Download Active Magnetic Bearings and Its Applications

Introduction to Magnetic Bearings
Lecture presented in Quality Improvement
Program (QIP’08) at Indian Institute of
Technology Guwahati
Overview of the Presentation
• Introduction
• Design of Active Magnetic Bearings
• Control Engineering of Magnetic Bearings
• Control of Rotor by using Magnetic Bearings
Jagu Srinivasa Rao, (Research Scholar)
Department of Mechanical Engineering
Indian Institute of Technology Guwahati
• Conclusions
December, 2008
Radial Magnetic Bearing
Introduction
• An active magnetic bearing (AMB) system supports
a rotating shaft, without any physical contact by
suspending the rotor in the air, with an electrically
controlled (or/and permanent magnet) magnetic
force.
• It is a mechatronic product which involves different
fields of engineering such as Mechanical, Electrical,
Eight-Pole Radial Magnetic-Bearing
Control Systems, and Computer Science etc.
Test Apparatus for rotor control
Thrust Magnetic Bearing
Introduction to Active Magnetic Bearings
Rotor Disc
Left AMB
Coil Winding
Upper AMB
Rotor shaft
Rotor shaft
Left AMB
Horizontal shaft
Lower AMB
Vertical shaft
Typical Actuator – Controller unit of an AMB
1
Introduction to Active Magnetic Bearings
• Magnetic Bearings are free of contact and can be utilized in
vacuum techniques, clean and sterile rooms, transportation of
aggressive media or pure media
Electro magnet
f
Rotor
Power
Amplifier
Advantages of Magnetic Bearings
Sensor
Controller
• Highest speeds are possible even till the ultimate strength of
the rotor
• Absence of lubrication seals allows the larger and stiffer
rotor shafts
• Absence of mechanical wear results in lower maintenance
costs and longer life of the system
Working principle of magnetic bearing
• Adaptable stiffness can be used in vibration isolation,
passing critical speeds, robust to external disturbances
Classification of Magnetic Bearings
According to
•control action
– Active
– Passive
– Hybrid
•Forcing action
– Repulsive
– Attractive
•Sensing action
– Sensor sensing
– Self sensing
•Load supported
– Axial or Thrust
– Radial or Journal
– Conical
Applications of Magnetic Bearings
•Turbo molecular pumps
•Test rig for high speed tires
•Blood pumps
•Magnarails and maglev systems
• Magnetic effect
– Electro magnetic
– Electro dynamic
•Molecular beam choppers
•Gears, Chains, Conveyors, etc
•Epitaxy centrifuges
•Energy Storage Flywheels
• Application
– Precision flotors
– Linear motors
– Levitated rotors
– Bearingless motors
– Contactless Geartrains
•Contact free linear guides
•High precision position stages
•Variable speed spindles
•Active magnetic dampers
•Pipeline compressor
•Smart Aero Engines
•Elastic rotor control
•Turbo machines
Fields of Applications of Magnetic Bearings
•Semiconductor Industry
•Maglev Transportation
•Bio-medical Engineering
•Precision Engineering
•Vacuum Technology
•Energy Storage
•Structural Isolation
•Aero Space
•Rotor Dynamics
•Turbo Machines
Electromagnetism
– Electromagnetic field
– Lorenz force
2
Electromagnetism
When a charged particle is
at rest it won’t emit
electromagnetic waves
rather it is surrounded by
electrostatic field
When the charged particle is
in uniform motion (i.e. the
motion with uniform velocity
in a direction) the
electrostatic field is
associated with
magnetostatic field.
Electromagnetism
When the particle is in
accelerated motion then
