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Design Principles of
Integrated Magnetic
Bearings
A dissertation submitted to the
Swiss Federal Institute
of Technology
Zurich
for the degree of
Doctor
of Technical
Sciences
presented by
Felix Betschon
Dipl. El.-Ing.
born
citizen
May
7th,
1970
ofLaufenburg, Switzerland
accepted
Prof
Prof
on
ETH
on
the recommendation
Dr. J.
Dr. C.
Hugel,
Knospe,
of
examiner
co-examiner
Zurich 2000
Preface
This work touches many different
processing electronics,
software,
etc.
power electronics,
It has been
and to compose them to
Prof. Dr. J.
Hugel,
a
a
areas
such
mechanics,
great pleasure for
me
to
actuators,
as
control
profit
signal
engineering,
in all these fields
terminated system. T wish to thank my
head of the Electronic
tory of Swiss Federal Institute of
pation
subject
Engineering
Technology,
and
for this very
supervisor
Design Labora¬
interesting
occu¬
and his active support.
During
my work Prof. Dr. C.
Knospe
made it
possible for me to visit the
Center for Magnetic Bearings of the University of Virginia in summer 1998.
During this period he introduced me to adaptive vibration compensation. I
wish to thank him for this and for his willingness to volunteer as second
advisor.
This work
supported by
the
companies Lust Antriebstechnik GmbH
and Sulzer Electronics AG. I am deeply indebted to Dr. R. Schob, manager
of the magnetic bearings group (Levitronix). He has supported me in all
practical and theoretical questions of this work.
was
I wish to thank Dr. J. Schmied of Delta
of rotor
Imlach,
dynamics.
JS, who advised
a
great part
questions
to the real¬
project.
I wish to mention my brother, Lukas Betschon. As mechanical
engineer
important contact for the construction of the rotor. He spent a
reading this work and he gave me many inputs for its readability.
was an
time
all
The students G. Foiera, M. Knaus, M. Holzner, M.
P. Jucker and D. DeMesmaeker contributed
ization of this
me on
It would not have been
to
other
I received many
employees
door and
during
of the
possible
laboratory.
the coffee breaks.
accomplish
he
lot of
this work without all the
tips
from the office next
Table
of Contents
Abstract
5
Kurzfassung
7
Symbols
9
and Abbreviations
1.1 The components of
a
17
simple magnetic bearing system
magnetically suspended
Position sensing
1.1.1 The
1.1.2
17
Magnetic Suspension
1. Introduction to Active
17
ball
18
21
1.1.3 Control electronics
amplifier
1.2 Extending the principle of the magnetically suspended
plete magnetic bearing
1.2.1 A single bearing axis
1.2.2 The complete support of a shaft
22
1.1.4 Power
2.
Motivation, Objectives and Structure of the Thesis
3.
Approaches
to
Reduce Costs
ball to
a com¬
25
26
28
31
33
of Magnetic Bearing Systems
3.1 Passive
33
3.1.1
magnetic suspension
Permanentmagnet bearings
34
3.1.2
Magnetic
3.2
Reducing
reluctance
the system
complexity
of active
3.2.1 Active magnetic bearing with
3.2.2
Magnetic bearing
3.2.3 Sensorless
34
bearings
with
a
35
magnetic bearings
minimal number of actuators 35
asymmetric premagnetization
.....37
magnetic bearings
39
3.3 Evaluation
3.3.1 Potentialities for cost reduction
3.3.2 Influence of current,
the cost of
39
voltage, power consumption and control
bearing systems
1
39
on
41
Table
of Contents
4. Elimination
4.1
of Bias
43
Current
Permanent-magnet-biased bearings
4.1.1 Homopolar configuration
4.1.2 Heteropolar configuration
4.1.3 Permanent-magnet-biased bearings operated by three-phase
43
44
45
drives
46
4.2 Losses
5.
43
4.2.1 Iron losses in the rotor
46
losses in stator
49
4.2.2
Copper
4.2.3
Comparison
50
of pole arrangements
Design of Permanent-Magnet-Biased Bearings
51
5.1 Fundamentals
51
5.1.1 The
Analogy between electric circuit
layout of the radial bearing
5.1.2
5.2 The
51
magnetic field generation
5.2.1 Construction and mode of
5.2.2 Definitions and
5.2.3
Winding layout
5.2.4
Layout
5.2.5
Stray fluxes
design
and
magnetic
circuit
56
56
operation
57
process
58
61
of permanent magnet circuit
62
63
5.2.6 Calculation of bearing parameters
66
5.3 The combined radial and axial
5.3.1 Construction and mode
5.3.2 Definitions and
design
5.3.3 Calculations of the
6. Elimination
bearing
of operation
66
68
process
69
bearing parameters
of Synchronous
73
Currents
73
6.1 Fundamentals
6.2
6.3
Approaches for cancelling speed-synchronous
6.2.1 Tracking notch filter
6.2.2 Adaptive FIR filter
Adaptive vibration control
6.3.1 Mode of operation and definitions
6.3.2 Theory of adaptive vibration control
6.3.3
Reducing
54
the number of
gain
matrices
6.3.4 Determination of transfer matrix T
2
currents
75
75
76
78
79
80
82
85
Table
87
7. Realization and Results
7.1
88
Magnetomechanical system
7.1.1
of Contents
90
Bearing system
7.1.2 Motor
92
7.1.3 Rotor
93
96
7.2 First control electronics
7.2.1 Structure
96
7.2.2 Software
99
100
7.2.3 First results
7.3 Miniaturized control electronics
103
7.3.1 The TMS320F240
103
7.3.2 Structure of the miniaturized control electronics
104
7.3.3 Software concept
108
7.3.4 Measurements and results with the miniaturized control electron¬
111
ics
8. Conclusion and Outlook
115
Appendix: Modelling
117
A. 1
117
Single-axis magnetic bearing
119
A.2 Mechanical model of the rotor
A.2.1 The
rigid
rotor
A.2.2 The
rigid
rotor
119
with
gyroscopic coupling
121
123
A.2.3 The flexible rotor
References
127
Curriculum vitae
133
3
|
"%. .f
fet
S*
I
\ S »Sa
-œ,
fei
11 ;<-! tl
4
U?
Abstract
Magnetic bearings operate contactlessly
and
wear.
and
are
They
are
operational
suited for
largely
in
immune to
vacuum.
and
are
therefore free of lubricant
heat, cold and aggressive substances
Because of their low
bearing
losses
they
are
with
high rotation speeds. The forces act through an
air gap, which allows magnetic suspension through hermetic encapsula¬
tions. Ferromagnetic bodies can be placed in any position within the
mechanical limitations. Vibrations of the housing or the rotor can be
actively suppressed. Applications for magnetic bearings are for example
pumps,
applications
turbocompressors, milling spindles
and turbomolecular pumps.
Because of the
complex system structure the prices of magnetic bearings are
high compared with conventional bearings (ball, sliding bearings etc.).
Therefore their
application
is limited. One
possibility to reduce costs is the
integration of the control electronics. It is easy to integrate the signal pro¬
cessing electronics with today's integration technology. However the inte¬
gration of the power stages is strongly dependent on the bearing currents.
For this they must be minimized.
This work presents two
bearing types,
and thus
small control currents. Moreover
known
consume
only
which
are
permanent-magnet-biased
a
control
method,
the
Adaptive Vibration Control, is applied which eliminates the
speed-synchronous parts of the control currents. It was developed by Kno¬
spe and Hope [HoKn/94] and allows a force-free suspension of the rotor.
as
Permanent-magnet-biased bearings
were
the main elements in the
and
adaptive vibration compensation
development of a magnetically suspended
high-speed spindle. With the first electronics developed a speed of
90'OOOrpm was reached with very small bearing currents. The knowledge
gained from this was incorporated in the development of a miniaturized and
inexpensive control electronics. A complete digital control allowed reduc¬
ing the number of devices, and the low bearing currents led to a closepacked power electronics. The electronics volume was reduced by 70%.
5
&*.&
;;sïïssa,
a
6
Si'
Auf
/
Kurzfassung
Magnetlager
vollständig berührungslos und sind damit Schmiermit¬
tel- und verschleissfrei. Sie sind weitgehend unempfindlich gegen Hitze,
Kälte, aggressive Medien und sind auch im Vakuum betriebsfähig. Auf¬
grund ihrer geringen Lagerverluste eignen sie sich besonders für hohe Dreh¬
zahlen. Die Lagerkräfte wirken über einen Luftspalt und auch durch
hermetische Kapselungen hindurch. Mittels aktiver Magnetlager können
ferromagnetische Körper innerhalb der mechanischen Begrenzungen prak¬
tisch
arbeiten
beliebig positioniert
werden.
Vibrationen des
Gehäuses
oder des
Rotors lassen sich aktiv unterdrücken.
rem
Anwendungsgebiete sind unter ande¬
Pumpen, Turbokompressoren, Frässpindeln und Turbomolekularpum¬
pen.
Aktive
Magnetlager
sind
aufgrund ihrer komplexen Systemstruktur vergli¬
chen mit konventionellen Lagern (Kugel-,Gleitlager, etc) relativ teuer. Des¬
halb ist ihre Verbreitung vorläufig eingeschränkt. Eine Möglichkeit zur
Kostenreduktion bietet die Integration der Magnetlagerelektronik. Für die
Signalverarbeitung ist das bei den heutigen technischen Möglichkeiten kein
Problem. Hingegen sind die Integrationsmöglichkeiten der Leistungsstufe
abhängig
stark
von
den
Lagerströmen.
Dazu müssen die Ströme minimiert
werden.
In dieser Arbeit werden zwei
Lagertypen behandelt,
tisch vorgespannt sind und daher
Zudem wird ein Verfahren
nur
Control" und
ermöglicht
Dadurch werden die
noch kleine Steuerströme
vorgestellt,
941 entwickelt wurde. Es ist bekannt
eine
die permanentmagne¬
das
unter
Lagerung
von
benötigen.
Knospe und Hope [HoKn/
dem Namen
"Adaptive
des Rotors frei
drehzahlsynchronen
von
Vibration
Rüttelkräften.
Anteile des Steuerstromes elimi¬
niert.
Die
Prinzipien
der
permanentmagnetisch erregten Magnetlager und der
adaptiven Vibrationskompensation wurden zur Entwicklung einer magnet¬
gelagerten Hochgeschwindigkeitsspindel verwendet. Mittels einer ersten
Elektronik wurde eine Drehzahl
Lagerströme
äusserst
gering
von
90'000U/Min
erreicht, wobei die
blieben. Dank dieser Erkenntnisse wurde eine
neue, miniaturisierte und damit
kostengünstige
7
Elektronik entwickelt. Eine
Kurzfassung
vollständig digitale Regelung ermöglichte eine Reduktion der Anzahl Bau¬
teile und durch die geringen Lagerströme konnte die Packdichte der Lei¬
stungselektronik erhöht werden. Dadurch verringerte sich das Volumen der
Magnetlagerelektronik um 70%.
8
Symbols
and Abbreviations
Symbols
Vectors and matrices
different
nent
properties,
written in bold letters. Subscribed indexes denote
are
such
for radial,
"r"
as
"§ri"
magnet. The subscripts
"a"
"gr2"
and
for axial
or
"PM"
for perma¬
stand for the active air gap and
the air gap between rotor and back iron. The
superscript
"
denotes trans¬
posed.
