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Investigation of an Electrodynamic Magnetic Levitation Device
Conference Paper · July 2020
DOI: 10.1109/EIT48999.2020.9208281
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Investigation of an Electrodynamic
Magnetic Levitation Device
Attila Lendek, StudentMember, IEEE
Constantin M. Apostoaia, Member, IEEE
Dept. of Electrical and Computer Engineering
Purdue University Northwest, Hammond, IN
alendek@pnw.edu
Dept. of Electrical and Computer Engineering
Purdue University Northwest, Hammond, IN
apostoai@pnw.edu
Abstract — This study is done in the area of magnetic
levitation, with future application for transportation
electrification. The type of the device designed and implemented
in this work, is known as an electrodynamic magnetic levitation
system (EDS). First, a simulation model of the EDS is developed,
which employs Halbach arrays, that are a special arrangement of
permanent magnets, on rotational discs (rotors), placed above a
conductive plate. The finite element analysis model of EDS is
designed, and simulated using ANSYS Maxwell™ software.
Second, an electrodynamic levitation device was designed, built
and experimentally investigated. Proof of concept was carried out
through modeling, simulation and experiments, on a scaled-down
built physical model, using speed controlled permanent magnet
AC motors, to drive the Halbach rotors.
Keywords — electrodynamic magnetic levitation system,
Halbach array, magnetic lifting force, simulation, finite element
analysis, experimental model.
I. INTRODUCTION
The advanced technology of magnetic levitation has been
already employed in various applications such as:
-in industrial equipment in fans, pumps, compressors,
motors, generators, using magnetic bearings without physical
contact
-in transportation systems, for high speed magnetically
levitated (Maglev) trains, personal rapid transit
-in space launch vehicle (NASA’s Space Flight Center), to
break free from Earth’s gravity
-in clean energy for frictionless wind turbines
-in civil engineering for elevators
-in nuclear energy for centrifuges of nuclear reactors, and
even for implantable heart pumps.
Common to all these applications is the lack of contact (no
wear and friction), and thus, increased efficiency with reduced
maintenance costs, overall increasing the life of the systems.
Maglev is certainly considered as a solution for the world
needs of future clean energy applications.
The rotational, or translational motion of a permanent
magnet above a copper or aluminum conductive plate, creates
an electrodynamic levitation, and drag force. At low speeds,
the drag force dominates [1]. Because of this dominating drag
force, it is not suitable for low speed operations [2].
Significant reduction in the drag forces can be achieved by the
null current approach [3].
The implementation of EDS in transportation, is an
attractive idea because of the Halbach Arrays unique property
of producing a strong periodic magnetic field, concentrated on
one side[4] [5] [6].
There is a critical speed that must be reached, for the
implementation of the magnetic levitation. Until this speed is
achieved, other means must be implemented, then magnetic
levitation can take over [7] [8] [9].
Magnetic levitation systems can be divided into two groups:
the attractive systems, which are called electromagnetic
suspension, and the repulsive systems, which are called
electrodynamic suspension [10].
This paper research work is focusing on the study of
electrodynamic magnetic levitation, and the development of
an experimental prototype, for test.
The paper is organized in six sections. In the Introduction
section, the motivation of the EDS device development, and
investigation is presented. The work in progress is discussed
elsewhere. In the second section, a brief background is
presented, referring to the theoretical concepts involved in
magnetic levitation. In the third section, a detailed analytical
mathematical model of the EDS system is developed. In the
fourth section, the modeling, and simulation of the EDS
device is examined in two steps: first, the model based on
FEA design is presented, and in a second step, the overall EDS
models simulation results are discussed. The fifth section
presents the development, and realization of the experimental
device, at a reduced scale, for the system’s experimental validation. The concluding remarks are outlined in the last section.
II. ELECTRODYNAMIC LEVITATION BACKGROUND
At the core of the electrodynamic levitation device is a set
of permanent magnets, with a specific arrangement in a so
called Halbach array. As proposed by the physician Klaus
Halbach in 1985 [11], a Halbach array of magnets allows
concentration of the magnetic field in a desired direction. The
array has a spatially rotating pattern of magnetization, which
almost cancels the field on one side but, boosts it on the other.