the magnetic field will be
oscillating.
3d electrostatic field
surrounding a
charged particle
In electromagnetic
waves both the electric
and magnetic fields are
oscillating and harmonic.
Magnetostatic field
Feed back loop of electromagnetism
The electric and magnetic
fields are generated by
electric charges
Charges generate electric
fields
Movement of charges
generate magnetic fields
The electric and magnetic
fields interact only with each
other
Changing electric field acts like
a current, generating vortex of
magnetic field
Changing magnetic field
induces (negative) vortex of
electric field
Feed back loop of electromagnetism
The electric and magnetic
fields produce forces on
electric charges
Electric force which is
generated by the electric field
and is in same direction as
electric field
magnetic force which is
generated by the magnetic
field and is perpendicular both
to magnetic field and to
velocity of charge
The electric charges move in
space
The electric charges move in
space when they are acted
upon by field forces
The four fundamental forces
Strong nuclear force
Weak nuclear force
which holds atomic
nuclei together
which causes
certain forms of
radioactive decay
The electric and magnetic
fields are generated by
electric charges
The electric
charges move in
space when they
are acted upon by
field forces
The electric and
magnetic fields
interact only with
each other
The electric and magnetic
fields produce forces on
electric charges
The four fundamental forces
Gravitational force
Which causes the
masses to attract
each other
Electromagnetic force
Which is caused by
electromagnetic fields on
electrically charged
particles
3
The four fundamental forces
Force between two electrically charged particles
All the other forces are derived from
these four fundamental forces
Electro-magnetic force is one of these
four fundamental forces
r
Coulomb force (Static)
fc =
q1q2
r
4πε 0 r 3
fc
q1
q2
Lorenz force (Dynamic)
 γq q r 
 γ q q v×r 
fl =  1 2 3  + v ×  1 2 2 3 
 4πε 0 r 
 4πε 0 c r 
Electric and magnetic components
of Lorenz force
Comparison Electric and magnetic
components of Lorenz force
If q1=q then
v × B v2
1
≤ 2 ≤ 23
E
c 10
F = q (E + v × B )
E=
γ q2r
4πε 0 r 3
r= r;
Electric flux;
γ=
B=
1
1 − (v / c)
2
γ q2 v × r
Magnetic flux;
4πε 0 c 2 r 3
Lorenz factor;
ε 0 = 8.854 ×10−12 C 2 / J-m Electric permeability of vacuum;
1
ε 0c 2
=µ 0 = 4π×10 -7 N/A 2
Three conclusions:
• Magnetic component of Lorenz force is at least smaller by a factor of 1023!
But we don’t face the effect of electric field in conductors because protons
and electrons are equal in number and generate equal and opposite electric fields
canceling each other
• Protons have no motion with reference to conductor and there won’t be
magnetic component from them. Thus the magnetic component observed is
the relativistic effect of electrons only
• When the conductor is moving with reference to another frame both the
protons and electrons will move with the same velocity thus the relativistic
effects due to the velocity of conductor will be cancelled out
Magnetic permeability of vacuum;
Relations between E and B
Effective Lorenz force in macro calculations
For macro calculations Lorenz force is
reduced to the form
Bw
Gauss’ Law for
magnetism
B
F = q ( v × B)
Gauss’ Law for linear
materials
v
F
Faraday’s law of
magnetic induction
∇⋅E =
q
1
∫ E ⋅ ds = ε ∫
ε0
S
∂B
∂t
Ampere’s law and
∂E 