A
Gain matrix
A(£l)
Speed-dependent gain
Aq
Speed-independent part
AjQ,
Speed-dependent part ofA(Q)
Acon
Cross-sectional
ACur
Copper
Ag
Pole face
^hb ^5a
Active
Agr2
Pole face
ctp
ß;
area
area
matrix
of A(Q)
of coil
(radial bearing)
area
pole
face
area
(radial, axial bearing)
of back iron
area
(radial bearing)
Fourier coefficients
Afc
Optimal gain
AN
Gain matrix of notch filter
Aopt
Optimal gain
ApM
Front face of permanent magnet
Awp Awa
Winding
B0, Bc
Bias, control induction
B()8rl> B()8a
Nominal bias induction in active air gap
bpt>2
Damping
Bc&rJ> Bc8a
Nominal control induction in active air gap
Bcoil
Coil induction
area
matrix for
speed Q^
matrix
ring
(radial, axial bearing)
(radial, axial)
of shaft
9
(radial, axial)
Symbols
and Abbreviations
BS
Air-gap
B
"max
Maximum
BPM
Permanent magnet induction
Br
Remanent flux
®s
Saturation induction
Ô
Air gap
d
Lamination
Am
Unbalance
"•outer
Outer diameter of
5P5a
Nominal active air gap
8r7, 5,,
Nominal active air gap, air gap between rotor and back iron
flux
density
induction
magnetic
density
length
thickness, damping
modelling
mass
bearing
(combined bearing)
(radial bearing)
dr
Vector between ends of flexible and rotation axis
1,2
^rotor
Rotor diameter
dxjo, dyj2
Components
8
Inverse of
E{...}
Expectation
f
Winding
of
dr12
adaptation
convergence rate of notch filter
value
factor
Magnetic
flux
®08rl> %8a
Nominal bias flux in active air gap
f
Force
Fi> Fa
Magnetic bearing
F0>> F0a
Magnetic disturbing
force
(radial,
axial
bearing)
(radial, axial bearing)
force
(radial,
axial
bearing)
Bias, control flux
®c8rl> ®c8a
Control flux in active air gap
fd
Disturbing
F
1
•*
x
m4->
m+>
fr
F
1
F,res
fs
Magnetic
m
*
i
m-
force
(small signal)
force
Amplified,
reduced
magnetic force
Remagnetization frequency
Resulting
(radial,
force
Sampling frequency
10
axial
bearing)
Symbols
Fsb Fs2
Bearing
acting
on
shaft
of force
on
shaft at
forces
Fsx,yb Fsx,v2 Components
and Abbreviations
bearing 1, bearing
fu
Update frequency
Fx, Fy
Components
y
Rotational
G
Gravitational force
Gq
Disturbance transfer function
Gj
Transfer function of
a
Gp
Transfer function of
position
Hc
Coercive force
H§
Magnetic
field
strength
in air gap
H§ri, H§r2
Magnetic
field
strength
in air gaps
Hpe> HpM
Magnetic
field
strength in iron, permanent magnet
/
Identity matrix
id, ic
Bias, control
lc
Effective control current
ic+, ic.
Amplified,
hmax
Maximum control current
iCP ica
Control current
id
Desired current
ie
Eddy
ITR
Current
'x/,W,x2,y2,z
Control currents
j
Complex
(p
Rotational
J
Performance function
Jg, J]
Moment of inertia
Lg
Angular
k
Stiffness of
kj, kj
Stiffness of shaft
2
of force
angle (z- axis),
inverse of convergence rate
spring-damper system
controller
(radial bearing)
current
reduced
bearing
current
(radial, axial bearing)
current
through transistor (1DS for FET's, IEC for BJT's)
unit J^i
angle (about
(radial, axial direction)
momentum
a
y- axis)
spring
11
Symbols
and Abbreviations
Kpp> %dp
Proportional,
kv ki
Force-displacement,
Kpc> Kjc
Proportional, integral gain
fyp kia
Force-current factor
Kkp Km
Force-displacement factor (radial bearing)
L
Inductance
I1J2
Distance between centre of
Ljj,Lj2,Lj3
Winding
inductance
^2bF22'F23
Winding
inductance
hearing
Bearing length
IpM
Length
IpMmin
Minimum
lwr
Mean
Lwr
Winding
(I
Adaptation step
m
Mass
\ig,\ipe,\ipM
Permeability
u,r
Relative
TV
Number of turns
na
Number of actuators
np
Number of
performance signals
nc
Number of
compensation signals
Nr Na
Number of turns
ns
Number of
Pg
Center of
P/, P2
Ends of flexible shaft
0
Magnetomotive
force
Or
Magnetomotive
force
r[ij
Discrete reference
Pc«
Specific
differential
gain
length
position
force-current
controller
factor,
of current controller
(radial, axial bearing)
gravity
of permanent magnet
length
of
and
bearing 1,2
ring
of permanent magnet
ring
per turn
inductance
size of LMS
algorithm
of vacuum, iron, permanent magnet
permeability
(radial, axial bearing)
position
sensors
gravity
(radial bearing)
signal
resistance of copper
12
Symbols and Abbreviations
R8rb R8a
Magnetic
resistance of active air gap
R8rl+> R8rl-
Magnetic
resistance of
j?
j?
t^w 1XWP
j?
,xwz
Winding
enlarged,
(radial bearing)
reduced air gap
resistance
s
Complex angular frequency
s
Maximum allowed current
S j, S2
Ends of
rigid
density
shaft
Magnetic leakage factor
t
Time
T
Transfer matrix, torque
J>
lR
Tp\VM
nri
*
nrr
x?
y
Real parameter matrix of notch filter
Cycle
time of PWM
Components
synchronous
Vector of
u
of torque
Fourier coefficients of feedforward
signals
u(t)
Vector of feedforward
u[k]
Discrete
UA,
uA
signals
in time domain
compensation signal
Amplifier output voltage,
small
signal
Udc
Intermediate circuit
UTr
Voltage
over
v* vPM
Air gap
volume, volume of permanent magnet ring
VPMniin
Minimum volume of permanent magnet
Q
Angular frequency
w[kj
Vector of FIR filter coefficients
Wj, w2
Weight
Qa]I
Speed-dependent part oïA(Q)
x, z
Radial, axial deflection
X
Vector of
of
voltage
transistor
adaptive
(Ups
for FET's,
of rotational
UE(^
for
ring
speed
FIR filter
synchronous
Fourier coefficients of
signals
%
Rotational
x(t)
Vector of
x[k]
Discrete
BJT's)
angle (around
x-
axis)
performance signals
performance signal
13
in time domain
performance
Symbols
and Abbreviations
Vector of
x0
synchronous Fourier coefficients
of uncontrolled
vibration
x
1,2^1,2
Deflection of shaft in x-direction in
bearing 1,2
position
Xd
Desired
xjt)
Vector of unfiltered
position signal
Abbreviations
digital
A-D
Analog
ASIC
Application specific integrated
AVC
Adaptive
BJT
Bipolar junction
CC
Current control
CMOS
Complementary
Com
Communication
Dj
Diode
234
to
circuit
vibration control
transistor
metal oxide semiconductor
analog
D-A
Digital
DMOS
Double-diffused metal oxide semiconductor
DSP
Digital signal
EEPROM
Electrically
FEM
Finite element model
FET
Field effect transistor
FIR
Finite
GEVP
Generalized
Inter
Interpolation
LMS
Least
MCDS
Magnetic
MIMO
Multiple input multiple output
PC
Position control
PCB
Printed circuit board
PID
Proportional- integral-
PWM
Pulse width modulation
R.M.S.
Root
S
Switches
1,2,3,4
to
processor
erasable and
impulse
mean
response
eigenvalue problem
square
control
mean
programmable
development
studio
derivative
square
14
read
only
memory
Symbols
hold
S/H
Sample
SMD
Surface mounted device
SRAM
Static random
T
Time
TTL
Transistor- transistor
access
memory
delay, torque
logic
15
and Abbreviations
ft
16
.-»-x
S
#
r
*
(,
'"
-
\
fi
V
„
/
'x
I /•
k
5>
w
f,-1
**% Ufa
Ci F
I
I. Introduction to Active
Magnetic Suspen¬
sion
The purpose of this
provide an introduction to the structure of a
magnetic bearing system. The magnetically suspended ball represents the
simplest active magnetic bearing system. Unfortunately this system works
chapter
is to
just by virtue of an external force field (e.g. gravity). However all elements
of a complete bearing can be found in the magnetically suspended ball too.
Therefore it is presented and its components are discussed in this chapter.
Afterwards the principle of the magnetically suspended ball will be
extended so that a rotating shaft can be supported independently of an exter¬
nal force field in five degrees of freedom.
This
chapter
nology.
enced
does not enter
It is intended to
secondary
deeply into the theory of magnetic bearing tech¬
give an overview. The later chapters and the refer¬
literature will
give
more
detailed
explanations
on
this
subject.
1.1 The components
1.1.1 The
The
magnetically suspended
magnetically suspended
ment
ball,
of
an
ball
ball
(Fig. 1.1) represents the simplest embodi¬
an active magnetic bearing system. It consists of a ferromagnetic
electromagnet, control electronics, amplifier and a displacement
measurement
The
of a simple magnetic bearing system
with
a sensor
and
an
evaluation unit.
the
position of the body. The control electronics then
calculates the right current to suspend the ball. This current is set by the
amplifier. The resulting force is within limits proportional to the square of
the current and inversely proportional to the square of the position (see Eqn.
sensor measures
1.1).
17
Chapter
1: Introduction
Active
to
Magnetic Suspension
.2
F
(LI)
~~
2
m
X
forces
Only attracting
or
in
vacuum.
can
Therefore
necessary to obtain
The
which
measuring
a
ferromagnetic
field,
in this
as
circuit within air
example gravity,
is
Magnetically suspended
ball
sensing
or on
methods
Inductive
sensors
Inductive
sensors
through
in
external force
1.1:
of the different
use
through
an
generated
stability.
Figure
1.1.2 Position
be
which
are
families
sensor
they
are
presented
consist of
a
netically
conducting1
depends
on
at the
following
coil with
by
[Gemp/97].
to measure
in the
it. The coil is surrounded
depends strongly
a
an
ferrite
the distance to the rotor. This
core.
18
can
The different
current
passing
The rotor must be mag¬
The
be
the media
sections.
alternating
measuring position.
on
impedance of the coil
evaluated electronically.
The
frequency
chosen low
4)
be
can
often
alternating
of the
enough
so
below
(typically
100kHz)
eddy-current effects in the magnetic
reduce the eddy-current effect in the
laminated at the
are
current
that
To
neglected.
Magnetic Suspension
I: Introduction to Active
Chapter
sensor
must be
circuit
(Sect.
rotors, these
positions.
.
Ferrite
"-\
-Coil
<^
t
ir
v
X
J
1.2: Inductive
Figure
-Rotor
sensor
Eddy-current sensors
Eddy-current
tive
sensors.
sensors are
The two
essentially
sensor
choice of the rotor material.