The main advantages of Halbach arrays are, that they can
produce about 1.4 times stronger magnetic fields on one side,
whilst creating a very small (stray) field on the opposite side.
In the beginning of this study, a linear Halbach array was
investigated.
In Fig.1, the simulation of the magnetic field lines,
generated by a linear arrangement of five 1-inch neodymium
type cube magnets, is shown. The physical realization of this
linear Halbach array is described later in section V.
Fig.1 Magnetic field lines simulation of a Halbach array using ANSYS
Maxwell software.
Note that a weak magnetic field is present on the top side of
the device, whilst on the bottom side, a strong magnetic field
is created. This array arrangement uses four blocks per spatial
period, with the magnetization axis rotating by 90 degrees, in
each subsequent block. The arrows in Fig.1 indicate the
magnetization vector’s direction.
In the next step of this study, the magnets of Halbach array
are assembled on a rotational disc (rotor), which is placed on
top of a conductive plate. Fig.2 shows the rotor containing a
radial Halbach array magnet arrangement, which creates six
poles, with four magnets per spatial period (South poles are
marked with a cross, while the North poles are marked with
dots).
The motional magnetic field from the Halbach array
magnets, induces an electromotive force (emf) in the shorted
circuit of a conductive plate, and thus eddy currents are
induced in the plate.
In order to solve for the eddy currents, substitute Eq. (3) into
Eq. (2) to obtain:
ωΦ
ω
(4)
In (1), (3) and (4), the excitation frequency ω is related to the
mechanical rotational speed of the Halbach rotor ω , by the
number pairs of poles:
ω
ω
(5)
The steady state sinusoidal response
, is conveniently
found using the electric circuit theory, as shown next. In the
input-output (I/O) model of EDS given in Eq. (4), the
following Laplace transforms are identified:
≡
ωΦ
(6)
≡
Fig.2 Halbach magnet array on a rotor.
The eddy currents create an induced (secondary) magnetic
field, opposing to the change of magnetic fields of the
permanent magnets. In this way, a repelling, levitation (lifting)
magnetic force is created. And if the rotor of the permanent
magnets remains in motion, above a minimum speed of
rotation, the rotor will be stably levitated.
Correspondingly, the I/O model of the EDS based on the
transfer function concept is given by:
(7)
Then, the steady-state solution for eddy currents is given as in
[12]:
III. EDSMATHEMATICAL MODEL
ωΦ |
The schematic representation of the EDS concept is shown
in Fig.3, with a Halbach array rotating above, and close to a
conductive plate.
where,
|
ω | cos ω
θ ω , (8)
ω |
θ ω
(9)
atan
Finally, substitute (9) in (8) to obtain:
cos ω
Fig.3 Halbach array rotor above a thin conductive plate.
The Halbach array is canceling the magnetic field above it,
while producing a nearly purely sinusoidal varying periodic
magnetic field below the array:
ω ,
where,
(1)
is the peak value of the flux.
The analysis of the magnetic levitation is simplified here
through lumped parameter electric circuit theory, and basic
laws of electromagnetism. Thus, the voltage circuit equation
relating the voltage (emf) induced, and eddy currents in the
conducting base plate is written as:
,
(2)
where, L, R are the inductance and resistance parameters
of the conductive circuit.
The magnitude of
, in accord to Faraday’s law is:
ωΦ
ω
(3)
sin ω
(10)
At the limit, with ω
ω ≫ , the sinusoidal steady-state
solution is obtained from (10) as:
cos ω
90°
(11)
Note that in (11) the current is shifted 90° , in phase with
is in
in (3), so that
respect to the inducing voltage
phase with the magnetic flux given in (1), maximizing the lift
force.
Using the theory of plane-electromagnetic wave
propagation, the wave number of the magnetic field of the
Halbach array is defined as in [13]:
,
(12)
where, λ(m) is the wavelength of the array, ω is the
is
excitation frequency of magnetic field given in (5), and
the magnets apparent linear velocity (
ω
, in the zdirection in m⁄s .
