Fig. 1. TEAM Workshop problem 28: An electrodynamic levitation device
Institut f ur Theorie der Elektrotechnik, Universit at Stuttgart, Pfa enwaldring 47, 70550 Stuttgart, Germany.Description of TEAM Workshop Problem 28:Hans Karl, Joachim Fetzer, Stefan Kurz, G unther Lehner and Wolfgang M. Rucker
An Electrodynamic Levitation Device
Abstract - This paper presents a new TEAM workshop problem
taking into account moving bodies. An
electrodynamic levitation device which consists of a conducting plate
over two exciting coils shall be ex-
amined. The aim is to determine the dynamic characteristics of the
levitating plate. A coupled solution
of the electromagnetic and the mechanical problem is necessary for
that.
1. Introduction
The modelling of electromechanical devices, i.e.
the solution of transient coupled electromechanical problems
taking into account moving bodies is gaining signi cance. For this
reason it is necessary to de ne a benchmark which
allows to compare di erent approaches regarding their advantages and
disadvantages for the solution of such prob-
lems. Typical diculties are the treatment of motion, strategies
for remeshing, force calculation, weak versus
strong electromechanical coupling, ecient time stepping schemes, etc.
Up to now only TEAM workshop problem 9 and TEAM
workshop problem 17 deal with moving bodies. In prob-
lem 9 a moving body with given constant velocity is considered. The
problem is axisymmetric and in nitely ex-
tended in the direction of the velocity. Therefore it is stationary
and lacks the feature of electromechanical cou-
pling. For problem 17 no measured data are available.
Only the description of the underlying experiment
is given and therefore the treatment of the problem is dicult. In
contrast, the new TEAM workshop problem 28 is a transient problem with
electromechanical coupling and mea-
sured data being available.
It is intended to split the new problem into di
erent packages with increasing level of diculty. In this pa-
per, Model A is presented. Model A is an axisymmetric problem without
signi cant eddy current reaction to the
exciting coils. In the future, two additional packages will be provided.
Model B will be very similar to Model
A, but the levitating plate will have an eccentric bore which disturbs
the axisymmetry and requires a full 3D
modelling. Model C will have a di erent geometry and operating frequency.
The exciting coils will be voltage
driven and the reaction due to the motion has to be taken into account.
This is an important feature because many
actuators are voltage driven and show a typical current drop
due to the motion of the armature.
II. Description of Model A
One of the earliest papers on levitation by elds
at power frequencies is that due to Belford, Peer and Tonks
[1]. Electrodynamic levitation is based on the induction of eddy currents
in conducting materials. These eddy cur-
rents can be induced by a time varying magnetic eld.
This is the case for the device shown in Fig. 1.
A cylin-drical aluminium plate ( = 3:40 10 7 1= m, m = 0:107
kg) is located above two cylindrical coils. All three parts are aligned
coaxially. The inner coil has w1 = 960 and
the outer coil w2 = 576 turns. The dimensions of the device are shown
in Fig. 2. The levitation height z refers
to the distance between the lower edge of the plate and the upper edge
of the current carrying area (z = 0). For
t 0 the plate rests above the coils at a distance of z = 3.8 mm due
to the thickness of the winding form.
Fig. 1. TEAM Workshop problem 28: An electrodynamic levitation
device
Fig. 2. Dimensions (in mm) of the electrodynamic levitation device
1
Both coils are connected in series, but with di erent
sense of winding. The device is operated directly be-
tween two outer conductors of the three-phase supply network. With
the help of an electronic switch, the instant
of switching on is synchronized onto the driving supply voltage in
a way that there is no electrical transient. For
t 0, sinusoidal currents i(t) ow in the coils in opposite directions,
i(t) =^{sin(2f0 t); ^{ = 20A; f0 = 50Hz: (1)
Due to the induced eddy currents a repulsive force
is exerted on the plate. After some damped oscillations, the
plate attains a stationary levitation height of z = 11.3 mm. A possible
reaction of the induced eddy currents
can be neglected. For this reason, the current in (1) can be regarded
as impressed.