100kHz and 2MHz. The
electrical
The
eddy
Typically
rotor
the excitation
material should be
sensor
Capacitive
sensor.
eddy-current
Because of the
is
sensors
types have about the
design
this arrangement is
1.
is driven
by
between the
a
»
7
measure
higher
on
aluminium,
for the gap width
excitation
frequency
than that of inductive
immunity
of
a
capacitive
to
the
sensors.
disturbance.
an
alternating
two
,
materials with this chaiactenstic
The
amplitude of the resulting
is proportional to the distance between
sensor is very high (0.2\im with a mea¬
current.
electrodes
Capacitive
equivalent circuit of
capacitors (Fig. 1.3 right). The
sensor.
series circuit of two
range of 0.5mm).
\ir
a
larger
same
them. The resolution of this type of
suring
frequency lies between
nonmagnetic with high
sensors
1.3 shows the
voltage
the induc¬
currents induced in the aluminium extract energy from the electric
bandwidth of
sensor
as
satisfies these conditions.
between rotor and
Fig.
way
the measurement is done
excitation circuit. This energy represents
Both
same
types differ in the excitation frequency and the
conductivity. Usually
ideally
which
built up in the
The
sensors are
arc
dtcrmed
19
sensitive to
"fciromagnetic"
pollution,
which
Chapter
1: Introduction
influences the
to
Active
permittivity
Magnetic Suspension
in the air gap.
Electrodes
(a)
(a)
C
x
Rotor
c
(b)
Figure
Magnetic
In
a
1.3:
Capacitive
sensor
sensors
ferromagnetic
circuit with
an
air gap,
a
constant current
generates
a
magnetic field. The magnetic flux through the front faces of the air gap
depends on the length of the air gap. It could be measured by a Hall probe.
Magnetic sensors are by definition sensitive to external magnetic fields.
Optical sensors
The
principle
of
a
simple optical
sensor
is shown in
Fig.
1.4.
Rotor
Transmitter
Detector
Figure
A transmitter sends
a
and rotor
in such
are
placed
light
1.4:
Optical
beam towards
a
beam. Therefore the received
a
sensor
detector. Transmitter, detector
way that the rotor
light quantity
is
partially interrupts
optical techniques
All
optical
Therefore
are
based
methods of
they
are
on
just
light
for the
position of
position sensing. Other
a measure
the rotor. Reflections interfere with this method of
the
these reflections.
position sensing
unsuited for many
are
sensitive to contamination.
applications.
20
Chapter
1: Introduction to Active
Magnetic Suspension
1.1.3 Control electronics
The heart of
is often built up
Advantages
-
-
of
is the control electronics. It
digitally.
digital technology
Flexibility
Complex
magnetic bearing system
active
an
in the
are:
development, belated changes
controller structures,
even
can
be made
time variant systems
simply
are
realiz¬
able
-
-
-
-
All
quantities
Calibration
Fast
are
of the
accessible for
monitoring
or
other processes
control parameter
adaptation of the control software
to
new
systems
Cheap reproduction of the software
structure of a
Fig. 1.5 presents the rough
digital
control system. A
quantity,
A-D
Converter
Figure
e.g. the
position,
rithm, which
verted into
is
1.5:
sampled.
runs on
an
analog
control system
Components of a digital
The data
serves
as
input
of the control
the processor. Then the calculated desired value is
form.
According
to
the
application
algo¬
con¬
the data width lies
between 10 and 64 bits.
Digital Signal
are
capable
cessor
rithms.
of
clock
Processors
calculating
cycle
More
functions. For
and
(DSP)
an
often used for such control tasks.
are
addition and
and therefore allow fast
more
example
DSPs
there exist
This allows the
design
multiplication
processing
available
are
already
converters, non-volatile memory and
integrated.
a
a
with
in
21
single
of the control
additional
pro¬
algo¬
controller
ICs where the DSP core, A-D
control unit for power electronics
of control electronics with
number of components.
a
They
a
are
minimized
Chapter
1: Introduction to Active
1.1.4 Power
Magnetic Suspension
amplifier
amplifiers in a magnetic bearing system serve to generate the control
current ic which corresponds to the desired signal id (usually voltages). Like
Power
signal processing electronics
the
two
can
be divided into
families:
-
-
Analog power amplifiers
Switched power
Linear
Fig.
the power electronics
amplifiers
amplifier
1.6 shows the structure of
analog amplifier.
an
Central element of the
Transistor
hi
Current
+
M
*»
controller
/,,
Current
sensor
\
Bearing
coil
Figure
circuit is
by
a
a
1.6: Linear
transistor. The output
closed-loop system
be considered
to
amplifier
voltage UTr
of the transistor is controlled
maintain the desired current
ideally
ic.
The
bearing
smoothed.
cur¬
Unfortunately analog power
amplifiers have large power losses. An impressing example of this fact is
presented in lZhan/95]. A single bearing axis consumes a total power of
rent
can
as
2480W of which alone 2448W
are
losses
produced by
the
amplifier!
I: Introduction to Active
Chapter
Switched power
Magnetic Suspension
amplifier
U
DC
ld
Current
+
controller
Figure 1.7:
Switched
amplifier (full bridge)
Fig. 1.7 shows the Full Bridge. Instead of a transistor, which is operated lin¬
early, this type of switched amplifier consists of a structure with 4 switches.
S j and S4 respectively S2 and Sj are simultaneously open or closed. The two
pairs
are
driven in
over
the
bearing
decreases
push-pull operation,
coil
according
to
Lw
between
Eqn.
A'<-
For
triggering
which allows
+Ujx;
and
-UDC.
switching
the
voltage
The current rises
or
1.2:
I
t/w
=
L
•
Ar
(1.2)
W
the switches, different modulation methods
are
available;
each method has its merits and also its
pop¬
ular is the method of Pulse Width
This
method allows
controlling
the desired current
the current
(Fig. 1.8).
The
tions. The individual switches
Because the transistors
cantly
1.
smaller
Thcic
ate
compared
only
two
disadvantages. Simple and very
Modulation (PWM) [JeWii/951.
are
so
that its
bearing
are
mean
current
can
corresponds
to
flow in both direc¬
realized with transistors and diodes.
operated nonlinearly
with the linear
opciatmg states
value
amplifier.
the tiansistois conduct (
,
the losses
are
The losses emerge
Ult=0)
ot
they
block
(7(
signifi¬
mainly
=0)
Chapter
during
1 : Introduction
the
switching
to
Active
transient.
TPWM
Figure
Magnetic Suspension
3TPWM
1.8: Current
generation
with
pulse
width modulation
A
disadvantage of the switched power amplifier is the current ripple. The
peak-to-peak value of the current ripple is directly proportional to the
switching cycle Tpy/M- Therefore the current ripple can be reduced by
increasing
the
switching frequency.
losses which in
These
principle
expensive
fully designed
could be avoided
means are
and
bearing
trol effort rises
Switched
magnetic
the
amplifier
not
ripple
is to introduce
amplifier
is
care¬
in the power circuits.
passive filter between
achieves good results the con¬
a
severely [Frit/99].
force
to
depends
generate
one
on
current direction
the
squared
currents in both
cussed
ation
higher switching
snubber networks.
necessary if the
coil. While this method
amplifierfor only
The
normally
by special
with minimum stray inductances
Another way to reduce the
amplifier
But this results in
current
The
ability
of
directions is useless for the dis¬
bearing type because the force has always
of the full bridge is shown in Fig. 1.9.
24
(Eqn. 1.1).
the
same
direction. A vari¬
Chapter
]d
1: Introduction to Active
Current
+
->»
controller
Figure 1.9: Homopolar
S2
the current
is forced to flow
ing
over
ic
and
S3
conducting
are
through
ideal diodes is
coil therefore becomes
-Upyc-
only
homopolar
switched
amplifier
If the two switches
the diodes
zero
and the
D2
and
D3.
voltage Uw
are
The
over
open,
voltage
the bear¬
The current is reduced until it reaches OA.
At this moment the diodes block and
The
switched
replaced by diodes.
The switches
drop
Magnetic Suspension
amplifier
no
is
two active switches. It is used in
negative
more
a
current can be
built up.
reasonable because it
large variety
of
requires
magnetic bearing
systems.
1.2
Extending the principle of the magnetically sus¬
pended hall to a complete magnetic hearing
To illustrate the mode of
operation
bearing type was chosen which repre¬
sents a direct extension of the magnetically suspended ball. For further
information on this type, see references [Schw/94] and [Habe/84j. Later
chapters
a
will discuss other types of active
25
magnetic bearings.
Chapter
1: Introduction
1.2.1 A
of
Active
single bearing
Differential driving
Fig.
to
Magnetic Suspension
axis
mode
1.10 shows the structure of
an
a
single bearing
axis which is
independent
mainly
opposite
so-called differential driving
external force field. The structure consists
electromagnets
mode. Both
are
electromagnets
control current
is increased
magnetic
which
id
operated
in the
carry the
same
bias current
is added to the bias current. The
(Fm+)
because of the
of two
In magnet
attracting magnetic
higher magnetic
force of the second magnet
i0.
(a),
a
force
flux in the air gap. The
(b) is reduced (FmJ
as
the
same con¬
trol current is subtracted from the bias current. The interaction of these two
forces
yields
a
resulting
force
Ftes,
which has the direction of magnet
For this
bearing type a homopolar amplifier
amplifiers must be chosen to drive a maximum
Figure
1.10:
Complete bearing
axis with
26
(a).
is sufficient. Note that the
current
of
/0+ic.
differential driving
mode
I: Introduction to Active
Chapter
For small deflections
x
the
bearing
Fres(^K)
called the
kj is
id equals
spring
zero,
shows
constant is
a
and
on x
ij.
K-id+K'X
=
force-current factor
Fies
depends linearly
force
Magnetic Suspension
and
kl
the
similar behavior
(M)
For
force-displacementfactor.
spring
to a
force except that the
often called
negative in sign. Therefore &x is
negative stiff¬
ness.
Differential
Coils
Figure
1.11 shows
Fig.
coils. The coils
a
configuration
L0I
passed through by
they
coils,
of
are
are
1.11:
and
L02
Differential
in which the
are
coils
electromagnets
connected in series and
the bias current.
They generate
often called the premagnetization cods.
again
winding
Ljj
and
Lj2,
of the coils is such that the control flux is added to
with radial direction. This
configuration
has the
the differential drive mode, that the bias current
constant current source.
must
the control
connected in series and carry the control current. The
(b). The consequence is again
they
permanently
are
the bias flux. Therefore
tracted from the bias flux
but
each have two
The
also be able
of this type is the
coupling
to
amplifiers
must
a
(a)
sense
and sub¬
force
resulting
advantage, compared
can
drive
be
generated by
only
a
with
cheap
the control current,
carry both current directions. A
disadvantage
between the coils since control and bias flux
27
use
Chapter
the
I. Introduction to Active
same
iron
path.
Magnetic Suspension
In Sect. 4 and 5 alternative
bearing types
are
presented
which have separate back irons for bias and control flux. These arrange¬
ments
allow the
use
generating
of permanent magnets
the bias flux.
complete support of a shaft
1.2.2 The
magnetic suspension of a rotor in all five degrees
radial bearings and one axial bearing are used.