Since we calculate the lifting force on a magnet array, we
replace ω by
in (10) and (11), with k given in (12).
The coordinate system used here, has its origin situated on
the upper surface of the conductive plate, with y-axis pointing
towards the magnet array, x-axis pointing across the plate, and
the z-axis in the direction of apparent (linear) motion along the
plate. The separation distance between the plate and the lower
surface of the Halbach array is denoted by the gap, g (Fig.3).
The calculation of the generated lift and drag forces,
requires the expressions of the longitudinal-z and vertical-y
components of the magnetic field of Halbach array. These are
given with a close approximation as in [14] [15]:
sin
(13, a)
cos
(13, b)
where,
is the peak flux density at the surface of the
Halbach array (at y = g).
The expression of the induced flux Φ, is derived by
integrating (13a) over y, between
0 and
(Fig. 3),
where d is the thickness of the conductive plate [14]:
Φ
sin
1
(14)
where,
(m) is the width of the magnets in the xdirection, transverse to the direction of motion above the plate.
Inserting peak value of the flux from (14) into (10), the
, is found as:
expression for the current in the x-direction
sin
cos
(16a, b)
Averaging the expressions of the forces’ magnitudes over the
wavelength,
then, the average
forces are found as follows:
Lift force:
(17)
Drag force:
(18)
A good measure of the efficiency of the magnetic levitation
system is given by the lift-to-drag ratio, dividing (17) by (18):
(19)
Note that
The conductive plate thickness d, it is assumed to
considerably exceed the standard skin-depth , which is given
by:
δ
(21)
where, ω 2 is the operating excitation frequency,
4π 10 (H/m) is the magnetic permeability of
μ μ
the conductive plate, and σ its electrical conductivity.
5.88 10 S⁄m)),
For a copper conductive plate (σ
and a Halbach array with six poles, at a rotational speed of
4875 (rpm), the skin depth is δ
4.2(mm); while for an
3.5 10 S⁄m)), the skin depth is
aluminum plate (σ
δ
5.4(mm).
The induced eddy currents flow only in this relatively thinskin layer, not occupying the whole thickness d, of the plate.
Therefore, the Joule (heat) losses are expected to increase.
In the limit case δ ≪ 1, in practical situations when the
skin depth is small compared to the wavelength, the
expression for the lift-to-drag ratio (when using a copper
plate) reduces to:
2.425
(22)
(15)
This current then interacts with the rotor magnetic field, to
produce the levitating force in the vertical y-direction, and the
drag force in the horizontal z-direction:
,
1.0). With the definition of K in (20), the transition
1.0.
velocity , is defined as the velocity where
/ ratio increases with the tangential velocity,
, while the drag force is varying inversely with
IV. SIMULATION AND PERFORMANCE INVESTIGATION OF
ELECTRODYNAMIC LEVITATION SYSTEM
The goal is to determine the best design solution, to achieve
a lifting force capable to levitate a physical prototype. The
main objectives addressed, are to enable the electrodynamic
levitation system to (a) achieve the expected lifting force by
maintaining at minimum the number of magnets, their size and
weight, and (b) stably operate in dynamic and stationary
conditions.
The approach applied here to achieve these objectives, is
using computational electromagnetics model-based design
technique. Due to the specific 3D nature of this application,
where phenomena such as the depth of penetration of varying
electromagnetic fields, and eddy currents effect are present,
the classic analytical calculations are difficult to use for
simulation, and visualization of the mentioned phenomena.
The solution chosen in this paper is to employ the finite element analysis (FEA) based software, ANSYS Maxwell™
[17]. In this way, the value of the lift force produced by the
Halbach array, mounted on a spinning rotor, at high speed, is
accurately calculated, and extracted for further analysis.
The Maxwell 3D design of two preliminary variants of
Halbach magnet rotor models used in this paper, are shown for
exemplification, in Fig. 4.
velocity.
