A numerical study of forces for sandwiched maglev stage

Abstract

A novel sandwiched maglev positioning stage is proposed in this paper. Mathematical model and finite element models are made up to analyze levitation force and driving force on maglev stage in two different structures—ordinary and sandwiched, which can verify the better performance of sandwiched structure. Also, the analysis of the relation between current input and working forces help a lot to the control of the maglev stage.

Keywords

Sandwiched maglev finite element analysis

1. Introduction

Traditional guide rail contact positioning platform has developed very mature, but problems like friction and abrasion still exist. However, precision manufacturing area, such as microelectronics processing, has an increasing demand on the equipment performance and processing environment. So the magnetic levitation positioning stage rises in response. Magnetic levitation technique is an advanced technology with advantage of zero friction, no wear, no need of lubrication, long life, low power consumption, noiselessness and so on.

There are different kinds of maglev stages, such as with or without rail support. To realize multidegree freedom motion, a maglev stage with rail support [1, 2] will own a redundant structure, which will cause low system response. Maglev stages without rail support [3, 4, 5, 6, 7, 8, 9] make use of the electromagnetic force to achieve levitation. As it is called, levitation performance is of vital importance. And to ensure the precision positioning, positioning control is also very important. This paper introduces one maglev stage with a special structure – sandwiched maglev stage, that has better levitation performance.

The sandwiched maglev stage is made up of one moving platform, four Halbach permanent magnet arrays, four stator windings, as shown in Fig. 1. The four stators lay upper and lower of the moving stage, with the Halbach permanent magnet arrays embedded upper and lower of the moving platform, forming the sandwiched structure. Compared to the ordinary structure, the key feature is that the four Halbach permanent magnet arrays are installed on different surfaces of the moving platform. Also, the four stator windings lay in different planes in correspondence with the magnet arrays.

Figure 1.

Schematic diagram of the sandwiched maglev stage.

So far, the sandwiched maglev stage has been set up, shown in Fig. 2. The main body of the maglev stage is made of aluminium alloy. The sandwiched maglev stage is an electromagnetic system with double driving. Though the stator windings are mounted on different planes, forces on $X$ , $Y$ axis work on the moving platform together, forming a coplanar structure which can eliminate effect of the abbe error. To detail the platform’s function and decrease the bulk size, moving platform is consisted of mobile platform and working platform designed in symmetrical structure, connected by the flexible support. In this way, the disadvantage of uneven distribution of heat on the platform can be overcome under the working condition. The stator windings need a high flatness in case not to affect the precision of positioning in vertical direction, but traditional circular coil can’t achieve high geometric accuracy because of the manufacturing processes. Plane coil is proposed to be used as the stator winding. The machinability of plane coil, design of loop current line, feasibility of multilayer, processing and manufacturing process are studied and experimented, realizing the high flatness of stator winding. Since the maglev stage is nearly an undamped system, which makes control difficult, an unconnected permanent magnet eddy current damper is designed to promote the control quality and increase the control stability.

Figure 2.

Photo of the sandwiched maglev stage.

The maglev stage is actuated by the electromagnetic force related to the magnetic field and the three-phase symmetrical current input. In this paper, a mathematical model is developed to describe the linear motor in the sandwiched maglev stage. Finite element analysis can visualize electromagnetic phenomena and help grasping changing rule of working forces, which makes a reference guide to the positioning control. In addition, comparison of levitation forces in both sandwiched maglev stage and ordinary maglev stage is done, in the working range of platform.

2. Mathematical model of sandwiched maglev stage

2.1 Mechanical structure of maglev stage

As shown in Fig. 3, Halbach permanent magnet arrays in $X$ axis are mounted on the lower surface of the platform, with the normal vector of the working face in opposite direction of $Z$ axis. The corresponding stator wingding fixed on the pedestal is underneath the permanent-magnet arrays’ working face, constituted repulsive linear motor that can provide vertical levitation forces $f_{\textit{ZX1}}$ , $f_{\textit{ZX2}}$ and horizontal driving forces $f_{\textit{X1}}$ , $f_{X2}$ in $X$ axis, as shown in Fig. 4. And $f_{\textit{ZX1}}$ , $f_{\textit{ZX2}}$ compose $f_{\textit{ZX}}$ , $f_{\textit{X1}}$ , $f_{\textit{X2}}$ compose $f_{X}$ .

Figure 3.

Structure and sketch map of the moving plate.

Figure 4.

Stress analysis diagram of the moving plate.