Maxwell's
∇ × B = µ0  J + ε 0

∂t 
extension

V
qdv
∫ B ⋅ ds = 0
∇⋅B = 0
∇×E = −
0
S
∂
∫ E ⋅ dl = − ∂t ∫ B ⋅ ds
L
S
∂E 

∫L B ⋅ dl = µ0 ∫S  J + ε 0 ∂t  ⋅ ds
Lorenz force acts perpendicular to both velocity
of charged particle and magnetic flux
These relations are called simplified Maxwell's relations who formulated
the original relations from previous works
4
Specifications
Design methodology of
magnetic bearing systems
Mechanical design
Design of magnetic
actuator
– Bearing magnet
– Magnetic circuit
– Coil
no
Performance O.K?
yes
Magnetic actuator design
no
Performance O.K?
yes
Control system design
Simulation
Experimentation
Performance O.K?
yes
no
End
Magneto mechanical systems
Areas involved in the design of magnetic
bearing systems
Magnetic bearing
system design
Mechanical
design
Modal
frequencies
Magnetic actuator
design
Bearing magnet
design
Coil design
Topology
Admissible coil
temperature
Load
estimation
Number of
turns
Control system
design
Power
amplifier
design
Losses
Controller
design
Sensor
design
Stiffness
Position
sensing
Damping
Bandwidth
Stability
Magnetic
circuit
design
Winding
scheme
Balancing
Coil head
According to the known technology till
now, magnetic bearings can be classified
for their design according to the purpose
of the levitated object as
Velocity
sensing
Current
sensing
Flux
sensing
Self sensing
Magneto mechanical systems
– Precision flotors (precision stages,
isolation bases, isolation springs)
• Levitation force
• Propulsion force
Magneto mechanical systems
Linear motors
(Contactless sliders,
maglev trains and
conveyors)
• Levitation force
• Propulsion force
Levitation force
Propulsion force
A magnetic Precision Stage
Principle of a linear motor
5
Magneto mechanical systems
Magneto mechanical systems
Levitated rotors
Bearingless motors
(gas turbines,
energy storage
flywheels, high
speed spindles,
balancing and
vibration control
of rotors)
(canned pumps,
compact pumps, blood
pumps, spindle
drives,
semiconductor
process)
– Radial load
– Thrust load
Rotor levitated by Radial and
Axial Active Magnetic Bearings
Magneto mechanical systems
– Radial load
– Thrust load
– Torque
Bearingless Motor
Linear systems from rotary systems
Contactless Gears and
Couplers
– Regulated torque
transmission
Design of a thrust magnetic bearing
Macro Geometry of Thrust Magnetic Bearing
Inner wall
Coil
Back-wall
Outer wall
Space for shaft
Space for coil
Figure 1: Parts of Thrust Magnetic Bearing
6
Magnetic circuit
Optimal design
Optimal design is carried out
in two steps
• Modeling the magnetic
circuit
Actuator
Rg
Ra
Ni
Ni
Coil
– Determines the accuracy of
achieving the objective
Rl
gap
φ
Rg
Rl
Equivalent electric (dc)
circuit representation
Levitated object
• Optimization of the
parameters
Ra
φ
Magnetic circuit
– Determines the efficiency of
the achieving the objective
R=
Magnetic circuit analogy
with electric circuit
l fp
µA
=
l fp
µ0 µ r A
Ideal magnetic circuit model
∫
L
H ⋅ dl = ∫ J ⋅ nda
S
la
( Ampere's law )
2 H g lg + H a la + H s ls = ni
Magnetic circuit
Electric circuit
Magneto Motive
Force (MMF)
Electro Motive
Force (EMF) or
Voltage (V)
Magnetic Flux ( φ )
Electric Current (i)
Reluctance (R)
Resistance (R)
H
J
B = µ H or H = B / µ
lg
B
B 
2 Bg lg + µ 0  a la + s ls  = µ 0 ni
µs 
 µa
B
B 
if µ0  a la + s ls  is neglected
µs 
 µa
Bg =
ls
µ 0 ni
2lg
Flux density is used to find the force exerted
Extension of the ideal model
B
B 
if K a is added for µ0  a la + s ls 
µs 
 µa
as core loss factor and K i is added
Force by using flux density
la
as coil loss factor, then
lg
The model reduces to
2 K a Bg lg = µ0 Ki ni
Bg =
ls
B=φ
A
F =
µ 0 K i ni
2 K alg
∆F =
=
Ni µ0
2(lg − x )
B02
Ag
2µ0
2 Ag
µ0
i0 + ∆i
B0 + ∆B
i0 − ∆i
B0 − ∆B
B0 ∆B
Differential actuator
7
Terminology used in
magnetic circuit
Magnetic flux density, T
Linear range of flux density
Bsat
α max Bsat
α min Bsat
Linear Range
Quantity
Symbol
Permeability
of vacuum
µ0
Permeability
µ
Reluctance
R
1
ε 0c2
µ0 µ r
l fp
µA
=
Units Magnitude
Vs/Am
l fp
µ 0 µr wl
4π ×10 −7
Vs/Am
0.026
Vs/A
7.95e5 for air
3.97e4 for Fe
Magneto
motive force
ni
n×i
Aturns
1600
Magnetic flux
φ
Niµ 0wl
Ni
=
2Rg
2(g − x)
Wb
0.0010
Flux density
Magnetic force, N
Formula
Magnetic flux
linkage
B
λ
φ
A
=
Ni µ 0
2( g − x )
Nφ
T
10.05
Wbturns
0.1005
Hysteresis is assumed to be negligible
while setting the linear range
Different quantities used in
magnetic circuit
Quantity
Symbol
Formula
Current
density
J
Magnetic
inductance
L
λ=
Nominal
inductance
L0
L x=0 =
Magnetic
force by
inductance
F
L0i 2
