Abstract
Magnetic levitation (maglev) systems for nonmagnetic conductive metals generally generate only repulsive force between the stator and the levitated body. To overturn the conventional common sense, we have fabricated a new maglev device that can generate “attractive force” between the stator consisting of plural ac electromagnets and a nonmagnetic conductive metal ring. In this paper, we elucidate the electromagnetic phenomenon that appropriately arranged seven ac electromagnets continuously pulls up an aluminum (Al) ring, which has 120 mm outer diameter and 5 mm square cross-section, by using finite element analysis. The analysis results reveal the existence of an equilibrium gap where the electromagnetic pulling force balances with the weight of the Al ring without active control. Besides, the equilibrium gap was measured when different weights were added to Al ring to verify the validity of analysis results. The measured results show that the maglev device has a positive stiffness (i.e., electromagnetic spring) of 50 N/m.
Keywords
Introduction
Ac induction type magnetic levitation (maglev) exists as one of the methods for generating an electromagnetic force in a nonmagnetic conductive metal [1]. This type of electromagnetic levitation furnace and semiconductor wafer transfer table are already used in manufacturing processes [2,3]. In principle, the alternating magnetic flux generated by ac electromagnets induces eddy currents in a nonmagnetic conductive metallic object, and repulsive forces act between the electromagnets and the object. Therefore, the electromagnets are necessarily located under the nonmagnetic conductive metal object to balance the repulsive forces with the gravity on the object. This type of maglev method is not subject to the constraint of Earnshaw’s theorem [4]. Thus, the appropriately arranged ac electromagnets passively stabilize the object in all directions.
To further utilize this maglev method as industrial systems, it is necessary to develop a new device for grasping an object from above, such as a parallel link robot or an articulated robot. Thus, we have proposed a new maglev device that can electromagnetically pull up an aluminum (Al) ring [5]. This device, which is composed of seven ac electromagnets, continuously generated a force to pull up the Al ring by adequately adjusting the excitation phase of each electromagnet. However, passive stabilization of the Al ring, which should be present in the ac induction type maglev, has not been confirmed for the proposed pull-up type device.
First, this paper describes the configuration of this maglev device and excitation conditions. Next, we evaluate the eddy currents in the Al ring, the magnetic flux density distribution, and the electromagnetic force on the ring using a finite element analysis (FEA) model based on the device configuration. Furthermore, we reveal the characteristics of pull-up force and positive stiffness (i.e., electromagnetic spring) to the Al ring with an experimental device improved to verify the obtained analysis results.
Maglev prototype and principle of pull-up force.
Figure 1(a) shows an assembly of ac electromagnets for pulling up an Al ring. A pancake coil type electromagnet (EM1) is arranged above the Al ring, and six electromagnets (EM2) are arranged around it. Figure 1(b) illustrates the direction of magnetic flux, current, and electromagnetic force when all electromagnets are ac-excited. The EM1 generates induced current i e and induced repulsive force f I against the Al ring. The six EM2s generate electromagnetic forces f A on the Al ring upward when the EM2s act magnetic fluxes 𝜙2 with an appropriate phase against the induced current i e in the Al ring. If the electromagnetic force ∑ f A generated at six locations of the Al ring is greater than the sum of the Al ring’s weight and induced repulsive force f I , the Al ring is pulled up toward the electromagnet EM1. Figure 2 illustrates a bottom view cut to 1∕6 and a side view on the A-A’ cross-section passing through the center of the magnetic pole. The specifications of the Al ring are an outer diameter of 120 mm, an inner diameter of 110 mm, a thickness of 5 mm, and a mass of 24 g. The gap d shown in Fig. 2(b) is the vertical distance between the lower surface of the pancake coil and the upper surface of the Al ring. Figure 2(b) shows the case when d = 10 mm. The distance from the magnetic pole of EM2 to the ring is 5 mm. By default, EM1 current, EM2 current, and frequency are i1 = 3.0 Arms, i2 = 3.0 Arms, and f = 180 Hz, respectively. Besides, the phase of i2 is set to advance 60 degrees from that of i1. By exciting the prototype device with these settings, the Al ring was successfully attracted toward EM1 [5].
Dimensions of maglev prototype.
1∕6 cut model and FE analysis specifications
Figure 3 shows a 1∕6 cut model for FE analysis. In the maglev prototype, six side coils are connected in series, and all the magnetic poles are excited so that they have the same polarity as shown in Fig. 1(b); therefore, the 3-D FE model suffices for a cylindrical shape with a central angle of 60 degrees, including one EM2, with Dirichlet boundary conditions on both cut surfaces. The FE model, assuming that the Al ring has no horizontal movement or tilt in the r-direction, derives the electromagnetic field distribution and force due to the change in the gap d. Table 1 lists the FE analysis specifications. The RMS value, phase angle, and frequency of excitation currents are constant.
1∕6 cut 3-D FE model. (Air area: invisible).
Specifications for FE analysis
Electromagnetic forces are calculated with A −𝜙 method and Maxwell’s stress tensor.