Similarly, Halbach permanent magnet arrays in $Y$ axis are mounted on the upper surface of the platform, with the normal vector of the working face in accordance with $Z$ axis. The corresponding stator wingding fixed on the pedestal is above the permanent magnet arrays’ working face, constituted attractive linear motor that can provide levitation force $f_{\textit{ZY1}}$ , $f_{\textit{ZY2}}$ and driving force $f_{\textit{Y1}}$ , $f_{\textit{Y2}}$ in $Y$ axis, as shown in Fig. 4. And $f_{\textit{ZY1}}$ , $f_{\textit{ZY2}}$ compose $f_{\textit{ZY}}$ , $f_{\textit{Y1}}$ , $f_{\textit{Y2}}$ compose $f_{Y}$ .

2.2 Modeling of maglev stage

Repulsive linear motor composed by Halbach permanent magnet array in one magnetic cycle and the corresponding plane coil is modeled in Fig. 5. The distance from lower stator winding to the working surface of lower magnetic array is $z_{0}$ , and $a-h$ are surfaces at boundaries. A half-infinite region can be obtained if free space is assumed except the regions for magnet array and plane coil.

Figure 5.

Repulsive linear motor model.

To simplify the model of linear motor, assume all physical constants are homogeneous and end effect is neglected. Executed Maxwell’s equations [10] on this model, magnetic field solution can be achieved at surfaces of magnet array and plane coil. Ignoring higher harmonic of current density, according to Maxwell stress tensor method [10], the working forces on repulsive linear motor can be described in Eq. (1). In the same way, Eq. (2) can be obtained, which describes the working forces on attractive linear motor.

$\displaystyle\left[{{\begin{array}[]{*{20}c}{f_{X}}\\ {f_{\textit{ZX}}}\\ \end{array}}}\right]=\frac{1}{2}uM_{0}\eta_{0}N_{m}Ge^{-\gamma_{1}z_{0}}\left[% {{\begin{array}[]{*{20}c}{\cos(\gamma_{1}x_{0})}l&{\sin(\gamma_{1}x_{0})}\\ {-\sin(\gamma_{1}x_{0})}l&{\cos(\gamma_{1}x_{0})}\\ \end{array}}}\right]\left[\begin{array}[]{ccc}\sqrt{2/3}&-1/\sqrt{6}&-1/\sqrt{% 6}\\ 0&1/\sqrt{2}&-1/\sqrt{2}\\ \end{array}\right]\left[{{\begin{array}[]{*{20}c}{i_{\textit{AX}}}\\ {i_{\textit{BX}}}\\ {i_{\textit{CX}}}\\ \end{array}}}\right]$ (1) $\displaystyle\left[{{\begin{array}[]{*{20}c}{f_{Y}}\\ {f_{ZY}}\\ \end{array}}}\right]=\frac{1}{2}uM_{0}\eta_{0}N_{m}Ge^{-\gamma_{1}\left({h_{0}% -z_{0}}\right)}\!\!\left[{{\begin{array}[]{*{20}c}{\cos(\gamma_{1}y_{0})}% \hfill&{\sin(\gamma_{1}y_{0})}\\ {-\sin(\gamma_{1}y_{0})}\hfill&{\cos(\gamma_{1}y_{0})}\\ \end{array}}}\right]\!\!\left[\begin{array}[]{ccc}\sqrt{2}/3&-1/\sqrt{6}&-1/% \sqrt{6}\\ 0&1/\sqrt{2}&-1/\sqrt{2}\\ \end{array}\right]\!\!\left[{{\begin{array}[]{*{20}c}{i_{\textit{AY}}}\\ {i_{\textit{BY}}}\\ {i_{\textit{CY}}}\\ \end{array}}}\right]$ (2)

Where: $\mu M_{0}$ is the remanence of the permanent magnets; $\eta_{0}$ is the winding density; $N_{m}$ is the number of pitches; the absolute value of fundamental wave is $\gamma_{1}=\frac{2\pi}{l}$ , and $l$ is the period length of permanent magnet array; $z_{0}$ is the levitation height; $x_{0}$ , $y_{0}$ is the movement in horizontal direction; $G$ is a constant related with the winding; $h_{0}$ is the sum of gaps of upper and lower winding from the corresponding magnet array; $i_{A},i_{B},i_{C}$ are the three-phase symmetry current of the upper and lower winding. Executed the CLARKE transformation [11] and PARK transformation [12], the equations above can be described as Eqs (3) and (4), where $i_{D},i_{Q}$ is the current under two-phase synchronous rotary coordinate.