For the active
usually
two
of
freedom,
Figure 1.12: Schematic illustration of a completely supported shaft
Radial
bearing
The construction of
a
possible
a
bearing
realization of
actuators are
arranged
a
for two radial
radial
bearing.
in radial direction.
axes
is obvious.
The
pole
They
Fig.
1.13 shows
shoes of the different
could be
grouped
in axial
direction too.
Axial
Fig.
Bearing
1.14 shows the
screened
A
cross
section of
electromagnets. They
ferromagnetic
disk
the two magnets. The
is
are
mounted
an
axial
operated
on
bearing.
It consists of two
in the differential
driving
mode.
the rotor. The disk is located between
generation of the
28
axial force works in the
same man-
Chapter
ner as
J
Introduction
to
Active
Magnetic Suspension
described in Sect. 1.2.1.
Coil
Rotor
Stator
Figure
1 13- Radial
bearing
Pot magnets
Rotor
Coil
Figure
1 14 Axial
29
bearing
/
/
->
r, 11.^
'I'M
-..
v,^
I
i
30
2.
Motivation, Objectives and Structure of
the Thesis
Motivation
Magnetic bearings
have been used for
pumps, turbines, compressors, energy
The
advantages
-
-
-
-
-
large variety
textile
storing,
applications such as
and milling spindles.
of
are:
Contactless support
Free
of lubricant
Maintenance-free
Robust
Low
against heat, cold,
bearing losses,
strength of the
-
a
Bearing forces
rotor
act
vacuum, steam
and chemical substances
the maximum rotation
speed
is determined
by
the
materials
through
an
air gap, the rotor
can
be
encapsulated
hermetically
of
an
magnetic bearing system. It was shown that the support of a rotor
degrees of freedom typically requires the following components:
in
Sect. 1 gave
an
overview of the architecture and the mode of
operation
active
five
10
Electromagn ets
5
Displacement
1
Evaluation unit
(for
1
Control unit
(with A-D and D-A converters)
5
Power
1
Constant current
sensors
5
sensors)
(with 4 power switches each)
amplifiers
source
Especially the complex electronics makes the active magnetic bearing tech¬
nology expensive compared with conventional bearings (e.g. ball bearings,
sliding bearings).
31
and Structure
Chapter 2: Motivation, Objectives
of
the Thesis
Objectives
This work will
investigate
different
for
reducing the size and cost
of magnetic bearings and their electronics. It will help to open new fields for
magnetic bearing technology. Therefore existing suggestions for minimiz¬
ing size and costs shall be examined to form the basis of the development of
such a system. Proper operation will be proved on a high-speed spindle,
which has first to be designed.
Structure
of the
strategies
Thesis
In Sect. 3 different
approaches
reduce the costs will be
to
evaluated. It will be shown that cost reduction in
can
be achieved
integration
reduced by
by minimizing
the
bearing
presented
magnetic bearing systems
current, which allows
further
a
of the power electronics. The number of electronic parts
a
complete digital
control
In Sect. 4 various arrangements
bias current
the
by
power losses
can
be
(current control included).
presented
which allow
eliminating
of permanent magnets. The different
use
introduced to
are
are
and
provide
the basis for
choosing
sources
the
of
the
bearing
permanent-magnet-biased bearings.
A short
type.
Sect. 5 deals with the
radial
Based
are
bearing
on
the
performed
and
a
design
combination of radial and axial
required
to
of
maximum
design
these
bearing force,
bearing types.
After the elimination of the bias current, Sect. 6
retical
background
for
minimizing
the control current is the
Different alternative
tive Vibration
Compensation (AVC)
Sect. 7 presents the mechanical and
spindle system.
The system
electronics served
designed
as
can
be
introduced.
introduces the theo¬
current
caused
largest part of
by unbalance.
theory
of the
Adap¬
presented.
electromagnetic part of the high-speed
controlled by two electronics. The first
prototype for the miniaturized electronics, which
after the actual work
measurements
finally
will be shown before the
is
are
all necessary calculations
the control current. The
speed-synchronous
strategies
bearing
illustrate the
on
bearings
operation
and AVC
was
proved.
was
Several
of these two systems.
Sect. 8 summarizes this thesis and presents the outlook for further work.
3.
Approaches
netic
This
chapter
to Reduce Costs
of Mag¬
Bearing Systems
the basis for further
procedure with this work. In the
already existing approaches for reducing the costs of
sets out
first sections various
magnetic suspension are
reduce the complexity of
described. Common to all of them is that
the hardware system. The second part derives
alternative strategy. For this the different elements of the active
bearing technology
are
investigated concerning
reduction. It is shown that
ering
the
bearing
3.1 Passive
Passive
current
they
a
and
further cost reduction
by further integrating
the
can
an
magnetic
potentiality
for cost
be achieved
by
low¬
the power electronics.
magnetic suspension
magnetic bearings differ
from active
suspensions
in the lack of
an
electronic system. The rotor is stabilized
only by permanent uncontrolled
this, passive systems would be much less
magnetic
fields. Because of
expensive
than active controlled systems.
passively suspend
[Earn/42]. At least
Unfortunately it is impossible to
ferromagnetic body in all five degrees of freedom
a
one
axis must be stabilized with forces of different
example the combination of active and passive magnetic bear¬
practical, since at least the associated part of the expensive electron¬
nature. For
ings
ics
is
can
be saved. The
bearingless
slice motor is based
on
this concept
[Bari/
98].
Another
disadvantage of passive magnetic bearings is their very low natural
damping. This could cause stability problems. Therefore damping and dis¬
turbing forces of the ambient media have to be considered in the design of
the system. The exact calculation of permanent magnet
complex
and
requires
finite element methods. In
estimating bearing stiffness and force
bearings stability in one axis always
another axis.
33
is
bearings
[Yonn/81]
presented.
a
is very
method for
In permanent magnet
involves consequent
instability
in
Chapter
3:
Approaches
to
Reduce Costs
of Magnetic Bearing Systems
3.1.1 Permanent magnet
bearings
Permanent magnet
are
bearings
based
on
the
principle
attraction of two permanent magnets. Radial and axial
able
by
suitable arrangement [Yonn/81]. In
a
permanent magnet bearings
are
ring. Axially magnetized rings
than
rings
with radial
repulsion or
bearings are realiz¬
3.1 different structures of
shown. Their magnets all have the form of
preferred,
magnetization.
Radial
Fig.
of the
are
since
bearing
they
cost less to
Axial
a
produce
bearing
HOB
c
H
j
A
A
.*L
H
H Eh
m
s
-
>H
[-*
+-
ï
a,
CD
«
-+*_
-*
_
P
bû
JH
-31 Eh
Ö
A
o
I
H.
bù
H
C
O
1 ! fl-
CS
H Fh
<
Figure
Usually
vide
rare
earth magnets
especially high
maximum
3.1 : Structural forms
speed
are
used
as
of magnet bearings
material for the
rings
energy densities. In the absence of extra
is limited
by
the permanent magnets, since
since
they
pro¬
precaution
they
are
the
brittle
and lack mechanical strength.
3.1.2
Magnetic reluctance bearings
The reluctance
another type of
mode of
on
bearing is
operation is based
the
passive magnetic bearing. Its
attracting force of magnetized iron bod-
34
Chapter
3:
Approaches
magnetic circuit attempts
its magnetic resistance. Fig.
to
Reduce Costs
of Magnetic Bearing Systems
minimize its reluctance
in other
ies. The
to
words
3.2 shows different structures of reluc¬
tance
a
bearings.
The
magnetic
circuits
are
constructed
strong reluctance minimum. In the picture
on
so
that
the left the
or
they both
show
flux is
magnetic
generated by permanent magnets. A combination of an active and passive
magnetic suspension is shown on the right. While the active magnetic bear¬
ing stabilizes the radial axes, its magnetic field is strong enough to passively
stabilize the rotor in the axial direction.
Figure
3.2
Reducing
3.2: Reluctance
the system
bearing
complexity of active magnetic
bearings
3.2.1 Active
magnetic bearing with
a
minimal number
of actua¬
tors
shows
Fig. 3.3
magnetic
rotor
a
fully
with
a
controlled active
magnetic bearing system for
a
ferro¬
reduced number of actuators. It consists of six electro¬
magnets [Ueya/94]. The
rotor has
a
conical
shape
in the range of the
bearings. This arrangement allows generating radial and axial forces. The
control
axes are not
independent
of each other, which raises the
complexity
of the controller.
The
same
magnetic bearing system
is used in
[Bühl/95]. In addition
to
type of switched power amplifier is presented, which
sists of only six switches (one switch per actuator).
bearings
a new
35
the
con¬
Chapter 3: Approaches
to
Reduce Costs
of Magnetic Bearing Systems
Figure 3.3: Conical magnetic bearing system
The
bearing current for the conical system is homopolar. Its generation
shown in Fig. 3.4. A DC-DC converter divides the supply voltage UDC
half. One connector of each
bearing
coil is connected to this
potential.
is
in
The
DC-DCconverter
Figure
potential
3.4: Power
amplifier for
of the other connectors
can
conical
bearing system
be switched between
ground
+
+
Udc
and therefore the
voltage Uc
over
the coils
changes
between
and
U
DC
2
U
and
DC
.
2
This allows
raising
or
lowering the bearing
36
current
in the
same
Chapter
way
as
3:
Approaches
to
Reduce Costs
of Magnetic Bearing Systems
discussed in Sect. 1.
already noted, this power amplifier uses only one switch per actuator
plus a DC-DC converter, which is reasonable. Compared to the full bridge
with the same supply voltage, this approach will not reach the same dynam¬
ics, because only half the supply voltage is applied to the bearing coil. The
switches have to be chosen to carry the whole bearing current. This raises
the price of the power electronics as Sect. 3.3 will show.
As
3.2.2
Magnetic bearing with asymmetric premagnetization
A variation of the differential coil
(Sect. 1.2.1) is shown in Fig. 3.5. This
L-y..
L?v
Figure
3.5:
construction is
Magnetic bearing
operated
with
asymmetric premagnetization
in contrast to the differential coil
by
a
homopolar
control current.
While both
electromagnets
carry
a
control coil
37
L/v2v, only
one
of them
(a)
Chapter
has
a
3:
Approaches
bias coil
Lq.
to
The
Reduce Costs
sense
of
of Magnetic Bearing Systems
winding
of bias and control coils is
Therefore the control flux is subtracted in
as
<E>C
the control current
is of the
ic rises,
amount
same
as
the
(a) from the bias flux >0.
magnetic
control flux
<E>C
is
In
soon
the bias flux the maximum force in the
If
(b)
resulting
zero.
[ScHa/97] this arrangement is combined with
similar to the
amplifier
permits eliminating
The bias current is
one
shown in the
switch. It
generated by
uses
a
a
3.6: Power
amplifier,
which is
section. This combination
only
per
one
simple voltage
JDC-DC converter]
power
previous
realization the internal resistance R limits the bias
Figure
As
amplified.
direction is reached. With the control flux half of the bias flux, the
force is
opposite.
bearing
axis
(Fig. 3.6).
divider. In the described
current.