2lg
Magnetic
force by flux
density
F
B02
Ag
2µ0
Magnetic
force for diff
actuator
F
i
i
=
A wl
i
Ag
2µ
n 2 µ0 wl
2 ( lg − x )
(B
− B−2
Magnitude
A/m2
16e4
H=Wb/A
n 2 µ0 wl
2lg
2
+
Design vector for optimal design
Units
)
H
0.0063
N
804.2
N
804.2
N
19.84
All the other parameters are dependant
0
Input parameters taken for the design of thrust magnetic bearing
Parameter
Value
Parameter
Value
Inner radius of the
bearing
25.00mm
Specific gravity of the
stator iron
7.77g/cm 3
Operating air gap
4.00mm
Specific gravity of the
copper
8.91g/cm 3
Operating load
2025N
7.5g/cm 3
Specific gravity of
permanent magnet material
neodymium-iron-baron
Variation in the gap
±5%
Coil mmf loss factor
1.394
Variation in the load
±10%
Actuator loss factor
1.072
Saturation flux density
1.00T
Flux leakage factor
0.840
Packing factor
0.85
Saturation current
density
4.0A/mm 2
Maximum allowable coil
volume
820mm 3
Maximum outer radius
of bearing
120mm
Maximum height of bearing
70mm
Remnant flux density of 1.2T
bias magnets
Known parameters are
•Gap
•Inner radius of the bearing
•Outer radius of the bearing
Free parameters
•Inner radius of the coil space
•Outer radius of the coil space
•Height of the coil space
•Current density supplied
Eight pole radial magnetic bearing
Eight Pole AMB
8
Radial magnetic bearing
The component of force will be
at an angle of half of the angle
between two poles
F=
µ0 Ag ( K i ni ) 2
4( K a lg ) 2
Three pole radial magnetic bearing
cos
α
2
α
Magnetic Circuit for
three pole AMB
Three Pole AMB
Winding scheme
Coil design
• Admissible coil temperature is determined by
the choice of insulation type
• Number of turns are chosen such that it
generates maximum admissible magneto
motive force at the maximum current supplied
by the power amplifier
Coil
Permanent magnetic bearings
Permanent magnetic bearings
B
Br
Hc
Ha
BH max
BH
9
Introduction
MAGNETIC BEARINGS
CONTROL
• Control is the process of bringing a
system into desired path when it is
going away from it
• Earnshaw(1842) had shown that it is
impossible to hover a body in all six
degrees of freedom by using permanent
magnets
• But it is possible to maintain the body in
equilibrium condition by active control
Types of control systems
• Closed loop control systems
• Open loop control systems
U ( s)
G (s)
Y (s)
• The control in which the output of the system has
no effect on input is called open loop control
• Open loop control is used when the input is known
and there are no external disturbances
• An example of open loop control is washing
machine which works on time basis rather than the
cleanliness of clothes
Classification of controllers
• According to control action controllers are
classified as:
•
•
•
•
•
•
Types of control systems
Two-position or on-off controllers
Proportional controllers
Integral controllers
Proportional-integral controllers
Proportional-differential controllers
Proportional-differential-integral controllers
U ( s)
Y (s)
G( s)
x
H (s)
• If the control maintains a prescribed output and
reference input relation by comparing them and
uses their difference as controlling quantity, it is
called feedback or closed loop control
• Temperature control of a room or a furnace is an
example of closed loop system
Classification of controllers
• Two-position or on-off controllers
• The output of the controller y (t ) will be a
maximum or minimum according to the state of
error e(t ) as below:
y (t ) = y0
for e(t ) < 0
= y1
for e(t ) > 0
• y0 and y1 are minimum and maximum values of
output
10
Classification of controllers
• Proportional controllers:
Classification of controllers
• Integral controllers:
• The output of the controller y (t ) is proportional to
the magnitude of the actuating error e(t ) signal as
y (t ) = g p e(t )
• In integral control action, the value of the
controller output y(t ) is changed at a rate
proportional to the actuating error signal e(t )
• By Laplace transformation
Y (s)
= gp
E (s)
dy(t )
= g i e(t )
dt
t
y (t ) = g i ∫ e(t )dt
(or)
0
• By Laplace transformation
Y (s)
• g p is called proportional gain
=
gi
E (s) s
•
• g i is called integral gain
Classification of controllers
• Proportional-Integral (PI) controllers:
• Control action is a combination of both
proportional and integral action
y (t ) = g p e(t ) +
gp
Ti
Classification of controllers
proportional-differential (PD) controllers:
The control action is defined by
y (t ) = g p e(t ) + g pTd
t
∫ e(t )dt
0
By Laplace transformation
• By Laplace transformation
Y (s)
= g p (1 + Td s )
E (s)