Figure 4 shows the characteristics of induced current density and magnetic flux density vector in the hatched cross-section (φ = 0 deg.) of the Al ring shown in Fig. 3. “Magnetic flux density vector” represents the magnetic flux densities of the r-direction component B r and the z-direction component B z generated at the center of the hatched cross-section. The white circles in Figs 4(a) and 4(b) are values at time 0. In Fig. 4(a), as the gap d increases, the induced current density in the Al ring also increases, and the phase advances within approximately 90 degrees. Although the lead phase of I2 was fixed at 60 degrees throughout this paper, the electromagnetic force increases more effectively when the phase of the magnetic flux from EM2 is finely regulated for each leading induced current shown in Fig. 4(a). In Fig. 4(b), the major axis tilt and the rotation direction of the elliptical locus of the magnetic flux density vector change as the gap d increases. This result means that the phase difference between the magnetic fluxes of EM1 and EM2 generates a rotating magnetic field.
FE analysis results (induced current and magnetic flux density).
Figure 5(a) shows the locus of the electromagnetic force vector. The force F r (F z ) corresponds to the sum of the force of the r-direction (z-direction) component generated at each mesh element in the Al ring in the 1∕6 cut model shown in Fig. 3. The force vectors are generated in the + z and −r directions at d = 10 mm, 15 mm, and 20 mm. These force vectors are evaluated by pulling up the ring while pushing it toward the center. Compared to Fig. 4(b), the major axis of each elliptical locus is inclined 90 degrees. Besides, the number of markers per cycle is halved from 36 to 18. The results show that the electromagnetic force is generated by the outer product of induced current and magnetic flux density around the Al ring, and also that the vibration frequency of the force is twice the excitation frequency.
FE analysis results (electromagnetic force).
Figure 5(b) shows the analysis results obtained by multiplying average electromagnetic forces in the z-direction by six. The average electromagnetic force corresponds to the center of each ellipse in Fig. 5(a). In the range of gap d from 2.0 mm to 17.3 mm, the average electromagnetic force exceeds the Al ring’s weight shown by a broken line. Al ring’s weight and average electromagnetic force are equal at the two gaps of 2.0 mm and 17.3 mm. One is a stable equilibrium point, and the other is an unstable equilibrium point. A positive stiffness of about 50 N/m appears in the hatched range of d ≤ 9 mm including the stable point. This positive stiffness characteristic means that a kind of spring is generated electromagnetically in the z-direction.
Figure 6 shows an experimental setup for the maglev device to confirm the positive stiffness characteristic of the electromagnetic spring. The Al ring is installed on a disk-shaped transparent acrylic pedestal. A brass round rod, which is fixed vertically from the center of the pedestal surface, penetrates two thrust bearings. The thrust bearings are installed to coincide with the central axis of EM1. The configuration of the additional parts allows the Al ring to move only in the z-direction while keeping the ring horizontal to the ground. The lower end of the brass rod and a counterweight are connected with a non-elastic thread via a pulley. The counterweight is set equal to the weight of the pedestal and rod excluding the Al ring. The above setup virtually moves only the Al ring up and down in the air. Adding a mass Δm to the counterweight is the same as adding it to the center of the Al ring. A displacement sensor (omron, ZX-LD40) is installed to detect the vertical movement of the Al ring.
Experimental setup for verification of electromagnetic spring.
Figure 7 shows a step response characteristic of pulling-up operation when initial gap d0 was set at 16 mm near the unstable equilibrium point without additional mass. When a predetermined current value (i1 = i2 = 3.0 Arms) was applied stepwise to the electromagnet, the Al ring quickly reached the stable equilibrium point of d = 2.0 mm. Besides, the Al ring reached the same stable equilibrium point even in the case of the other three initial gaps (d0 = 7 mm, 10 mm, 13 mm). This result proves that the Al ring inevitably reaches the stable equilibrium point if the initial gap is such that the electromagnetic force exceeds the weight of the Al ring.
Step response characteristic of pulling-up operation when the initial value is set near an unstable equilibrium point without additional mass (Δm = 0).
Figure 8 shows the change in the equilibrium gap when the additional mass is added from Δm = 0 to 24 g. The measured results are expressed as the average and standard deviation (SD) of ten experiments. Although the friction at thrust bearings varies the gap measured with a certain additional weight, the tendency of the average value of each equilibrium point is in good agreement with that of FE analysis results. This experiment demonstrated the validity of the FE analysis results and revealed that the electromagnetic spring characteristic occurs in a wide gap.
Stable equilibrium points with various additional masses (Δm = 0 to 24 g).
In this paper, we evaluated the electromagnetic phenomenon when the proposed maglev device pulls up the Al ring by using quasi-static FE analysis set a 1∕6 cut model. The periodical behavior of the induced current, magnetic flux density and electromagnetic force was clarified, and the relationship between the electromagnetic pull-up force and the gap d was derived. An experimental device was fabricated to ensure the validity of analysis results, and the existence of electromagnetic spring characteristic was proved by changing the additional mass to the Al ring.
According to the Earnshaw’s theorem, the ac induction maglev can passively stabilize the levitated body in all directions. The Al ring pull-up technique by ac excitation of electromagnets presented in this paper will also provide passive stabilization in all directions. In the future, we will evaluate the passive stabilization of the Al ring in the r-direction by FE analysis and aim at passive stabilization in all directions in the pull-up technique.