V UDC
amplifier for magnetic bearing
R
systems
with asymmetric
premagnetization
The bias coil needs
constantly
the
peak
space and generates
a
large
number of ampere turns, because it has to
control excitation. Therefore it claims
large
copper losses. This
applications requiring large
forces
as
bearing
against small and
tems.
38
a
lot of
equal
winding
type is not suitable for
cost-sensitive sys¬
Chapter
3.2.3 Sensorless
Approaches
3:
Reduce Costs
to
of Magnetic Bearing Systems
magnetic bearings
operated as a generator, a magnetic bearing
can feed back electrical energy. Voltages are induced into the bearing coils
by the moving rotor. Their amount depends on its speed, and the displace¬
Just
as an
electric motor
ment can be
Another
derived
can
be
electronically
from theses
method, the modulation method,
uses
voltages.
the actuator
as
inductive
sen¬
high-frequency current is modulated upon the bearing current and the
inductivity of the coil can be determined. In [Okad/92] a similar method is
presented. The pulse-width-modulated output voltage of the amplifier
serves as modulation signal and the magnitude of the current ripple is a
measure for the position of the rotor.
sor.
A
Both methods have been examined in
[Kuce/97]. Kuçera stated that only the
signals. The complexity of the evalua¬
tion electronics is in both cases quite high compared to a position measure¬
ment with sensors. Sensorless bearings are suited for applications where the
modulation method delivers usable
number of wires has to be minimized.
3.3 Evaluation
3.3.1 Potentialities for cost reduction
Actuators
Electrical machines have
a
tradition. The know-how in the
long
production
widespread and established. Active mag¬
technology too. The price level is already
of electromotors and generators is
netic
bearings
are
based
very low and stable.
reducing prices
of
on
this
Therefore,
actuators as
no
long
outstanding
as
progress
can
be
expected
in
their size is not reduced.
Mechanical structure
Magnetic bearings usually
allow
higher
tolerances of the rotor surface
com¬
bearings. Another advantage of magnetic bearings is that
they allow higher unbalance. To reduce iron losses the rotors are laminated
in the range of the bearings (see Sect. 4.2.1). Usually the sheets are shrunk
onto the shaft. Unfortunately these shrink fits require high production tolerpared
with other
39
Chapter
ances,
3:
Approaches
especially
Sensors and
the
near
for
Reduce Costs
high
integration
of
speeds.
signal
future. State of the art
0.18\im (e.g.
Intel Pentium
produced theoretically.
mixed
rotation
of Magnetic Bearing Systems
signal processing
in the
Progress
to
It is
electronics is
in CMOS
III). ICs with
possible
technology
a
is
a
to
slow down in
structure width
100 million transistors
integrate
to
unlikely
very
complex
can
of
be
electronics in
signal technology-".
Power electronics
The
integration
can
be achieved.
of power switches
depends largely on the power that
has to be switched. Fig. 3.7 shows graphically the prices of different single
transistor devices versus the product of maximum blocking voltage and
maximum current. The figure shows that by reducing the power consump¬
tion of magnetic bearings other integration technologies and lower prices
14
12
<¥^
rate
i
-
10
00
8
4)
o
Ph
6
4
2
-
-
-
H
0
1
10
Maximum
Figure 3.7:
Price
blocking voltage
of a
transistor
blocking voltage
1.
As
2.
Mixed
things
stand
1000
100
January
x
Maximum current
depending
and maximum
10000
the
and
digital signal processing
40
[VA]
product of maximum
output current (1.2.2000)
on
2000
signal technology analog
100000
on
the
chip
Chapter
3.3.2
The
3:
Approaches
Reduce Costs
of Magnetic Bearing Systems
Influence of current, voltage, power consumption and con¬
trol on the cost of bearing systems
previous
section discussed the
aspect of the components. In the
potentialities for cost reduction from the
following the question is examined, how
different parameters influence the
their electronics.
Fig.
power and control
Changes
a
digital
design
of active
magnetic bearings
3.8 sketches the relations between
strategies
and the
components of
a
voltage,
and
current,
bearing system.
Influenced elements:
in
Figure
With
to
3.8:
Influences
on
bearing design
control the number of hardware devices could be reduced.
Especially the control electronics and the power stage are simplified. Actu¬
ally the complexity of the system is not reduced. All components of the sys¬
tem
still exist,
only
the controllers for
positions
and currents
are
realized in
software.
The
constitutes
important part of the total power consump¬
tion. It occurs mainly during normal operation. It determines therefore the
size of the power supply and the size and method of cooling of the
power
amplifiers.
r.m.s.
current
an
41
The maximum
expended,
bearing
current is
and it will be reached
bances. Therefore it has
cooling system
only
only
during
the
peak
take-off and
minor influence
but it determines the power
42
current which has to be
on
as
reaction to distur¬
the power
amplifiers
supply
and the
and the actuators.
4. Elimination
It
was
of Bias
noted in Sect. 3 that the
bearing
ther cost reduction. The bias flux
Another way to
produce
the
was
Current
current must
so
far
be minimized for fur¬
generated by
premagnetization is
the
use
a
direct current.
of permanent mag¬
nets.
This
chapter presents the state of the art in permanent-magnet-biased bear¬
ings. It provides the basis for choosing the suitable bearing type, which is
calculated in the next chapter. For this the first section presents different
realized arrangements. Then the
occurring
losses
are
that
especially
4.1
Permanent-magnet-biased bearings
4.1.1
In
discussed. It is shown
the iron losses in the rotor determine the
pole arrangement.
Homopolar configuration
Fig.
4.1
a
permanent-magnet-biased bearing
with
homopolar
Permanent magnet
Control flux
Figure
4.1:
Permanent-magnet-biased bearing
43
flux is
ring
\
with
homopolar flux
Chapter 4:
Elimination
of
Bias Current
sketched. It consists of two identical stator parts with 4
poles. A permanent
magnet ring is clamped between the two stator parts. It is axially magne¬
tized (Fig. 4.1 right) and generates the bias flux, which flows through one
through the second stator part. The control
flux of an axis does not flow through the permanent magnet rings. Therefore
control and bias flux do not influence each other. Fig. 4.1 left shows the
to the rotor and back
stator
part
cross
section
bias flux
through
always
one
with this characteristic
The
bearing
of
rotor. It can
emerges from the rotor in the
ings
sense
part and the
stator
coils of
winding
so
an
are
axis
that
known
are
as
same
seen
that the
radial direction. Bear¬
homopolar bearings.
connected in series.
they generate
be
the
same
They
have the
same
control flux. It is super¬
posed additively in (a) and subtractively in the opposite air gap (b). In
other stator this happens inversely. A force in direction (a) results from
the
the
different energy contents in the air gap.
4.1.2
The
Heteropolar configuration
homopolar bearing
is
relatively long.
Figure
rotor
dynamics.
In
4.2:
[Lewi/94]
a
This is
a
disadvantage considering
Heteropolar bearing
short
magnetic bearing
is
presented
heteropolar bias flux. With the use of this bearing type the
designed shorter with higher natural frequencies (App. A.2).
has
a
44
rotor
which
can
be
Chapter 4:
8-pole arrangement is sketched in Fig. 4.2
bearing type consists of only one stator. The
An
parts. Permanent magnets
manent
ond
flux is carried
pole.
other coils
pole.
(b) the
production
cost
for the
The
The coils
current
pole
explain the principle.
iron path is cut open in
This
four
between these stator parts. The per¬
and flows back
over
through the surface
designated with (a) carry
This allows
ix+ir
tolerances for
production
controlling
the rotor and
a sec¬
of the rotor
changes
the current
ix-iv
the force in
x-
the
and y~
rare
of such
a
earth magnets
are
quite large, raising
the
bearing.
Permanent-magnet-biased bearings operated by threephase drives
bearings
most
first
Current
independently.
The
4.1.3
a
of Bias
to
The direction of the flux
every second
direction
by
clamped
are
Elimination
discussed up to
now are
all driven
by
a
two-phase
of the electrical machines and their power converters
large variety of three-phase
they are relatively reasonable.
A
Figure
4.3:
power converters
Magnetic bearing
with its
45
are
are
drive. But
three-phase.
available. Therefore
three-phase power converter
Chapter 4:
For the
Elimination
of
use
of Bias Current
three-phase
rents must be zero. In
power converters the
[Schö/97]
sum
of the three output
cur¬
condition. The arrangement is similar to the
presented that fulfills this
previously presented homopo-
lar
bearing.
poles.
lar
premagnetization
so
Now the stators have
it is
bearing
called Park
system into
a
sum
construction is
only
possible
that the condition of the
effort of this
so
a
three
Because of the
to connect the three
of currents
being
is not much greater than for
transformation [Schö/93]
allows
zero
bearing
homopo-
coils in star
is met. The control
two-phase bearing. The
transforming a two-phase
a
three-phase system.
4.2 Losses
In
an
electromagnetic system
-
-
-
Copper
the
following types
of losses
occur:
losses
Hysteresis losses
Eddy
Hysteresis
losses
current
losses
and
occur
eddy current losses are often
mainly in the rotor, whereas the
united in the iron losses. Iron
copper losses dominate in the
stator.
4.2.1 Iron losses in the rotor
In the
following, only
the bias
flux is considered, the control flux is
neglected.
Iron losses in
inductions.
rotating machines are caused by cyclic changing magnetic
Fig. 4.4 shows the magnetic induction on the surface of the rotor
depending on the rotational angle cp for different premagnetization types. In
(a) the heteropolar and in (b) the homopolar arrangement is shown. Note
that the
amplitude
of the
synchronous
induction in
(b).
46
(a) is much larger than
in
Chapter 4:
Elimination
of Bias
Current
<?
LI
Figure
4.4:
Hysteresis
M
L„
Cp
Magnetic induction depending on the rotation angle (pfor het¬
eropolar (a) and homopolar (b) premagnetization
losses
For dia- and
tion
ear
are
paramagnetic materials the magnetic field strength and induc¬
connected linearly. Ferromagnetic materials show a strong nonlin¬
and not
unique
behavior. The state of
magnetization depends
the actual external
zation
curve
has
magnetic field, but also on the prehistory.
hysteresis character (Fig. 4.5).
Bs
Inner
Virgin
not
The
curve
hysteresis cycle
Figure 4.5: Hysteresis
curves
47
of a ferromagnetic
material
only
on
magneti¬
Chapter 4: Elimination of Bias Current
This effect
can
be illustrated
clearly
with the model of the elemental mag¬
rising external
curve. During this pro¬
material is magnetically
degaussed material is exposed for the first
magnetic field, its magnetization follows the virgin
nets
cess
.
If
a
the elemental magnets orient themselves. The
saturated
as
field. The
soon as
occurring
all magnets have the
same
time to
a
orientation
as
the external
induction is called the saturation induction
Bs.
