Joint Advanced Student School
2006
Magnetic Bearings
Jeff Hillyard
Technische Universität München
Overview
Magnetic Bearings
•
•
•
•
•
Introduction
Magnetism Review
Active Magnetic Bearings
Passive Magnetic Bearings
Industry Applications
Introduction
Magnetic Bearing Types
• Active/passive magnetic bearings
– electrically controlled
– no control system
• Radial/axial magnetic bearings
Introduction
Motivations
Advantages of magnetic bearings:






contact-free
no lubricant
(no) maintenance
tolerable against heat, cold, vacuum, chemicals
low losses
very high rotational speeds
Disadvantages:
Minimum Equipment for AMB
 complexity
 high initial cost
Source: Betschon
Introduction
Survey of Magnetic Bearings
Source: Schweitzer
Magnetism
Magnetic Field
south pole
north pole
magnetic
field line
iron filings
Pole Transition
Magnetism
Magnetic Field
Magnetic field, H, is found around a magnet or a current
carrying body.
H
 H ds  i
H
i
2r
i
(for one
current loop)
Magnetism
Magnetic Flux Density


B  H
multiple loops
of wire, n
B = magnetic flux density
 = magnetic permeability
H = magnetic field
  0  r
Meissner-Ochsenfeld Effect
0 = permeability of free space
r = relative permeability
diamagnetic
paramagnetic
ferromagnetic
ni
H
2r
 1
 1
  1
Magnetism
B-H Diagram
Ferromagnetic: a material that can be magnetized
Remanence, Br


B  H
magnetic saturation
B
Coercivity, Hc
H
area within loop represents
hysteresis loss
Magnetism
Lorentz Force


  
f Q EvB

f = force
Q = electric charge
E = electric field
V = velocity of charge Q
B = magnetic flux density
Magnetism
Lorentz Force
Simplification:


  
f Q EvB


 
E  v  B




 
f Q vB

Source: MIT Physics Dept. website
Magnetism
Lorentz Force
Further simplification:

 
f  Qv  B


i  Qv
Analogous Wire
B
i
f
  
f i B
force perpendicular to flux!
Magnetism
Reluctance Force
Force resulting from a difference between magnetic
permeabilities in the presence of a magnetic field.
 force perpendicular to surface!
The energy in a magnetic field with
linear materials is given by:
1
U   BHdV
2V
U = energy
V = volume
f 
U
l
2
B A
f 
2
Magnetism
Reluctance Force
Basic equation:
1
U   BHdV
2V
Energy contained within airgap:
1
1
U a  Ba H aVa  Ba H a Aa 2s
2
2
l Fe  2s
Aa
Magnetism
Reluctance Force
Evaluating the magnetic circuit for a simple system:
 Hds  l
l Fe
B
0  r
l Fe  2s
Fe
H Fe  2sH a  ni
 2s
B
0
 ni  NI
Aa
Assumption:

B  0
NI
 l Fe

  2 s 
 r

  BFe AFe  Ba Aa
BFe  Ba  B
Magnetism
Reluctance Force
Principle of virtual displacement:
U a
f 
 BH a Aa
l
Ha 
B
0
quadratic!
2