Y ( s)
1 
= g p 1 +

E ( s)
T
is 

Control Design
Classification of controllers
proportional-Integral-differential (PID)
controllers:
It has the advantages of all three actions. So this is
the most common type of industrial controllers
Mathematical form of PID action is
gp
t
de(t )
y (t ) = g p e(t ) +
e(t )dt + g pTd
Ti ∫0
dt
By Laplace transformation


Y (s)
1
= g p 1 +
+ Td s 
E ( s)
T
s
i


de(t )
dt
Input
System
Output
An over all system
U(s)
G(s)
Y(s)
Y ( s ) = G ( s)U ( s )
Transfer-function representation of a system
u(t)
x(t )
x(t )
y(t)
y (t ) = Ax(t ) + Bu(t )
State-space representation of a system
11
Control Design
System
Input
Methods of design and
analysis of controllers
Output
Methods of design and analysis
An over all system
Known
Known
unknown
Studying the behaviour of a system
Transfer-function method
Unknown
Known
Studying the characteristics of a system
Unknown
Unknown
State-variable method
known
known
Transient and
steady state
Response
analysis
Root locus
analysis
Frequency
response
analysis
(Classical control)
Pole-placement
analysis
Linearquadratic
optimization
(Modern control)
Designing of a control system of required behaviour
Transfer-function
method
Classical control method
Mechanical and electro magnetic
stiffness
State-space method
Modern control method
Used for single input single output Used for multi input multi output
(SISO) systems
(MIMO) systems can be used for SISO
also
It is useful for linear and simple It is useful for nonlinear and
systems only
complex systems also.
Frequency domain method
fs
x
x0
Equilibrium
position
Rotor
mg
Mechanical spring stiffness
mechanical spring
Time domain method
Analysis consists of single higher Analysis consists of system of n first
order differential equation
order differential equations.
Steady state and transient response Pole-placement method and Linearanalysis, Root locus analysis and quadratic optimization are the main
frequency response analysis are the methods of design and analysis.
main methods of design and analysis
fm
xm
Operating
position
x0
mg
magnetic displacement stiffness
Magnetic spring
Magnetic Bearing Control
Operating
current
i0
im
Instantaneous
current
fm
x0
x0
mg
Magnetic spring
Operating
position
• Equilibrium and Operating points
• For a mechanical spring there will be an
equilibrium point where the force resisted by the
spring is equal to the force applied on the spring
• For electro magnets there will be a quantity of
current corresponding to position of the object and
force applied. At this point the gravity force and
magnetic force will be equal. A slight movement
form this point will cause indefinite movement of
the body. This point is called operating point
12
Linearization at operating point
mg
i0
x0
Linearized formula around the operating point will be
f ( x, i ) = k x x + ki i
where
x is the displacement from the
operating position
i is the deviation of current
Linearization of current
from operating current
Linearization of displacement
i = im − i0
k x is displacement stiffness
x = x0 − xm
f = ki i
im is the instantaneous current
f = kx x
xm is the instantaneous position
ki is current stiffness
f
is instantaneous force
Magnetic Bearing Control
• Linearized equation is suitable for most of the
applications of magnetic bearings
• It is not valid in three occasions
• When x = x0 the rotor touches the bearing magnet
• When there are strong currents such that magnetic
saturation of the material occurs
• When i = −i0 or very small currents there won’t be
levitation of the rotor because of very small
magnetic forces.
f = k x x + ki i
By Newton's law
Active magnetic
bearing system
f = mx
Combing above two equations we get
k
mx − k x x = kii
f
x
c
m
Rotor
spring mass damper
system
If controlling current i is zero then
mx − k x x = 0
And the response grows exponentially thus
the rotor may fall down or touch the magnet
If we supply controlling current i such that
i ( x) =
k + kx
c
x + x
ki
ki
then it becomes
mx + cx + kx = 0
And the response is imitated to a spring mass damper
system by the magnetic bearing system
k
f
x
c
m
Rotor
Response of magnetic
bearing without control
spring mass damper system
Response of magnetic
bearing with control
13
PD controller model
k +k
c
x
( x) =
x + with
x
• The model is iPD-controller
proportional
ki
ki
and differential feed back
P=
k + kx
ki
D=
r
c
ki
+
−
• In design of controller we choose the stiffness
and damping to ensure the system come to
steady state in optimum time.
• The optimal stiffness suggested is
k =systems
kx
• The range of damping ratio for better
suggested is 0.1 to 1
Controller
r
+
−
e
ic = Pe + De
+
1
edt
Ti ∫
i = ic
ic = Pe + De
Amplifier
ic
i
i = ic
ki
+
f
1/ m
x
∫
x
∫
x
kx
y
y=x
Sensor
Block diagram of PD controller with
current control
∆fload
Amplifier
ic
Controller
e
i
ki
+
f
1/ m
x
kx
∫
x
∫
x
Control of rotors by using
magnetic bearings
y
y=x
Sensor
Block diagram of PID controller with
current control
Topics to be covered
Differences between mechanical and
magnetic bearing models
Mechanical bearing model
• Rigid rotor model
• Flexible rotor model
Magnetic bearing model
• Stiffness is very high
• Stiffness is very low thus
thus the vibration of the
the rotor can rotate freely
rotor will be transmitted
about the principal axes of
to foundation
inertia which results in a
vibration isolation system.
• Damping is directly
observed due to
hydrodynamic effects
• As the rotor is free in the
air there is no coulomb
damping acting on the
system. The control law
will have damping term.
14
Rigid rotor model
Rigid rotor model
If the variable vector is chosen as
x = [ x1
x2
x3
T
x4 ] = [ β
T
y]
x −α
Angular velocity vector can be expressed as
ω x 0   α cos Ωt + β sin Ωt 

