively, the modulation of excitation and armature currents, and their implications on traction and levitation forces, offer invaluable insights into optimization strategies for linear motor applications. The findings pave the way for further in-depth research into decoupling strategies for force control.

In the pursuit of understanding the electromagnetic field dynamics in LSM for the efficient operation of maglev trains, particularly those aiming for 600 km/h speeds, extensive numerical simulations were conducted. Employing both analytical methods and the finite element method to scrutinize the electromagnetic forces, salient insights were derived:

(1) Given an unaltered motor's air-gap magnetic field and a power angle maintained at 90°, the levitation force was discerned to be directly contingent on the excitation current. In parallel, the traction force was ascertained to be linked with the magnitude of the three-phase current.

(2) Examination of the magnetic field lines at the motor's terminus revealed no significant anomalies in distribution patterns. Consequently, for LSMs fabricated using silicon steel sheets as their iron core, both the end effect and the consequent eddy currents were deemed inconsequential.

(3) A consistent trend was observed wherein both the traction and levitation forces augmented progressively with rising excitation currents. Absence of any downward inflection in the waveform curve suggests that core saturation remains elusive at an excitation current proximate to 30A.

(4) An incremental pattern was discerned in the traction thrust, aligning with the surge in the three-phase winding current. Nonetheless, the influence of this current on the levitation force remained subdued. To illustrate, when juxtaposed against the amplified levitation force resultant from augmented excitation current, the influence factor disparity exceeded a magnitude of tenfold. Such findings subtly infer the potential to regulate traction force—by modulating the armature current, while simultaneously preserving the levitation force—as long as the power angle persists at 90°.

It is worth noting that the simulations undertaken in this inquiry exclusively incorporated electromagnetic field considerations, thereby excluding multi-physics field simulation analyses. In real-world scenarios, an elevation in train velocity would invariably culminate in amplified aerodynamic lift. A hypothetical threshold velocity could be envisioned where this aerodynamic lift equates to the gravitational pull on the train. To hone the precision of magnetic field analyses, it is imperative that aerodynamic loads are incorporated into the model, necessitating comprehensive multi-physics field coupling simulation analyses within tools such as ANSYS. This avenue is highlighted as pivotal for forthcoming research endeavors.