ni
 Aa cos 
f   0 
 l Fe  r  2 s 
i2
f k 2
s
0
inversely quadratic!
Active Magnetic Bearings
Elements of System
•
•
•
•
•
Electromagnet
Rotor
Sensor
Controller
Amplifier
Active Magnetic Bearings
Force Behavior
Magnetic Force
1
2
xs
fs
Force
~
Force
fm
Spring Force
Distance
xs
Distance
xs
Active Magnetic Bearings
Force Linearization
Magnetic Force
fm
~
Spring Force
1
2
xs
fs
mg
mg
x0
xs
x0
xs
Active Magnetic Bearings
Force Linearization
Operating Point (constant current)
Redefining distance:
fm
f
x
f  ks x
x  xs  x0 
x
x0
xs
ks = force-displacement factor
f m, s i
m i0
 ks x
Active Magnetic Bearings
Force Linearization
Operating Point (constant position)
f m ,i
im
xs  x0
 ki i
i  im  i0
ki = force-current factor
fm
~ im
2
fm
f  ki i
i
mg
i0
im
i0
im
Active Magnetic Bearings
Force Linearization
im
Linearized equation:
f x, i   f m, s i
f m, s i
f m ,i
m i0
xs  x0
m i0
 f m ,i
xs  x0
x
 ks x
i  im  i0
 ki i
f x, i   k s x  ki i
x   xs  x0
Not valid for:
- rotor-bearing contact
- magnetic saturation
- small currents
Active Magnetic Bearings
Closed Control Loop
Open Loop Equation:
Basic System
f x, i   k s x  ki i
i
Controller function?
- Provide force, f
Controller signals?
- Input: position, x
- Output: current, i
 i = i(x)
x
x
Artifical damping and stiffness:
f  kx  dx 
k
x
d
Active Magnetic Bearings
Closed Control Loop
Solving for controller function:
ks x  ki i  kx  dx
Basic System
i
k  k s x  dx
ix   
x
ki
x
To model position of rotor:
f  mx
f x, i   k s x  ki i
mx  k s x  ki i
Just like for the spring system!
mx  dx  kx  0
Active Magnetic Bearings
Closed Control Loop
System characteristics:
x(t)
m2  d  k  0

    j
Ce t
with
k
d2


m 4m 2

d
2m
General solution for position:
xt   Ce t cost   
Eigenfrequency:
0   2   2  k m
t
Active Magnetic Bearings
Closed Control Loop
Controller Abilities:
1)
2)
3)
4)
k, d can be varied in controller
air gap can be varied in controller
specify position for different loads
rotor balancing, vibrations, monitoring...
Active Magnetic Bearings
Closed Control Loop
Differential driving mode
Linearization:
magnetic force was
determined to be
2
f 
1
i
 0 n 2 Aa   cos 
4
s
i2
 f  k 2 cos 
s
where
1
k   0 n 2 Aa
4
 i0  ix 2 i0  ix 2 
 cos 
f x  f   f   k 

2
2 
 s0  x  s0  x  
s  s0  x
s  s0  x
Active Magnetic Bearings
Closed Control Loop
Differential driving mode
Linearization:
f x
fx 
ix

x0
f x
 ix 
x
x

x0
 4ki0

 4ki0 2

f x   2 cos  ix   3 cos   x
 s0

 s0

ki
f x  ki i x  k s x
ks
linearized for
differential driving
mode
Active Magnetic Bearings
Bearing Geometry
Radial Bearing
Axial Bearing
Active Magnetic Bearings
Bearing Geometry
B circumferential to
rotor axis
B parallel to rotor axis
- similar to electromotors
- rotor requires lamination
- hysteresis loss low
- lamination avoided
Orientation:
magnet pole pairs are often lined up with the principle
coordinate axes x and y (vertical and horizontal)
 control equations are simplified
Active Magnetic Bearings
Sensors
Position Sensor
• contact-free
• measure rotating surface
+
– surface quality
– homogeneity of surface material
– various values
Other Sensors
• speed
• current
• flux density
• temperature
• …
…other concerns:
observability
placement
cost
sensor
Active Magnetic Bearings
Sensors
“Sensorless“ Bearing
- calculate position
- less equipment
- lower cost
Source: Hoffmann
Active Magnetic Bearings
Amplifier
Converts control signals to control currents.
Analog Amplifier:
Switching Amplifier:
- simple structure
- low power applications
P<0.6 kVA
- lower losses
- high power applications
- remagnetization loss
Active Magnetic Bearings
Electrical Response
There is an inherent delay in the electrical system
 inductance
di
voltage drops: u L  L
dt
and
uR  Ri
Total voltage drop:
di
u  Ri  L  ku x
dt
ku = voltage-velocity coefficient
velocity within magnetic field
induces a voltage
Active Magnetic Bearings
Control Equations of Motion
Block diagram with voltage control:
di
u  Ri  L  ku x
dt
f ( x, i)  ks x  ki i
mx  f
Source: Schweitzer
Active Magnetic Bearings
Current vs. Voltage Control
Voltage Control:
- more accurate model
- better stability
- low stiffness easier to realize
- voltage amplifier often more convenient
- possible to avoid using position sensor
Current Control:
- simple control plant description
- simple PD or PID control
Flux Control:
- very uncommon
Active Magnetic Bearings
Addressing of Assumptions
Uncertainties in bearing model
- leakage flux outside of air gap
- air gap is bigger than assumed
- iron cross section is non-uniform
Active Magnetic Bearings
Types of Losses
Air Losses
- air friction  divide shaft into sections
Copper Losses (Stator)
2
- wire resistance  PCu  RCu i
Iron Losses (Rotor)
- hysteresis (higher w/ switching amplifier)
- eddy currents
Active Magnetic Bearings
Copper Losses
For differential driving mode:
PCu ,max  2RCu imax
2
An K n  Ad n
NI max  PCu ,max
An K n
2 lm
An = slot area
Kn = bulk factor
 = specific resistance
lm = average length of turn
limit of permissible mmf!
Active Magnetic Bearings
Rotor Dynamics
Areas of Consideration
•
•
•
•
•
natural vibrations
forward/backward whirl (natural vibrations)
critical speeds
nutation
precession (change in rotation axis)
Source: Wikipedia
Active Magnetic Bearings
Rotor Dynamics
rotor touch-down in retainer bearings
- maintenance
- sudden system shutoff
- during system shutdown
 very difficult to simulate
cylindrical motion
conical motion
Source: Schweizer
Active Magnetic Bearings
Rotor Stresses
Radial
2 2
 2