  
ω = ω y 0  =  −α sin Ωt + β cos Ωt 

ω z 0  
Ω


y
y
y
α
Rotor mechanical bearing system
O
Ω
Angular velocity of shaft
α
β
Infinitesimal rotation about x axis
O
z'
z
z
z
y
y'
z
β
z'
α =
Infinitesimal rotation about y axis
dα
dt
β =
x
dβ
dt
x
x
x
x'
Motion about x- axis Motion about y- axis
Rigid rotor model
Rigid rotor model
Kinetic energy is expressed as
Equations (1) can be expressed in matrix form by rearranging
1
1
T = m ( x 2 + y 2 + z 2 ) + ( J x 0ωx20 + J y 0ω y20 + J z 0ω z20 )
2
2
Equations of equilibrium can be obtained as by using Lagrange’s principle
M
x + ( G + C ) x = F
M is the inertia matrix (M = M T )
G is the gyroscopic matrix(G = -GT )
d  ∂T  ∂T
= Fi

+
dt  ∂xi  ∂xi
C is the damping matrix (C = C T )
F
th
Fi is the generalized force corresponding to i variable
can be expressed as
F = −(K + N )x
K is conservative force matrix (K = K T )
N is non-conservative force matrix(N = - N T )
Rigid rotor model
• Conservative forces include
Rigid rotor model
– forces due to stiffness
• Non-conservative or circulatory forces
include
–
–
–
–
–
Internal or structural damping
Steam or gas whirl in turbines
Seal effects
Process forces such as in grinding
Unbalance, etc
• Damping include
• From Eq. (2) and (3) we get
M
x + ( G + C ) x + ( K + N ) x = 0
• If the non-conservative and gyroscopic
forces neglected, we have
M
x + Cx + Kx = 0
– Coulomb damping due to hydrodynamic effects
15
Natural modes
Natural modes
• The solution of the equations (5) gives
four modes, for there are four degrees
of freedom considered
Translation mode
Forward whirl
Backward whirl
Rotation mode
Forward nutation
Magnetic bearing model
Backward nutation
Weight considerations
• In a magnetic bearing if we neglect the
conservative, non-conservative, and
damping effects, we will have
mg
M
x + Gx = F
• For small rotations gyroscopic effects
can be neglected and the equations in x
and y directions can be decoupled
M
x=F
α
f g = mg = ki i0
Imbalance considerations
fg =
mg
= ki i0
cos α
Magnetic bearing model
It can be written as
Ω
mx = f k + f c − f g + fω
where
e
Ω2 cos ( Ωt + θ )
fω = me
m is the imbalance mass
e is the eccentricity of
imbalance mass
f k = kx x
f c = ki i
f g = mg = ki i0
Ω2 cos ( Ωt + θ )
fω = me
θ is the angular position
of imbalance mass
16
Magnetic bearing model
Rigid rotor with magnetic bearing
• Three steps involved:
– Formulation with respect to centre of gravity
– Transformation with respect to the bearing
coordinates
– Transformation with respect to the sensor
coordinates
• It will be
Ω 2 cos ( Ωt + θ )
mx = k x x + ki ( i − i0 ) + me
• i at any instant will be
i = i0 +
Centre of gravity
Bearing
x
Ω 2 cos ( Ωt + θ )
mx − kx x − me
ki
O
z
y
Sensor
Why with respect to sensor
coordinates
With respect to centre of gravity
• In slow role x and y directions can be decoupled
p
• Sensors cannot be
arranged directly in the
magnetic actuator.
• This requires certain
gap between the
magnet and the sensor.
• The displacements with
respect to sensor
coordinates will be
transformed to bearing
coordinates
With respect to bearing coordinates
• Forces are transformed as
p
A
B
x
f
x
O
z
y
f ax
a
c
f = f ax + f bx
p = af ax + bf bx
β
1
=
 a
y
O
z
β
y
f ax
fbx
a
b
c
d
f