The
magnetization follows the limit cycle if it has reached saturation once. The
remanent flux density Br and the coercive force Hc both lie on this curve. Br
remaining magnetization with disappearing external field and
remaining magnetic field force at zero inductance. No magnetiza¬
describes the
Hc
is the
possible beyond the limit cycle. If the material is
saturation the magnetization follows an inner hysteresis.
tion is
not
steered up to
Hysteresis losses can be described as "frictional heat of the elemental mag¬
nets" [Fisc/92] during a magnetization cycle. The area surrounded by the
hysteresis
corresponds to the energy expenditure during such a cycle.
due to hysteresis is approximately proportional to the square
curve
The power loss
of the maximum induction and
proportional
to
remagnetization
the
fre¬
quency:
2
Pu~B
H
Eddy
(4.1)
-f
s
> r
max
current losses
alternating magnetic field induces volt¬
ages. If the material is an electrical conductor these voltages cause currents
in the exposed material [Kall/94].
Following
the law of induction
an
pe~ys;-<b2
(4.2)
Rel. 4.2 shows that the power losses caused
tional to the square of
inversely proportional
l.
The
fragments
of
a
by eddy
remagnetization frequency
to the
magnetic
are
propor¬
flux and
are
electrical resistance. A constructive
specific
broken petmanent magnet
and
currents
are
still
idea that all magnets consist of molecular magnets
48
magnetized.
[Huge/08]
This etfcet led to
Ampere's
Chapter
measure
to
reduce the
eddy
currents is
Figure
Stacked iron sheets
lated
against
The losses due to
sheets.
so
eddy
that the
field
the
magnetic
4.2.2
The
RM
and
eddy
currents is
are
iso¬
inhibited.
a
(4.3)
magnetic
field. It acts
dynamics
currents
of the
against
delay
the initial
the set-up of
electromagnetic system.
in stator
electric circuit of
an
of
d2
Therefore, eddy
field and reduce the
equivalent
tance
to Lenz's law.
Copper losses
currents
propagation
-
themselves generate
according
4.6:
proportional to the thickness of the
(down to 150p.m) are used for stator and rotor.
Therefore, thin sheets
currents
Eddy
Fig.
Current
currents are
Pe
Eddy
illustrated in
of Bias
used instead of solid material. The sheets
are
each other
4.6:
4: Elimination
inductance
bearing coil consists of an internal resis¬
LM (Fig. 4.7). The copper losses are nothing else
a
than the ohmic losses of the internal resistance:
Pr
=
I2
R
49
(4.4)
Chapter 4:
Elimination
of Bias
Current
Rw
U
A
I
Figure
Equivalent
4.7:
Besides the ohmic losses,
Rw
circuit
I
of a bearing
coil
limitation of the maximum
causes a
bearing
current:
A
cmax
The
voltage Uw
over
This effect limits the
the inductance
dynamics
and the copper losses, the
(4.5)
R.
Lw
of the
winding
is reduced
bearing
as soon as
flows.
current
force. To reduce these effects
space must be raised since the internal
resistance is diminished.
4.2.3
Comparison ofpole arrangements
It
shown in the
was
previous
sections that
hysteresis
and
eddy
current
losses
increase with the square of the
once
again.
the rotor
The
was
curve
of
illustrated.
magnetic induction. Fig. 4.4 is referenced
the magnetic induction over the circumference of
In the heteropolar arrangement (a) the magnetic
induction reaches values between
induction in the
+B0
homopolar bearing (b)
fore the iron losses of the
the
the
and
is
-Bq during
always
a
cycle,
between 0 and
whereas the
+B0.
There¬
homopolar bearing will be smaller than those of
heteropolar bearing, especially for high speeds. This is the reason why
homopolar bearing is preferred.
50
5.
Design of Permanent-Magnet-Biased
Bearings
It
was
demonstrated in the
previous chapter that homopolar bearings show
comparable heteropolar bearings. A disadvantage of
lower iron losses than
the
homopolar bearing is its length. In this chapter a homopolar bearing is
examined that has only one stator part. A construction of a radial and an
axial bearing is shown, which both use the same bias flux.
5.1 Fundamentals
The mode of
ing
is based
current
and
eration
are
operation
on
a
two
presented
two
first. The later calculations
analogy
between the electric and
Hpe,&ç0ji
field gen¬
simplified through the
the magnetic circuit.
circuit with air gap 6 is sketched in
coil has N turns and carries the current
length
are
magnetic
circuit
simple electromagnetic
The
methods of
magnetic field generation
Electromagnetic
A
permanent-magnet-biased active magnetic bear¬
superimposed fluxes, which are generated by an electric
a
permanent magnet. These
introduction of the
5.1.1 The
of
of the iron
path
is
lFe
and the
ic.
Its cross-sectional
pole
describe the induction and the field
air gap and the coils.
face
area
strength
is
Fig.
5.1. The
area
is
A^0,y.
A§. BFe>§,Coii
in the iron
path,
and
the
By means of Ampere's law (Eqn. 5.1) and the assump¬
tion that the magnetic flux in the circuit is constant (Eqn. 5.2), the air
gap
induction B§ can be calculated (Eqn. 5.3)[Huge/98].
51
Chapter
5:
Design of Permanent-Magnet-Biased Bearings
N
'Acoil
\wsx
<V
n
A<
Figure
5.1: Circuit
of an electromagnet
(5.1)
®c
=
BCoificoil
B5A5
=
=
(5.2)
nst
Ni^
The fact
was
made
large compared
strength HFe
use
to the
in iron
of that the
ws
pole
\i0 of
disappears (p*Fe
=
W§
vacuum, so
-10
4
5
...
\i0). Eqn.
10~ x
in the air gap.
that the
V§
magnetic
field
5.4 shows the
(5.4)
=
W§
of iron is very
is the air gap volume.
^s.äsä8 ^vs.^
The derivation of the energy
the
magnetic permeability nF(,
permeability
calculation of the energy content
(5.3)
for 5 results in the force
F§,
which acts
on
faces:
Fs(g
=
jô
=T'l'ô
52
^0
(5.5)
Chapter 5: Design of Permanent-Magnet-Biased Bearings
The permanent
magnetic
circuit
Permanent magnet
PM
5.2: Circuit
Figure
The
same
way to calculate the air gap induction
chosen here. The
instead
a
of a permanent magnet
magnetic
circuit
now
has
a
as
before
can
basically
permanent magnet
coil. The lack of electric currents influences
Ampere's
be
(APM, lPM)
law
as
fol¬
lows:
0
Again,
the
=
JFeHFe
+
5#5 + lPMHPM
high permeability
of iron
«
5^S
causes
+
its
JPMH.FM
magnetic
(5.6)
field
strength
to
vanish.
The
following equation
describes the conditions in the permanent magnet:
BpM
The
-
magnetic flux density can
stant (Eqn. 5.8 and Eqn. 5.9):
°0
B5
=
=
Br + \ipMHpM
be calculated
BPMAPM
B
=
ßoA8
=
(5.7)
assuming
const
lPM^0APiM
1PAf^0A8
+
53
^PMAPM
the flux to be
con¬
(5.8)
(5.9)
Chapter
5:
Design of Permanent-Magnet-Biased Bearings
The force
on
the
faces is
pole
calculated via the derivation of the
again
energy content of the air gap.
F
^
5
=
2u
•
5.1.2
Analogy
between electric circuit and
In the
following
table the most
netism
summarized to
are
nomena
(5.10)
-A8 Bl
—
db
magnetic
circuit
important definitions of electricity and mag¬
point out the analogy of these two physical phe¬
[Kall/94]:
/
Electric current
\\SdA
=
O
Magnetic flux
<r->
=
A
Electrical volt¬
U
=
age
A
fi
JEdl
comparison
circuits
can
age
shows that the
be used for
resis¬
magnetic
circuits too. For
The permanent magnet is modeled
source
QPM
and
a
R
-®
methods for the calculation of electric
same
illustration, the computa¬
previously discussed permanent magnetic
equivalent network (Fig. 5.3).
age
JHdl
tance
tion of the
an
=
r
force)
Magnetic
<-»
0
(magneto¬
motive
y
tance
The
<->
U
=
volt¬
Magnetic
r
Electrical resis¬
^BdA
as
circuit is
series connection of
magnetic resistance RPM
that
are
a
repeated
with
magnetic
volt¬
presented
in
Eqn.
5.11 and 5.12.
0
^PM
RPM
-
l-^-B
~
..
MpM
üi
PM
-
\iPMA PM
54
(5.11)
(5.12)
5:
Chapter
Design of Permanent-Magnet-Biased Bearings
RPM
®n
'0
A
R Fe
©PM'
Rs
Figure
The
5.3:
magnetic
the air gap
on
network
Equivalent
resistances
RFe
and
R§
describe the influence of the iron and
lFe
Because of the
V-fAfc
5
(5.14)
=
large permeability
can
(5.13)
0
_
Fe
Rb
the resistance of iron is
negligible.
°
Inserting Eqn. 5.11,
result
The
be calculated using Ohm's law:
ei
to the same
circuit
the circuit:
p
magnetic flux
diagram of permanent magnet
RPM
+
5.12 and 5.14 into
as
in
58
=
Eqn.
B
(5.15)
*l
Eqn. 5.15
and
dividing
it
by Ag
leads
5.9:
lPM^l0APM
/PM^0AS
+
55
^PMAPM
(5.16)
Chapter
5:
Design of Permanent-Magnet-Biased Bearings
5.2 The
layout of the
radial
5.2.1 Construction and mode
Figure
5.4:
5.5: Flux
A
4-pole homopolar bearing was
parts. A passive back iron replaces
permanent magnet flux
the
is
path of
is
radial
bearing
described in Sect. 4.1. It had two stator
here the second stator part
conducted
premagnetization
now
by
homopolar.
of
(Fig. 5.4).
The
this back iron, the rotor and the
The
bearing
coils of
an
axis have
winding and carry the control current. Therefore the force
generated by lowering and increasing the magnetic flux in the oppo-
same sense
is still
of operation
Permanent-magnet-biased homopolar bearing
Figure
stator. The
bearing
56
Chapter
5:
Design of Permanent-Magnet-Biased Bearings
site air gaps.
5.2.2
and
Definitions
design process
For the further calculations
that the
"2"
In
Fig. 5.6
defines all
geometrical symbols.
stands for "radial", number
subscript "r"
"/'
Note
for active air gap
and
for the air gap between rotor and back iron.
[Alla/90]
bearings.
the
parameters
Here the
the active air gap
are
derived from the
starting point
specified
is the
existing
dimensions of the
maximum force
Fspec
and
Ô? j.
I
A
5 tl
PM,
A PM
N- h
vw
4-br2
—
As,SrF
TT! X*H
'
V
"A 5/2
"V\ T A
1
!
r*.
A
Figure 5.6: Geometrical definitions
pole
To prepare the calculations the active
face
area
A§7j
is determined
(Fig.
5.7). Then the winding is defined geometrically and electrically. The perma¬
nent
magnet circuit is independent of the electrical design and
designed simultaneously. During
1
Active
air
gap the
an
its
layout stray
gap. wheic the toi ces
aie
57
actively
fluxes
genet ated
must
can
be
be taken into
Chapter
5:
account.
Finally
Design of Permanent-Magnet-Biased Bearings
Their amount is estimated and must be checked at this
bearing parameters
specifications. The parameters
the
are
are
It should be
that this
ning
make
the
pointed out
engineer has to
corrected later
during
the
calculated and
with additional
necessary for the controller
design
face
ace
design.
process is recursive. At the
assumptions
layout.