$\displaystyle\left[\begin{array}[]{c}{f_{X}}\\ {f_{\textit{ZX}}}\\ \end{array}\right]=\frac{1}{2}uM_{0}\eta_{0}N_{m}Ge^{-\gamma_{1}z_{0}}\left[% \begin{array}[]{c}{i_{\textit{QX}}}\\ {i_{\textit{DX}}}\\ \end{array}\right]$ (3) $\displaystyle\left[{{\begin{array}[]{*{20}c}{f_{Y}}\hfill\\ {f_{\textit{ZY}}}\hfill\\ \end{array}}}\right]=\frac{1}{2}uM_{0}\eta_{0}N_{m}Ge^{-\gamma_{1}\left({h_{0}% -z_{0}}\right)}\left[{{\begin{array}[]{*{20}c}{i_{\textit{QY}}}\\ {i_{\textit{DY}}}\\ \end{array}}}\right]$ (4)

The levitation force is composed of attractive force $f_{\textit{ZY}}$ provided by upper winding and repulsive force $f_{\textit{ZX}}$ provided by lower winding. Imagine $i_{\textit{DX}}=i_{\textit{DY}}=i_{D}$ , the resultant force in $Z$ direction is:

$\displaystyle f_{Z}=f_{\textit{ZX}}+f_{\textit{ZY}}=\frac{1}{2}uM_{0}\eta_{0}N% _{m}Gi_{D}\left[{e^{-\gamma_{1}z_{0}}+e^{-\gamma_{1}\left({h_{0}-z_{0}}\right)% }}\right]$ (5)

The Taylor series expansion in the bracket of Eq. (5) about operating point $z_{0}=\frac{h_{0}}{2}$ , where is the expected levitation height, is shown in Eq. (6). And $\Delta z$ is the tiny variation around the operating point, an object to be controlled.

$\displaystyle\!\!\left[{e^{-\gamma_{1}z_{0}}+e^{-\gamma_{1}\left({h_{0}-z_{0}}% \right)}}\right]$ $\displaystyle=\left[\!\!{\begin{array}[]{l}\left.{e^{-\gamma_{1}z_{0}}}\right|% _{z_{0}=\frac{h_{0}}{2}}+\left({-\gamma_{1}}\right)\left.{e^{-\gamma_{1}z_{0}}% }\right|_{z_{0}=\frac{h_{0}}{2}}\cdot\Delta z+\frac{\gamma_{1}^{2}}{2}\cdot% \left.{e^{-\gamma_{1}z_{0}}}\right|_{z_{0}=\frac{h_{0}}{2}}\cdot\left({\Delta z% }\right)^{2}+\cdots\\ +\left.{e^{-\gamma_{1}\left({h_{0}-z_{0}}\right)}}\right|_{z_{0}=\frac{h_{0}}{% 2}}+\gamma_{1}\left.{e^{-\gamma_{1}\left({h_{0}-z_{0}}\right)}}\right|_{z_{0}=% \frac{h_{0}}{2}}\cdot\Delta z+\frac{\gamma_{1}^{2}}{2}\cdot\left.{e^{-\gamma_{% 1}\left({h_{0}-z_{0}}\right)}}\right|_{z_{0}=\frac{h_{0}}{2}}\cdot\left({% \Delta z}\right)^{2}+\cdots\\ \end{array}}\right]$ (6) $\displaystyle\approx e^{-\gamma_{1}\cdot\frac{h_{0}}{2}}\left[{2+R_{2}\left({% \Delta z}\right)}\right]$

The last right term of Eq. (6) is the higher-order term of Taylor expansion, which is exponentially small enough to be omitted. In this way, terms in the bracket of Eq. (5) can be a constant approximately. In other words, levitation force worked on the moving plate $f_{Z}$ is approximately linear to current $i_{D}$ , unaffected by levitation height $z_{0}$ , which illustrates that the moving plate will have a stable levitation under the steady and lasting levitation force.

The forces acted on the maglev stage are analyzed above. According to the feature of sandwiched structure and Maxwell stress tensor method, the relation among working forces, displacement and three-phase symmetry current is given in the form of formula. The stable of levitation force is verified on the mathematics level, which will be described more detailed in the following chapter.