ri ra
1
2
2
 r  3    ri  ra  2  r 2 
8
r


Tangential


2 2


ri ra
1
2
2
2
 t   3   ri  ra  3   2  1  3 r 2 
8
r


Source: Schweizer
largest stress is at inside
radius of disc with hole!
Active Magnetic Bearings
Rotor Stresses
Implications of max stress:
 max velocity (full disc)!
vmax  ra  
8 S
3  
s = max tensile strength
Material
steel
brass
bronze
aluminium
titanium
soft ferro. sheets
vmax (m/s)
576
376
434
593
695
565
Actual reached speeds (length 600 mm, dia. 45 mm):
vmax  300 m
s

max  120,000rpm
Source: Schweizer
Passive Magnetic Bearings
Permanent Magnets
Common Materials:
1)
2)
Relative Sizes
neodymium, iron, boron (Nd Fe B)
samarium, cobalt, boron
(Sm Co, Sm Co B)
3)
4)
ferrite
aluminium, nickel, cobalt
(Al Ni, Al Ni Co)
Issues:
- material brittleness
- varying space requirements (B-H)
- operating temperatures
(equal H at 10 mm)
Passive Magnetic Bearings
Permanent Magnets
at least one degree of
freedom unstable!
increase in stiffness with
multiple rings
caution: misalignment!
reluctance bearings:
- non-rotating magnets
- resistance to radial
displacement
Passive Magnetic Bearings
Permanent Magnets
High Potential
- economical
- reliable
- practical
 already replacing some active magnetic bearings
- smaller size equipment and systems
- systems with large air gaps
Source: Boden
Applications
Turbomolecular Pump
École Polytechnique Fédérale de Lausanne, Switzerland
- eliminates complicated lubrication system
- high temperature resistance
- reduction of pollution
- vibrations, noise, stresses avoided
- improved monitoring (unbalances, defects, etc.)
Status: suboptimal design
 overheating at load (> 550°C)
 increase life span
 optimize fill factor
 reduce cost
 simplify manufacturing
Applications
Flywheel (‘97)
New Energy and Industrial Technology Development Organization
(NEDO) – Japan‘s Ministry of International Trade and Industry (MITI)
•
•
•
•
T=½J2  speed has larger influence than mass (better energy density)
fiber-reinforced plastics for high strength
fracture into small pieces upon failure  above ground
combination of superconductor and permanent magnet bearings (hsys = 84%)
Applications
Flywheel (‘97)
Current Development Goals (NEDO)
•
•
•
•
increase load force
reduce amount load force decrease with time (magnetic flux creep)
reduce rotational loss
increase size of bearings for larger systems
Applications
Maglev Trains
Maglev = Magnetic Levitation
• 150 mm levitation over guideway track
 undisturbed from small obstacles (snow, debris, etc.)
• typical ave. speed of 350 km/h (max 500 km/h)
 what if? Paris-Moscow in 7 hr 10 min (2495 km)!
• stator: track, rotor: magnets on train
Source: DiscoveryChannel.com
Applications
Maglev Trainsx
Maglev in Shanghai
- complete in 2004
- airport to financial district (30 km)
- world‘s fastest maglev in commercial operation (501 km/h)