III. Measurement of the levitation height
The levitation height is measured by means of laser
triangulation. The principle is shown in Fig. 3. A laser beam
emitted by a laser diode is directed near the center of the plate.
The re ected beam is detected by a position sens-
ing semiconductor detector (PSD). A displacement zof the plate yields
a displacement sof the re ected beam.
With this principle, a high resolution limited only by ampli er noise
can be achieved.
In practice, however, there are some undesired e
ects which complicate the measurements. The rst problem
is that the device can never be perfectly axisymmetric. Any radial
displacement of the plate, nonhomogeneous
winding of the coils and other similar e ects disturb the symmetry.
This causes additional oscillations of the plate
around a radial axis. Luckily the experiments showed that these oscillations
have only a minor e ect on the measured
results. Fig. 4 shows the measured levitation height of four di erent
measurements and gives an idea about the
reproducibility. The diference of the respective results is
Fig. 3. Measurement of the levitation height by means of laser triangulation
Fig. 4. Measured levitation height of four di erent measurements.
acceptable. The average values of these measured data are given in Table
I on the next page and should be used
for the comparison with numerical results. It is important to make
sure that the plate and the coils
have ambient temperature when the measurement starts.
Due to the ohmic losses the temperature of the device
raises signi cantly during operation. This would increase
the resistance and cause inaccurate results.
Another diculty is related to the exact modelling
of the coils. For numerical purposes the coils are represented
by domains which carry a homogeneous azimuthal current density. The
cross section of these domains is given by the
rectangular cross section of the winding space. The coils are made
of copper wire with 1.2 mm diameter and con-
tain insulating layers. The real current density is neither strictly
homogeneous nor sharply bounded by a rectangle.
These e ects may in uence the plate during its initial lift o phase.
It is difcult to estimate the influence of this efect.
IV. Comparison of Measured and Computed
Results
For the sake of completeness we include a comparison between the measured
data according to Table I and com-
ig.4 Response to a current turn-off
Flaw 3 characteristics (length*heigth) : 1*4mmputed results in this
description, see Fig. 5. The computed results have been obtained with the
help of the BEM-FEM
code described in [2], [3]. The discrepancy of the maximum levitation
height during the rst half period might
be traced back to the modelling of the coils as explained in the previous
section.
V. Concluding Remarks
This description of TEAM problem 28 superseeds the preliminary description
[4]. The results in Table I
can be supplied on request by e-mail (hans.karl@ite.unistuttgart.de).
It is hoped eventually to have the entire