3. Finite element analysis in Comsol

3.1 Comsol model of magnetic array

The finite element model of Halbach magnetic array is conducted in Comsol, as shown in Fig. 6. And Fig. 7 shows the distribution of strengthened magnetic field in the vertical direction $Z$ , on the surface 0.5 mm away downward from the working surface of magnetic array. By analyzing the coordinates and size of the magnetic array, it can be found that the magnetic field starts from the middle of the first magnetic block. There are three periods and each period lasts $l$ .

Figure 6.

Sketch map of magnetic array.

Figure 7.

Distribution of magnetic array’s magnetic field on $Z$ axis.

3.2 Comsol model of stator winding

Plane coil is consisted of 60 copper wires, the adjacent two wires compose one phase. The Comsol model of plane coil is constructed as in Fig. 8. The stator winding needs three-phase symmetrical current as follows, $i_{A}=\sqrt{2}I\cos\left({\theta+\phi}\right)$ , $i_{B}=\sqrt{2}I\cos\left({\theta+\phi+120^{\circ}}\right)$ , $i_{C}=\sqrt{2}I\cos\left({\theta+\phi+240^{\circ}}\right)$ , where $\phi$ is the initial angle of current, $\theta$ corresponds to platform’s displacement $x_{0}$ , whose relation is $\theta=\frac{2\pi}{l}x_{0}$ .

Figure 8.

Sketch map of plane coil.

Figure 9(a) shows the distribution of plane coil’s magnetic field on $Z$ axis when setting the current parameter as $\theta=0^{\circ},\phi=30^{\circ}$ . The magnetic field shown in Fig. 9 has 5 periods, each lasts $l$ , same with the magnetic array’s. In this way, stator winding can provide sufficient space for the platform’s movement. And Figs 9(b)–(d) correspond to the parameter $\phi=120^{\circ},210^{\circ},300^{\circ}$ respectively, where $\theta=0^{\circ}$ . It can be observed that the distribution of magnetic field changes with the different phase of current input.

Figure 9.

Distribution of plane coil’s magnetic field on $Z$ axis.

3.3 Comsol model of linear motor

Figure 10 shows the Comsol model of linear motor, composed by one magnetic array and one stator winding, with a gap of 0.5 mm. The dotted line shows the superposition of the starting point of magnetic array’s first magnetic field and the starting point of plane coil’s second magnetic field. When input current with $\theta=0^{\circ},I=4.5∼{}A,\phi=30^{\circ}$ , the Lorentz force of plane coil is $-$ 33.028403 $∼{}N/m$ in $X$ component and $-$ 59.745091 $∼{}N/m$ in $Z$ component. Since it is a 2D model, the length of copper wire is not defined, so the unit of force is $N/m$ . Figure 11 shows the distribution of Lorentz force density in $X$ axis and $Z$ axis.

Figure 10.

Sketch map of linear motor in Comsol.

Figure 11.

Distribution of Lorentz force density.

Under the above premises, vary $\phi$ from 0 ${}^{\circ}$ to 360 ${}^{\circ}$ and collect the Lorentz force data. The tendency of horizontal force and vertical force are shown in Fig. 12, where the line with triangle represents horizontal force , the line with circle represents vertical force and the solid line is a zero line to be contrasted as a reference. From the calculation, horizontal force can be expressed as $F_{Lx}=-A\cos\left({\phi-30^{\circ}}\right)$ , and vertical force can be expressed as $F_{Lz}=A\sin\left({\phi-30^{\circ}}\right)$ , where $A$ is the maximum of force.

Figure 12.

Force of plane coil.

From the analysis, it can be known that when the platform is fixed in one place, different levitation force and driving force can be get, whose value and direction change as the trigonometric function, if current with different $\phi$ is input. Since the magnetic array needs upward levitation force, according to Newton’s third law, plane coil has to provide downward vertical Lorentz force. In other words, to ensure the platform’s levitation, $\phi$ should be chosen from $0^{\circ}-30^{\circ}$ or $\mbox{21}0^{\circ}-{36}0^{\circ}$ .

When the platform, or magnetic array, moves to one position ( $\theta\neq 0^{\circ})$ , the model should also move, whose relation is $x_{0}=\frac{\theta}{2\pi}l$ . Figure 13 shows the comparison when magnetic array moves $\frac{l}{4}$ left and right separately, corresponding to $\theta=90^{\circ}$ and $\theta=-90^{\circ}$ . Input current with the same $\phi$ and different $\theta$ to the corresponding model. The calculating result shows that $F_{Lx}$ and $F_{Lz}$ are the same as the result of $\theta=0^{\circ}$ model. So is the linear motor on $Y$ axis that provide attractive levitation force.