- service speed of 430 km/h
Source: www.monorails.org
Applications
Maglev Trains
Noise Reduction
by Frequency
Noise Reduction
by Speed
Source: Moon
Magnetic Bearings
References
1.
Betschon, F. Design Principles of Integrated Magnetic Bearings, Diss. ETH. Nr. 13643, ETH
Zürich, 2000.
2.
Boden, K. & Fremerey, J.K. Industrial Realization of the “SYSTEM KFA-JÜLICH“ Permanent
Magnet Bearing Lines, Proceedings of MAG ‘92 Magnetic Bearings, Magnetic Drives and Dry
Gas Seals Conference & Exhibition. Lancaster: Technomic Publishing, 1998.
3.
Electricity and Magnetism. Hyperphysics. Georgia State University, Dept. of Physics and Astronomy.
1 Apr. 2006 <http://hyperphysics.phy-astr.gsu.edu/Hbase/hph.html>.
4.
Fremery, J.K. Permanentmagnetische Lager. Forshungszentrum Jülich, Zentralabteilung
Technologie, 2000.
5.
Hoffmann, K.J. Integrierte aktive Magnetlager, Diss. TU Darmstadt. Herdecke: GCA-Verlag 1999.
6.
Lösch, F. Identification and Automated Controller Design for Active Magnetic Bearing Systems,
Diss. ETH. Nr. 14474, ETH Zürich, 2002.
7.
Maglev Monorails of the World: Shanghai, China. The Monorail Society Website. 1 Apr. 2006
<http://www.monorails.org/tMspages/MagShang.html>.
8.
Maglev Train Explained, DiscoveryChannel.ca. Bell Globemedia 2005
<http://discoverychannel.ca/interactives/japan/maglev/maglev.html>.
9.
Magnetic Bearings & High Speed Motors, S2M. 1 Apr. 2006 <http://www.s2m.fr/chap3/>.
Magnetic Bearings
References
10. Moon, F.C. Superconducting Levitation: Applications to Bearings and Magnetic Transportation.
New York: John Wiley & Sons, 1994.
11. Research and Development for Superconducting Bearing Technology for Flywheel Electric
Energy Storage System. New Energy and Industrial Technology Development Organization
(NEDO). 1 Apr. 2006
<http://www.nedo.go.jp/english/activities/2_sinenergy/1/p04033e.html>.
12. Schwall, R. Power Systems – Other Applications: Flywheels. Power Applications of
Superconductivity in Japan and Germany. WTEC Hyper-Librarian 1997
<http://www.wtec.org/loyola/scpa/04_02.htm>.
13. Schweizer, G., Bleuler, H., & Traxler, A. Active Magnetic Bearings: Basics, Properties and
Applications of Active Magnetic Bearings. Zürich: Hochschulverlag AG an der ETH, 1994.
14. Widbro, L. Magnetic Bearings Come of Age. Revolve Magnetic Bearings Inc. 2004.
MachineDesign.com. 1 Apr. 2006
<http://www.machinedesign.com/ASP/strArticleID/57263/strSite/MDSite/viewSelectedArticle.asp>.
15. Wikipedia contributors (2006). Hysteresis. Wikipedia, The Free Encyclopedia. April 1, 2006
<http://en.wikipedia.org/w/index.php?title=Hysteresis&oldid=45621877>.
16. Wikipedia contributors (2006). Magnetic field. Wikipedia, The Free Encyclopedia. April 1, 2006
<http://en.wikipedia.org/w/index.php?title=Magnetic_field&oldid=46010831 >.
Questions?
Applications
Crystal Growing System