-(Estimation of pole
compared
which
are
JV
point.
begin¬
checked and may be
Agr/
L_
of
Layout
Design
V
winding
6,,A,,,„ N„LpRr
lI1g/
of permanent magnet
APM, lP^, br2, Agr-
circuiyrcuiy
i
Verification of
leakage
factor j
ctor)
g,.
-(Bearing parameter, verification*; itionV
Figure
5.2.3
5.7:
Design
1F
?
_
process
Winding layout
Active pole
face
area
and inductions
In order to guarantee linear
control induction
Bc$rj
operation (Sect. 5.2.6)
and bias induction
than the saturation induction
B0§r]
the
sum
of maximum
in the air gap must be less
Br-
B08rl+Bc8rl<B,
Eqn.
Frmax> kir kxr
(5.17)
5.18 describes another condition, which is derived from the demand of
58
Chapter
linearity
too.
5:
Design of Permanent-Magnet-Biased Bearings
force being
magnetic
It results from the
square-dependent
on
the total air gap induction.
&W
0<B0Srl-Bc8rl
the control and the
These two conditions give the limitations for defining
to
density
bias flux
F>-
=
in the air gap.
^-^B0in*Bc&rl)2-lBOM-BrM)2)
magnetic force,
The connections between
shown in
area are
solved for the
Eqn. 5.19.
pole
face
area
A8ry
=
(5.19)
air gap inductions and
It is derived from
Eqn.
pole
5.9 and 5.10 and
face
can
be
A§r/:
n0
-
1
/specß—
(5-20)
aclrl
a0hrl
Winding
assumption that the magnetic flux is constant (Sect.
the basis for the layout of the winding. Also the magnetic resis¬
iron is neglected. A correction factor cq > 1 is introduced to con¬
Ampere's
5.1.1) are
tance of
law and the
sider iron losses. This allows
calculating
the minimum
magnetomotive
force:
®r
Ar
Cur
The
required
copper
M-0
2CQ
®r
=
area
(5-2D
~-Bc8rAl
=
—
c
=
—-
.,
•
c
ACm. depends
ß
e
,5
(5.22)
,
corl^rl
on
6r
l
^^
and the maximum allowed
(Eqn. 5.22). The required copper area does not define the
insulation has not been consid¬
space for the real winding, because space for
ered yet. With the introduction of the filling factor /'the space used for the
current
density
S
59
Chapter
Design of Permanent-Magnet-Biased Bearings
5:
winding Awr
real
Kr
The
ing
winding
of
a
bearing axis
be estimated:
^f p^|-^8^5'-/'wlth/<7
(5-23)
=
=
area must now
equations define
three
can
the number of turns
Nr is
tive force
(Eqn. 5.21)
amplifier
used.
be distributed between both
poles.
The follow¬
the electrical parameters of the coils. In
calculated. It
depends
and the maximum current
Nr
=
on
the
Eqn.
5.24
required magnetomo¬
which is
lcmax,
given by the
j-2-
(5.24)
cmax
The resistance
and the inductance
Rwr (Eqn. 5.25)
linearly proportional
p
Kwr
_
~
to the
n
Lwr (Eqn. 5.26)
are
both
square of the number of turns.
Nrhvr
PCtMo
SNVWr
Ar„
2c~
B^rA.i
'
PCu
"0
lCu
(5.25)
uc8r]url
"r
L
=
un-I—£ll
t)
Resistance and inductance influence
force
as
rt
strongly
in Sect. 4.2.2 stated.
60
(5.26)
the
dynamic
of the
bearing
Chapter 5: Design of Permanent-Magnet-Biased Bearings
5.2.4
Layout of permanent magnet circuit
HPMlPM
+
H8rlKl
PM
ö
r+ ^PM
=
=
<5
estimated
as
aAPMBPM
A8rJ^08rl
is inserted in
4
4
Transformation of
5.8
Eqn.
Eqn.
to
length 1PM
magnetic
and the
5.27 and 5.29.
o~
leak¬
has to be
(see Sect. 5.2.5).
5.27-5.29 leads to the following
Eqn.
describes the condition for the
manent
leading
5r2
the exact geometry is not known
as
2Ç)
{5{D'Zy)
_
~
~
long
(5'2?)
(5.28)
5.6 is extended with the additional air gap
age factor
°
PM
AhrlBOhrl
-AB
^OÔrl
Eqn.
H5r2^r2
+
equation,
and the front face
APM
which
of the per¬
magnet ring.
7
'PM
Pm
_
'
m
'
a
.
H
or J
orzri
•
'a
~
_t->
i
uori
ri'
r>
a
,~
j^s
[J.jU)
APM'cBr-4-AhrTB0hrl
Aor2
Minimum permanent magnet volume
Eqn.
general
5.30 represents the
magnet rings for
derivation of
a
specified
bias induction in the active air gap.
aAPM
solving
Taking
the
VPM
—_:
and
solution for the geometry of the pennanent
it for
APM
^^(//w.ApM)
(5.31)
clAPM
results in the
optimal
front face for the minimum
magnet volume:
APM,mn
=
8
^Ag,,
61
(5.32)
Chapter
5:
Design of Permanent-Magnet-Biased Bearings
Introducing APMmin
1
_
'PMmin
-
into
o
Z
"
Eqn.
5.30 leads to the
^PM B0Srl fR
'
d
\
ür
H0
5.2.5
4'A
*
..
optimal magnet length:
rl
orl
(5.33)
+
a
Ahr2
Stray fluxes
R
ySl
RpM
<pPM
Jl,
Rbr2\
%2
p
_*_
lvsn-l
R si
6 PM\
R
X
Rs2
Figure
To
analytically
determine the
resistances
can
age
the four fluxes
source
be
5.8:
magnetic flux factor
a
network of
magnetic
generated (Fig. 5.8). With the permanent magnet
<[>§ri through
mined. The accuracy of this method
resistances. Therefore the electrical
rough
Stray fluxes
the four air gaps
depends
equivalent
very much
circuit is
might be
on
just
as
volt¬
deter¬
the number of
used for
a
rapid
draft.
Another
method, the finite element method, is shown in Fig. 5.9. A three-
dimensional
magnetic
field
plot
is illustrated. This method allows visual
examination of the geometry.
This method is
usually
used
Unexpected stray fluxes are easily recognized.
for designing optimized magnetic circuits.
62
Chaptei
5
Design of Petmanent-Magnet-Biased Bearings
**
*
H J f ff*
1
SOOOe+Ono
1
4j82e+nno
1
i7n1e+M0
1
H4Sf HOJ
l
5'f7e
001
7
1081c
0
h
„lOCfc
Oil
4
5"
7e
001
3
05346
001
1
4350p
001
1
t3 30e
00?
31
V,
**.~-"2>
Figure
5.2.6 Calculation
5.9:
Magnetic field of
radial
bearing
of bearing parameters
The force-current factor
fy
®thl
<E>nR05; 1
*&.
N, h
%i
Figure
5.10:
0> 00/1
Magnetic equivalent network diagram for
the bias
flux (radial
bearing)
For the calculation of the force-current factor the
magnetic control flux for
not deflected. The
air gaps
are
one
bearing
axis is
magnetomotive force
modeled
by resistors.
acts
equivalent
set up
as
a
network of the
(Fig. 5.10).
voltage
The rotor
source.
is
The two
The influence ol the permanent magnets
63
Chapter 5: Design of Permanent-Magnet-Biased Bearings
is simulated
by
two identical current sources
fluxes and the control flux
The
bearing
force
<I>cgr; superimpose
Fr depending
<E>o§r7 arranged
in the two resistors.
the control current
on
that their
so
icr
be derived
can
from this model:
vr
B0brlAhrlNr
,
a
°rl
The derivation of
Fr
l
k.r
after
ia
à
-
(5.34)
.
leads to the force-current factor:
vr
(5.35)
_
-jçt^cr)
The force-displacement factor
B0^rlAbtJNr
,
-
brj
kx
4®ohr\
RpM
R8r2
<t>XvÔr/ +
®M
R8rl+L RSrl-
Figure
5.11:
Displaced
rotor
and
equivalent
network
RSrl
R8n
of permanent magnet
circuit
Only
the permanent magnet circuit is considered for the
computation
of the
force-displacement factor. Fig. 5.11 left shows a situation where the rotor is
displaced only in x-direction and the control flux is set to zero. The force
For
is drawn in with the
equivalent
Eqn.
same
direction
as
a
spring force.
The
network of the permanent magnet flux is shown
5.36-5.38 define the
appropriate
on
the
right.
magnetic resistors. The "+" in the subscript desig-
64
Chapter
nates the
large air
Design of Permanent-Magnet-Biased Bearings
gap and the "_" the small air gap.
R
Sri
R5rl +
must be much smaller
1
°rl
V>0
A8rl
J
8ri
8rl-
«
<3>PM
=>
X
(5.3S)
=
4
can
®obrl
magnetic
force-displacement
be considered to be constant:
~
const
(5.39)
bias flux in the air gap. The mag¬
netic
the nominal
~~
netic
<ï>0gr/ designates
8r/
(5.37)
for the derivation of the
factor. Then the permanent magnet flux
x
+ x
A5rl
^0
Sr;
than
(5.36)
A5rl
MO
hrl
R
x
5:
disturbii
disturbing
in
brl+
and
force
on
the rotor
can
be calculated via the two bias fluxes
8r7_
FoM)
8
Li
Msr 1
The linearization of
ment factor of the
k
=
xr
Eqn. 5.40 at
radial bearing:
cj)".
x
=
0 leads to the
negative force-displace¬
•o;R
0tK
x
=
(5.40)
(2ô;;-.o
—Fn(x)
dx
.
Mô/Ai
0
65
08 r 7
(5.41)
Chapter
5:
Design of Permanent-Magnet-Biased Bearings
5.3 The combined radial and axial
5.3.1 Construction and mode
Figure
The combined
with
only
one
bearing
of operation
5.12: Combined radial and axial
bearing
enables the active
suspension
bearing
of
a rotor
in three
axes
permanent magnet flux. It has the advantage of its short size
compared with arrangements using
axial bearing.
a
separate bias flux for the radial and the
previously discussed radial bearing, only
that the back iron is replaced by an axial bearing unit. It consists of two sta¬
tor parts enclosing the control coil (Fig. 5.12). In the inner radius they leave
The construction is similar to the
open space for the radial disk of the rotor.
In
Fig.
5.13. the
paths
of control and bias fluxes
the bias flux of the radial
known radial
bearing
bearing
66
shown. The control and
path as in the already
they work identically.
unit have the
of Sect. 5.2. Therefore
are
same
Chaptet
Figure
Mode
5
5.13: Schematic structure
Design of Permanent-Magnet-Biased Bearings
of combined radial and axial bearing
of operation of axial suspension
The
magnetic
two
parts. One part flows clockwise, the other part flows counterclockwise
around the
bias flux
bearing
generated by
the permanent magnet
bearing
is
split
into
coil. In the axial disk these two fluxes recombine in the
axial disk and flow back via the radial
The axial
ring
coil
produces
bearing
the control
unit to the
source.
flux, which is carried by the
axial stator parts and the axial disk of the rotor. Bias and control flux
super¬
in
air
and
in
impose additively
(b). The different
gap (a)
subtractively
energy contents in the
air
gaps
cause a
compensating
force in axial direc¬
tion.