Using (19), a parameter K is defined by the expression [14]:
(20)
The parameter K in (20) measures the levitation efficiency
of the system. Note that the units of measure for K is (s/m),
reciprocal of velocity units, or (N/watts).
Another useful design quantity is the “transition velocity”,
defined as the velocity at which, the lifting force has reached
on half of its asymptotic value, and at the same time, is the
velocity at which the lift and drag forces are equal ( /
(a)
(b)
Fig.4 Preliminary Halbach rotor design:(a) one array of NdFe61 magnets;
(b) NdFe55 magnets in three-concentric arrays.
In Fig.4 (a), the rotor contains a circular shape Halbach
magnet array, using NdFe61 magnets, with a magnetic
coercivity of -890 (kA/m). The array geometry data is: outer
diameter of 120 (mm), inner diameter 70 (mm), magnet axial
thickness 8 (mm). The magnetization arrangement of the
Halbach array (Fig.2), creates strong six poles field directed
axially towards the copper plate, placed underneath, at 8 mm
gap separation. The simulation is performed in Maxwell 3D,
using a transient solution with a rotor velocity of 4,400 (rpm).
The returned results are the calculated flux density 0.8 (T) in
the gap, and the lifting force magnitude of 288 (N). The
simulation of the induced eddy currents in a copper plate,
generated by the rotor design shown in Fig. 4(a), is displayed
in Fig. 5.
generated by the final rotor design, are shown in Fig. 8 and
Fig. 9, respectively.
The practical construction of the experimental device, to test
and validate the simulation of the final rotor model, and to
demonstrate the proof of concept, is presented in the next
section V.
Fig.7 Final design of the Halbach rotor with two concentric arrays of NdFe52
magnets.
Fig.5 Eddy currents generated by a 12-magnet rotor.
In Fig. 4(b), the rotor contains three concentric Halbach
arrays, using NdFe55 magnets, with the following data: most
outer diameter 120 (mm), most inner diameter 37 (mm),
magnet axial thickness 13 (mm). The Maxwell 3D transient
solution, returned the calculated flux density of 1.1 (T) in the
gap, and the lifting force magnitude of 1296 (N). The
simulation of the induced eddy currents in the conductive
plate, generated by this rotor design (Fig. 4(b)), is shown in
Fig. 6.
The final design variant has been decided based on the
analysis of simulation results, obtained with the two
preliminary models of the Halbach magnet rotors (Fig. 4), and
verifying the objectives mentioned at the beginning of this
section.
Fig.8 Magnetic flux density of the Halbach rotor final design (rotor’s bottom
side facing the gap shown).
Fig.9 Eddy currents induced using the Halbach rotor final design.
Fig.6 Eddy currents induced using three arrays of 12 magnets each.
The final model design shown in Fig.7, consists of a rotor
with two concentric Halbach magnet arrays, each one using 12
magnets with four magnets per spatial period, which create six
magnetic poles. The following is the final design data (see
Fig.7): inner diameter of the inner magnet array 52 (mm),
outer diameter of the outer array 105.2 (mm), NdFe52 cube
magnets with 0.5-inch (12.7 (mm)) long edges. The array’s
thickness of 12.7 (mm), is equal to one quarter of the spatial
period, and thus this array is easily fabricated (see section V)
since the blocks are identical, and are simply rotated as the
array is assembled.
The simulation of the final rotor design model (Fig.7), is
performed in Maxwell 3D, at a rotor velocity of 4,400 (rpm),
and using a transient solution. The solution returned is for the
calculated flux density of 1.0 (T) in the gap, and the lifting
force magnitude of 687 (N). The simulation results of the flux
density, and the induced eddy currents in the conductive plate
V. EDS EXPERIMENTAL DEVICE
The goal of building an EDS device was the proof of
concept, and the ability to physically make measurements that
can be related to the software simulations.