 p 
With respect to bearing coordinates
• Displacement vector can be
transformed as
x = TB x B
x
x= 
p
B
x
 β 
f
 f ax 
1

 ,f = 
b  B  f bx 
In matrix form as
M
x=f
where
m 0 
x
M =
,x =  
0 I y 
 β 

f =
f = Tf fB
Tf
d
f
x
mx = f
I β = p
B
x
A
fbx
b
A
O
x
z
y
xa
a
c
xb
β
TB =
1 b

b − a  −1
−a 

1 
x 
xB =  a 
x
 b 
b
d
17
State feed back
With respect to sensor coordinates
xS
p
A
f
x
z
y
xc fax
x s = As x s + Bsu
x 
=  c
 xd 
TS
β
fbx xd
a
x
1 c 

1 d 
x= 
β
=
 
• We do not know the velocity components
directly from sensors. So a state observer is
required to find the velocities
x S = TS x = TS TB x B
b
c
xS 
=T x
S  s b
x
0 
TS TB
Ts = 
TS TB 
 0
xs = 
d
xs
State feed back
x s = As x s + Bsu
x S = Cx s
−
u
+
Bs
x s
d
dt
F
As − Bs F decides the closed loop
dynamics of the system
Model at high speeds
 fx 
 p 
f = y 
 fy 


 − px 
C
As
x s = ( As − Bs F ) x s
0
0 0 
m 0

0 Ix 
xS
xs
∫
u = − Fx s
0
0 0
0 0
G=
0 0

0 −1
xB
0 0
0 1

0 0

0 0
 xa 
 
x
=  b 
y
 a
 yb 
= Cx S
x s is the full state vector
x S is the vector from the sensor
Model at high speeds
• The whole closed loop system can be shown as
block diagram
m 0
0 I
y
M =
0 0

0 0
As = Ts AbTs−1
Bs = Bb
• The control vector is found by using control
law
u = − Fx s
B
O
x
State space form with respect to sensor coordinates
• At high speeds the gyroscopic effects cannot
be neglected, thus the model becomes
M
x + Gx = F
• The displacements in x and y directions no
longer decoupled, so four forces and four
displacements should be taken into
consideration simultaneously.
• The same procedure is to be followed as for
the slow rotation
Conclusions on rigid rotor model
– There is an optimal design for each speed
– The optimal design at higher speed may not
be stable at lower speeds, for the gyroscopic
effects are reduced.
– The optimal design at zero speed may not be the
optimal at higher speeds
– The gyroscopic effects will not destabilize the system
which is stable at lower speeds.
– Further more the design at lower speeds is decoupled
and easier to design. Decentralized designs for lower
speeds can be implemented
18
Conclusions on rigid rotor model
– Thus for stability considerations and other
advantages systems are designed for
lower speeds and with decentralization
Flexible rotor model
• Rigid rotor can be
defined by two
points
uxa = − Fxa xa
uxb = − Fxb xb
• Flexible rotor has
infinite degrees of
freedom. One
cannot define
uniquely by some of
the points
u ya = − Fya xa
Decentralized control mode scheme
u yb = − Fyb xb
Flexible rotor model
Flexible rotor model
• By substituting we get
z
• Equation motion of
an Euler-Bernoulli
beam is given by
EI
L
∂4 y
∂2 y
+m 2 =0
∂z 4
∂t
 d 4Y ( z ) 
 d 2 q (t ) 