Table 1.Measured results on the levitation height
t (s) z(mm) t (s) z(mm) t (s)
z(mm) t(s) z(mm) t (s)
z(mm) t(s) z(mm)
0.0 3.7
287.4 11.9 574.9 11.6
862.3 11.7 1149.8 11.5
1437.2 11.4
9.9 4.0
297.4 10.4 584.8 11.0
872.3 11.4 1159.7 11.4
1447.2 11.4
19.8 4.9 307.3
9.3 594.7 10.4
882.2 11.2 1169.6 11.3
1457.1 11.4
29.7 6.9 317.2
8.7 604.6 10.0
892.1 11.1 1179.5 11.2
1467.0 11.3
39.6 9.7 327.1
8.7 614.5 9.9
902.0 11.0 1189.4
11.1 1476.9 11.3
49.6 12.8 337.0 9.2
624.5 10.0 911.9 11.0
1199.4 11.1 1486.8 11.3
59.5 15.6 346.9 10.2
634.4 10.3 921.8 11.1
1209.3 11.1 1496.7 11.3
69.4 17.4 356.8 11.4
644.3 10.8 931.7 11.2
1219.2 11.1 1506.6 11.3
79.3 18.0 366.7 12.4
654.2 11.3 941.6 11.4
1229.1 11.2 1516.5 11.3
89.2 18.1 376.7 13.2
664.1 11.7 951.6 11.6
1239.0 11.2 1526.4 11.3
99.1 18.2 386.6 13.6
674.0 12.1 961.5 11.7
1248.9 11.3 1536.4 11.3
109.0 17.8 396.5 13.7
683.9 12.3 971.4 11.8
1258.8 11.4 1546.3 11.4
118.9 16.4 406.4 13.3
693.8 12.3 981.3 11.8
1268.7 11.4 1556.2 11.4
128.9 14.1 416.3 12.7
703.8 12.2 991.2 11.8
1278.6 11.4 1566.1 11.4
138.8 11.5 426.2 11.8
713.7 12.0 1001.1 11.7
1288.6 11.4 1576.0 11.4
148.7 9.0 436.1 10.9
723.6 11.6 1011.0 11.5
1298.5 11.3 1585.9 11.4
158.6 7.2 446.0 10.1
733.5 11.3 1020.9 11.4
1308.4 11.3 1595.8 11.4
168.5 6.7 456.0 9.6
7 43.4 1 1.0 1030.8
11.2 1318.3 11.3
1605.7 11.4
178.4 7.3 465.9 9.4
7 53.3 1 0.8 1040.8
11.1 1328.2 11.2
1615.7 11.4
188.3 8.8 475.8 9.6
7 63.2 1 0.7 1050.7
11.0 1338.1 11.2
1625.6 11.3
198.2 10.7 485.7 10.1
773.1 10.8 1060.6 11.0
1348.0 11.2 1635.5 11.3
208.2 12.6 495.6 10.8
783.0 10.9 1070.5 11.0
1357.9 11.2 1645.4 11.3
218.1 14.3 505.5 11.6
793.0 11.2 1080.4 11.1
1367.9 11.2 1655.3 11.3
228.0 15.6 515.4 12.2
802.9 11.5 1090.3 11.2
1377.8 11.3 1665.2 11.3
237.9 16.2 525.3 12.7
812.8 11.7 1100.2 11.3
1387.7 11.3 1675.1 11.3
247.8 16.3 535.2 13.0
822.7 11.9 1110.1 11.4
1397.6 11.4 1685.0 11.3
257.7 15.8 545.2 12.9
832.6 12.0 1120.1 11.5
1407.5 11.4 1694.9 11.3
267.6 14.8 555.1 12.7
842.5 11.9 1130.0 11.5
1417.4 11.4 1704.9 11.3
277.5 13.5 565.0 12.2
852.4 11.8 1139.9 11.5
1427.3 11.4 1714.8 11.4
Fig. 5. Comparison between the measured (Table I) and the computed (BEM-FEM) levitation height
description available on a WWW TEAM page.
References
[1] B.D. Bedford, L.H. Peer, and L. Tonks, \The electromagnetic levitator,"
Gen. Elect. Rev., vol. 42, no. 6, pp. 246{247, 1939.
[2] S. Kurz, J. Fetzer, and G. Lehner, \Threedimensional transient
BEM-FEM coupled analysis of electrodynamic levitation prob-
lems," IEEE Transactions on Magnetics, vol. 32, no. 3, pp. 1062{1065,
May 1996.
[3] S. Kurz, J. Fetzer, G. Lehner, and W.M. Rucker, \A novel formulation
for 3D eddy current problems with moving bod-
ies using a Lagrangian description and BEM-FEM coupling," Submitted
to the COMPUMAG 1997.
[4] H. Karl, J. Fetzer, S. Kurz, G. Lehner, and W.M. Rucker, \Preliminary
proposal for a new TEAM workshop problem: An elec-
trodynamic levitation device," in Proc. of the TEAM Workshop, Graz,
Austria, Sept. 1996, pp. 41{42. 4