The first device built in this work is a linear Halbach array,
which is comprised of five 1-inch neodymium type cube
magnets, shown in Fig.10. The magnetization vectors of the
cube magnets point from the South pole toward the North
pole. The magnets are positioned so that the magnetization
vector is rotating in the same plane, 90 degrees from one
magnet to the next. Due to this specific arrangement, the
adjacent magnets do exert forces on each other, and thus a
housing is needed around the array, which will keep them
aligned and compact in the above-mentioned setup, shown in
Fig. 10(a). The housing was designed in SolidWorks, and
printed out of ABS (Acrylonitrile butadiene styrene), on a 3D
printer. The strength of the magnetic field of this array can be
visualized using a magnetic viewing film, as displayed in
Fig.10 (b).
(a)
(b)
Fig.10 (a) Built linear Halbach array (b) Field visualization through magnetic
viewing film: on the left the weak side, on the right the augmented side
showing two large magnetic poles.
Next stage in the design, is the implementation of a Halbach
array that can be used in generating a dynamic magnetic field.
This is a radial Halbach array, in which, the magnets are
placed in an equal radial distance from the center of the base
disk. The magnets are rotated in the same plane, so that the
augmented side of the array is perpendicular to one of the
faces of the disk, leaving the other face to the weak side of the
magnetic field.
magnets, in radial arrangement, with each of the magnets
being rotated 90 degrees at each step, creating six poles
around the array. Several factors played a role in considering
the array fit for this project such as: the robustness, weight,
and size. The rotor in the middle (Fig.11) offered less poles
and light weight, by compromising the robustness of the array
at high speeds.
The rotor on the top right (Fig.11), offered strong fields
because of the number of magnets in the array. Comparing the
top right (white) variant, to the design variant on the top left
(orange), by measuring the weight of the whole array, there is
a compromise to be made in between the strength of the
magnetic field, to the weight of the array. As more and more
arrays of magnets are placed in radial direction, the number of
magnets cannot be increased in the rows further away from the
center of the array. This results in increased space, in between
the magnets, in azimuthal direction, that must be filled with
base structure material. This process will add more weight to
the array, while places the magnets further apart, weakening
the magnetic fields.
The experimental tests on these physical prototypes were
conducted in a safe environment, with eye protection. Using
the Bosch Hammerdrill 1199VSR that provided 3000 rpm, all
rotors were initially tested for lifting force evaluation, above a
copper and an aluminum plate, Fig. 12.
Several designs were drawn in SolidWorks, and printed out
of ABS, with the help of a 3D printer. The design variants
worth mentioning are shown in Fig.11, where the main
differences in the design can be seen.
Fig.12 Initial testing of the radial Halbach arrays.
Fig.11 Radial Halbach arrays (top), viewed through the magnetic viewing film
(bottom). Each array has its corresponding visible field underneath it.
The array on the top left (orange color in Fig.11), has two
sets of ½ inch cube magnets in radial arrangement, with each
of the magnets being rotated 90 degrees at each step. This way
the magnets create six poles going around the array. The array
in the middle (black in Fig.11), was created with rectangular
shape neodymium type magnets with their magnetization
vector being perpendicular to the axial length of magnets. This
disk array housing allows the magnets to be placed on the
edge and on the side, this way allowing the construction of an
array where the magnetization vector is rotated 45 degrees at
each step. This leads to a reduced number of augmented
magnetic poles, which are four in this case. The array on the
top right (white color in Fig.11), has three sets of ½ inch cube
Considering the tests findings, the implemented EDS variant
confirming the simulated final design (Fig. 7, 8, 9) was the
one on top left in Fig.11 (orange), with small modifications,
where the radii of the arrays and rotor disk were reduced to a
safe minimum, so the structure remained sound, and the arrays
are as tight as possible. The outcome of the final implemented
design can be seen in Fig.13.
The design accounted for the coupling of the rotor to the
axis of the driving motor, with the countersink in the middle
of the rotor. The rotor disk in Fig.13 was then loaded with ½
inch magnets glued into place.
Fig.13 Radial Halbach arrays housing final design, 3D printed from ABS
material.
The glue was applied to 5 sides out of 6 of each magnet,
accounting for a minimum 80% of the magnets surface in
contact and glued to the surface of the housing.