4
dt 2 
EI  dz 
=−
= ω2
m Y ( z)
q (t )
• By rewriting we get
• The variable
separable form is
d 4Y ( z )
− β 4Y ( z ) = 0,
dz 4
y ( z , t ) = Y ( z )q (t )
Flexible rotor model
z
L
β=
ω2
EI / m
d 2 q (t )
+ ω 2 q (t ) = 0
dt 2
ω2
• The mode shapes or
modal functions
depend on the end
conditions
• By applying initial
conditions and solving
we get the natural
frequencies ω
ω3
ω0
z
ω1
z
• By substituting the
Eigen values in (29) we
get the Eigen functions
or model functions Y ( z )
Flexible rotor modes
Rigid rotor modes
19
Actuator sensor location
Actuator sensor location
• We can conclude that the sensor can be
set at a place where we can get
information from each mode under
consideration
• Sensor should not be set at nodes
• Sensor and actuator should not lie on
opposite sides of a node
actuator
sensor
Modal reduction (mathematical representation)
Modal reduction
• While designing a flexible rotor system, we
can not consider all the modes of the system
for they are infinite
• Thus we consider first n number of modes
corresponding to first n natural frequencies
and neglect the remaining modes
• If we study the effect of the reduced modes
we can find the number of modes which we
can consider without destabilizing the system
• Mathematical model of the
x = Ax + Bu
– full system
y = Cx
– Divided system
y = [ CM
– Control spillover (By the
input)
– Interconnection spillover
(By the parameters of the
system)
– Observation spillover (on
the estimated output)
AMR   x M   BM 
+
u
AR   x R   BR 
x 
CR ]  M 
 xR 
x M = AM x M + BM u
– Reduced system
y = CM x
Modal reduction
Modal reduction
• The reduced modes
give three kinds of
effects on the system
called spillovers
 x M   AM
 x  =  A
 R   RM
Modeled modes
AM
 x M   AM
 x  =  A
 R   RM
y = [ CM
AMR   x M   BM 
+
u
AR   x R   BR 
u
BM
x 
CR ]  M 
 xR 
BR
Input
System
Output
Control
spillover
xM
+
∫
ARM
Interconnection
spillover
+
∫
xR
AR
CM
+
y
AMR
CR
Observation
spillover
Unmodeled modes
Control
spillover
Interconnection Observation
spillover
spillover
Block diagram of effect of model reduction
20
Conclusion on flexible rotor control
Conclusions
• Modal reduction is studied to consider
the number modes to be taken into
consideration for having stable control
• Mechanical design is studied for finding
the sensor actuator locations
• Magnetic bearings advantages and
applications have been discussed
• Electromagnetism and Control system
technologies have been introduced
• Design of thrust and radial magnetic
bearings have been studied
• Control of a rotor by rigid rotor and
flexible rotor models have been studied
Further References
Schweitzer, G., Bleuler, H. and Traxler, A., 2003, “Active
Magnetic Bearings: Basics, Properties and Applications of Active
Magnetic Bearings”, Authors Working Group, www.mcgs.ch
reprint.
Chiba, A., Fukao, T., Ichikawa, O., Oshima, M., Takemoto,
M. and Dorrell, D.G., 2005, “Magnetic Bearings & Bearingless
Drives”, Newnes, Elsevier.
Anton, V.L. , 2000, “Analysis and initial synthesis of a
novel linear actuator with active magnetic suspension”, 0-78038486-5/04/$20.00 © 2004 IEEE
Maslen, E., 2000, “Magnetic Bearings”, University of
Chee, K.L., 1999, “A Piezo-on-Slider Type Linear
Ultrasonic Motor for theApplication of Positioning Stages”,
Proceedingsof the 1999IEEE/ASME.
Groom N.J. and Bloodgood, V.D. Jr., 2000, “A
Comparison of Analytical and Experimental Data for a Magnetic
Actuator”, NASA-2000-tm210328.
Shyh-Leh, C., 2002, “Optimal Design of a Three-Pole
Active Magnetic Bearing”, IEEE TRANSACTIONS ON MAGNETICS,
VOL. 38, NO. 5.
Virginia.
Bloodgood, V.D. Jr., Groom, N.J. and Britcher, C.P., 2000,
“Further development of an optimal design approach applied to
axial magnetic bearings”, NASA-2000-7ismb-vdb.
21