Figure 13.

Comparison diagram of linear motor at specified location.

The distribution of magnetic array and stator winding with different current is visualized in this chapter. Based on the above work, the finite element model of linear motor of maglev stage is constructed and three-phase symmetrical current with different parameter is input. In this way, changing trend of working forces can be achieved, which can be described in trigonometric function. Also, value range of $\theta$ of the current is determined. To simulate the positioning condition of maglev stage, the plane coil should also move to the position $x_{0}$ , where the current parameter $\theta$ changes in the relation $\theta=2\pi\cdot\frac{x_{0}}{l}$ . The same working force can be get if current with changed $\theta$ is input.

4. Recommendation to control

The force analysis can provide method to the control of maglev platform: each time the controller output current, the parameter $I,\theta,\phi$ in the current equation should be identified.

4.1 Initializing

At the initializing of the platform, a located block is needed to help the platform located in a certain position to get $\theta$ . To be convenient, $\theta$ can be set as $0^{\circ}$ , where is the superposition of the starting point of magnetic array’s first magnetic field and the starting point of plane coil’s second magnetic field. Also, at the initializing, the platform needs levitation under the affect of levitation force but not driving force, or described as $F_{x}=0$ and $F_{y}=Mg$ , where $M$ represents the quality of the moving platform that can decide $I$ . From the changing curve of force in Chapter 3.3, the vertical force can be maximum and horizontal force be zero at the point $\phi=300^{\circ}$ , which meets the requirement of initializing.

4.2 Positioning control

In positioning control, by implementing feedback control on the desired position and the actual position, needed levitation force and driving force can be obtained. Here , define $\eta=\frac{F_{y}}{F_{x}}$ , and according to the analysis in Chapter 3.3, $\eta=-\tan\left({\phi-30^{\circ}}\right)$ . In this way, the required $\phi$ can be defined by transformation of trigonometric function. The actual position $x$ can decide $\theta$ , and Eqs (1), (2) can decide $I$ . Then the confirmed three-phase symmetry current can be input to the plane coil to realize precision control. The control flow chart deduced from above is shown in Fig. 14.

Figure 14.

Control flow chart.

Based on the above analysis of working force, the parameter of current at maglev stage’s initializing is determined. To realize the positioning control, parameter $\eta$ , related to working forces, is defined. And the control flow chart shows that $\eta$ plays an important role in positioning control.

5. Comparison evaluation of levitation force

The simplified model with only magnetic arrays and plane coils of sandwiched maglev stage and ordinary maglev stage model are constructed in Comsol shown in Fig. 15. In ordinary maglev stage model, the four plane coils are arranged in the same surface. Since the air gap varies from 0.4 mm to 0.6 mm, the magnetic array’s position is changed to simulate this variation. Both model are input the same constant current, and values of the total vertical force are collected.

Figure 15.

Comsol model of maglev stage.

Figure 16 is the comparison of total vertical force, where line with circle, nearly a straight line, represents the sandwiched maglev stage’s and line with triangle, obviously with a slope, represents the ordinary maglev stage’s. It shows that in ordinary maglev stage, the levitation force the platform received has a large variation range, and the higher the platform levitate, the smaller the levitation force is, which is bad to the stable levitation and work. But in sandwiched maglev stage, this phenomenon can be avoided as shown in Fig. 16. No matter whether the air gap is getting bigger or smaller in the working range, the levitation force remains almost unchanged.

Figure 16.

Comparison of total vertical force of two types of maglev stage.

Obviously, this comparison verifies that sandwiched maglev stage has a better performance in stable suspension, which fits with the numerical analysis in Chapter 2.2.

6. Conclusion

The presented force equations and finite element model show that the sandwiched maglev stage can overcome the exponential relation of levitation force and the suspension height. The sandwiched maglev stage can keep stable within an acceptable range, providing a better levitation ability. The finite element model clearly visualize the magnetic field distribution of magnetic array and stator winding with different currents. It also shows the variation tendency in trigonometric function of working force, which determine the range of current’s parameter value and contribute to the control strategy. According to the working forces’ changing trend, a coefficient $\eta$ that can determine the parameter of needed current to realize precision control is defined, which can be determined by the needed force corresponding to the expected position.

Footnotes

Acknowledgments

This work was supported by National Natural Science Foundation of China (No. 51375052).

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