In the next step of this work, to drive the assembled radial
arrays at high rotational speeds, the Axi 5330/24 outer runner
brushless direct current (BLDC) motor has been selected. The
selection was made on the several criteria such as: required
power sources, provided torque, and maximum speed.
The BLDC motors are open loop voltage controlled by the
Jeti Advance Opto 77 ampere controllers, with 1V/197 rpm.
An Arduino Uno was used to create 50Hz pulse with
modulation (PWM) signals, to simply act as the throttle for the
controller.
With the rotor driven by the motor, the testing of the
dynamic magnetic fields could be performed. The static and
dynamic magnetic fields where measured using a F.W. Bell,
Model 5070, Gauss/Tesla Meter (Fig. 14). The dynamic flux
density was measured at 4010 rpm, and yielded B = 0.29 (T).
The static flux was measured to be strongest across two
opposing poles, B = 0.602 (T).
(c)
(d)
Fig.15(c) CNC machine drilling), (d) manual countersink drilling.
The initial assembly of the four motors, controllers and
Halbach rotors on the frame is shown in Fig.16.
(a)
(b)
Fig.16 (a) Rotors with loaded Halbach arrays facing up, (b) the frame with
motors, controllers and Halbach arrays facing down.
(a)
(b)
Fig.14 Testing the strength of the magnetic fields, (a) dynamic and (b) static.
The chassis of the EDS device was machined from a 1 , ¼
inch thick aluminum plate with the goal to accommodate four
BLDC motors symmetrically placed, Fig. 15.
Because of the dynamic nature of the device operating at
high rate of speed, a safety barrier was designed around it as
shown in Fig.17 and Fig.18. It also plays the role of footing
for the device, this way the device can rest on the safety
barrier whenever not levitating. The safety barrier was design
in SolidWorks, printed out in ABS with a 3D printer.
(a)
(b)
Fig.15 (a) Cutting design in SolidWorks), (b) plasma cut plate.
(a)
(b)
Fig.17(a) Safety barrier components with the interlocking parts designed in
SolidWorks, (b) final assembly of the barrier around the device
The weight of the assembly is 6618 (g), with a 182 (g)
carrying tray that was designed to be attached on the top of the
device. The overall weight is 6800 (g), or 15 (lb.).
Tests were conducted above a copper, and an aluminum
plate, at different speeds. The air-gaps were measured and
recorded at different speeds (Fig. 20), and also at different
weights
EDS device has then been built in this work and successfully
validated the proof of concept.
The variation of the most important parameters in
electrodynamic levitation systems, which are the magnets
coercivity, their spatial arrangement, the magnet size and their
number, the excitation frequency determined by the
mechanical velocity and the number of poles, all have been
simulated using ANSYS Maxwell software. The influence of
parameters variation on the generated magnetic flux density,
induced eddy currents, and on the magnitude of the lifting
levitation force, were reported.
Fig.18 Final assembly of the EDS device
In Fig. 19 the magnetic fields can be observed with all four of
mounted Halbach arrays. This viewing represents the strong
side of the arrays.
Four outer rotor AC permanent magnet motors have been
interfaced with controllers and then used to build a physical
device of the designed EDS prototype. The implemented
electrodynamic system stable operation has proved the
concept of magnetic levitation by experiments and test results
were reported.
REFERENCES
[1]
[2]
[3]
[4]
Fig.19 Strong field side of the assembly with four Halbach arrays visualized
though the magnetic viewing film
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
Fig.20Air-gap measurement versus rotors speed, (top) for a copper plate, and
(bottom) for an aluminum plate
The copper plate yielded a larger gap, because the larger
eddy currents induced made a stronger opposing magnetic
field. Air gap levels topped off at a certain speed due to fast
decay of the flux.
VI. CONCLUSION
The work in this paper is meant to take a close look at the
challenges in designing and implementing an electrodynamic
levitation system. The model-based design method using FEA
and ANSYS Electromagnetics software has been employed to
develop and simulate the proposed system. An experimental
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