Because the permanent bias flux is constant with small deflections in radial
and axial
direction, the three
axes
of the combined
pling.
67
bearing
show
no
cou¬
Chapter
5.3.2
5:
Design of Permanent-Magnet-Biased Bearings
Definitions
and
design process
A 8/
!
Eva
i
wîîiWîC- \<
i
s
.
St——'
A
wa
PM
Figure
5.14:
Definitions
Fig.
5.14 defines the geometry of the combined
and
"a"
The
design
stand for the radial
are
bearing.
The
determined first. The
four times
unit and the axial
process for the combined
sented for the radial
ings
bearing
larger
than
Ag,
bearing
pole
pole
bearing.
area
bearing
is similar to the
face and
face
The
winding
A§a
because of the condition that the
tion in the air gap must be between
zero
unit
procedure
areas
of the axial
subscripts "/'
pre¬
for both bear¬
bearing
magnetic
and saturation induction
unit
is
induc¬
Bs (Eqn.
5.17 and 5.18). Then the permanent magnet circuit is laid out. The check of
the stray fluxes must be done for both units
cess
the
bearing parameters
The calculations for the
similar to those
repeated
ters
are
layout
Only
At the end of the pro¬
determined.
of the combined radial and axial
shown for the radial
in this section.
separately.
the
bearing.
computation
is discussed.
68
Therefore
of the axial
bearing
they
bearing
are
are
not
parame¬
Chapter
5.3.3 Calculations
5:
Design of Permanent-Magnet-Biased Bearings
of the bearing parameters
The force- current factor
kit
ia
$c8«
<ï> 05a
R8a
N
i
a
(
ca\
R8c
a> OSa
Figure
5.15:
Equivalent network diagram of the magnetic fluxes
air gaps
The rotor is not
equivalent
equivalent
bearing
network of the radial
and the
opposite
not
displaced)
in axial direction.
network for the axial
fluxes in the
Oögrt,
displaced
(rotor
air gaps
magnetomotive
Fig.
5.14 shows the
modeled
force N„
a
magnetic
unit. Its structure is identical to the
bearing (Fig. 5.10). Again
are
in the axial
/„„
as
as a
ca
the two bias
identical current
voltage
source.
sources
The axial
force is then:
Wca)
B08aA8aNa
(5.42)
=
5
The force-current factor is the derivation of the axial force again:
Jr
Kia
d
-
F
C
B0à«AàaNa
\
~fj~~ta^Icci'
1
s:
a
ca
69
(5.43)
5:
Chapter
Design of Permanent-Magnet-Biased Bearings
Force-displacement factor kxa
^PM
2O08a
*5«
n
K6a+
^
%5
R8a-
-
e PM
ik_CJa^«ÄÄ2.
%
RSi
Figure
5.16:
Equivalent
network
The
compared
A
illustrates the
this arrangement. The axial
to the
air gap
8Ö.
permanent magnets generate
flux
+ :
(5.44)
(5.46)
.
rotor
equivalent
With this
is shown. It is
circuit of the
displacement
a
(5.45)
±
-
(left) the axially displaced
right figure
bias
L. 6a-:
A5a
-
R8,
5.16
magnetic
A8a
V>0
R8a-
Fig.
the
%
~
K6a +
In
diagram of
ôa
/?
R8i
0/
is assumed to be very small
c
assumption
constant
radially centered.
magnetic bias flux of
it
is
guaranteed
that the
flux which is twice the amount of
the nominal bias flux in the air gaps:
8.
In
Eqn.
5.44-5.46 the
Opm
magnetic
flux of the smaller air gap and
With these basics
the
=
2
•
O06,
resistances
-
<
are
A'gö+ corresponds
equation
for the
70
(5.47)
onst
given. R§Ch
to
the
large
is the
magnetic
air gap.
magnetic disturbing
force
F Oa
Chapter
depending
on
the
displacement
F ».a
The derivation of 5.48 for
axial
bearing
z
5:
Design of Permanent-Magnet-Biased Bearings
z can
=
be formed:
-^f-
Vo'Aba
leads to the
f
<s-48>
°a
force-displacement factor
for the
unit:
Ka
=
4f0a(z)
dz
=
-—f-ß" <&L
•
\i0A8ada
7/
(5.49)
V--* I
72
6. Elimination
of Synchronous
Currents
largest part of the control
current. Therefore the elimination of speed-synchronous currents is exam¬
ined in this chapter. In the first section the cause of the synchronous currents
The
is
speed-synchron ou s
explained.
current
constitutes the
Then various methods for
eliminating
these currents
are
pre¬
adaptive vibration control and the asso¬
ciated theory. It is shown that this method yields good results in eliminating
the synchronous currents. The algorithm requires much memory space.
Therefore a method is presented for reducing the amount of memory space
sented. The main section introduces
needed.
6.1 Fundamentals
Figure
6.1:
Displaced
The axis of inertia of
a
axis
of inertia caused by unbalance
well-balanced rotor
corresponds
axis. The shaft rotates about this axis and the
deflection. The
magnetic bearing
sensors
does not generate any
73
to
the
geometric
do not detect any
speed-synchronous
Chapter
6: Elimination
of Synchronous
Currents
forces.
Fig.
6.1 shows the rotor in
small
be
mass
Am, which is attached
displaced
rotate
single plane.
a
from the
around the
detected. With
no
geometric
axis
to
Unbalance is modeled with
the rotor. It
causes
a
the axis of inertia to
axis. A rotor that is
of inertia.
extra measure a
A
freely supported would
synchronous deflection would be
bearing
must overcome the
unbalance
force to minimize the deflection. The unbalance force is
square-dependent
on the rotation speed. Because of the linear connection with the bearing
force (Eqn. 1.3) the currents can reach amplitudes so large that the amplifi¬
ers saturate. Unbalance can change during operation for example because of
changing speeds or contamination. Therefore eliminating speed-synchro¬
nous currents requires adaptive methods.
A
big advantage
bearings
of active
is that the rotor
magnetic bearings compared
with conventional
spin around axes other than the geometric axis.
speed-synchronous currents the shaft must turn
can
For the elimination of the
around the axis of inertia.
Orbit after
compensation
Position
Controller
I
Amplifiers/
Bearings
Elimination of
speedI synchronous part
;
7L,
Orbit
Figure
In
Fig.
The
6.2: Elimination
of the speed-synchronous parts of sensor signals
6.2 the
principle of synchronous current
periodic portions of the ns sensor signals are
elimination is illustrated.
eliminated. Therefore the
position controller does not act on the deflections of the rotor
no synchronous currents are generated. The rotor turns now
of inertia. The
same
result
can
any
longer
around its axis
be achieved if the elimination of the
74
and
speed-
Chapter
synchronous parts
is done
on
6. Elimination
of Synchronous Currents
the desired current behind the
position
control¬
ler.
The elimination
in the
bending
which
This
6.2
are
can
can
be done
over
the whole
speed
range of the rotor except
critical (see
Appendix ). The parts of the synchronous current
damping the first bending mode are quenched too.
necessary for
destabilize the system.
Approaches for cancelling speed-synchronous
cur¬
rents
6.2.1
Tracking
notch filter
sin(Qt)
sin(Qt)
I
u(t)
>
TR -Tj
-
lj
*
R
i
cos(Qt)
cos(Clt)
xc(t)
Controller
Notch filter
+
Amplifiers/
Bearings
Position
T_
x(t)
—
Sensors
Rotor
Figure 6.3: Principle of the generalized
notch
First
experiments for eliminating speed-synchronous
with
tracking
nately they
1
A
notch
filters1
influence the
hacking
notch filter
is a
filter
currents
in the feedback of the control
stability
filtet with
of the control system
spced-s\nchtonous
75
notch
so
hcqucno
were
made
loop. Unfortu¬
that they can be
Chapter
6: Elimination
of Synchronous
Cm-rents
employed in only a limited speed range [Knos/91]. A modified notch filter
is presented in [Herz/96]. The structure of this filter is shown in Fig. 6.3.
The position signals are collected and processed centrally. Heart of this
notch filter type is a gain matrix AN:
1
AN
=
R
~1
J
,
real with size 2nx2n
s
(6.1)
s
The transfer function of this notch filter is:
N(s)
xr(s)
=
-7-7x(s)
MIMO
=
((v2
(s2I
N(s) represents
a
independent
the choice of
is known
8,
Tp
on
the
as
and
Tj
computation
+
Ü2)
+
s-eTr
+
(6.2)
/
-:
Q2I-eQTi)
system. It is obvious, that N is equal
TR
and
Tj
for
s
=
jQ,.
can now
be chosen
so
that the control
of these three parameters is very
6.2.2
Adaptive
In the
following [k]
and
open-loop
loop
remains stable. The
and is based
complex
to
control.
annotates the k
sample
of
a
discrete
only
same
a
suitable
a
reference
compensation signal u\k] equal
in
signal
amplitude
synchronous component of \[A:J (Fig. 6.4). For that a
filter with adjustable weights u'; and w2 is used [Bets/98],
condition for the reference
frequency
as
the rotation
free. This reference sine
MIMO
signal.
the first
first-order FIR
The
on an
FIR filter
sini— k)
phase
and is
Therefore this method
The idea of the method described here is to generate with
=
zero
generalized notch filter.
accurate model of the
r[k]
to
multiple
input
can
signal
speed
is that it be
of the rotor.
a
sine
Amplitude
signal
and
with the
phase
are
be derived from the motor control and does
multiple output
76
Chapter
therefore not
require
expensive
an
r[k+l)
6: Elimination
of Synchronous
Currents
resolver.
r[k]
=
sinljk
Position
controller
x(t)
Figure 6.4: Principle of the
Bearing
FIR
filter-based
nous current
The filter
mean
1.
weights
are
adjusted
of
one
minimize the
to
=
error
e[k] using the least
wT[k)-r\k],
w[k]
Wj[k]
=
and
r[k]
r[k\
=
Wj[k]
Calculate the
e\k]
=
Adapt
x[k]
This
error s
-
r[k+J]
ignal:
u[k\
the coefficient vector
w[k+1]
1
of speed-synchro¬
axis
Calculate the output of the filter with the actual coefficients:
with
3.
elimination
(LMS) algorithm [MoDti/95J:
square
u[k]
2.
Amplifier/
Rotor
Sensor
algoiithm
=
w[k]
minimizes
+
,ua[A']
the least
by using
r[k]
mean
squate
77
enoi
the momentary
gradient:
Chapter
6: Elimination
of Synchronous Currents
p, is the step size of the
Therefore it influences
sen
The
adaptation algorithm and determines the update
the stability of the LMS algorithm and has to be
rate.
cho¬
appropriately.
of the FIR-based current reduction must be chosen very low.
Under this condition, this method has no influence on the stability of the
update
closed
rate
loop.
It
can
be shown that for
approach behaves
lowing advantages:
ter
-
-
-
6.3
No exact model
to
control system necessary
computing expenditure
speed-synchronous reference
phase can be chosen freely
one
Adaptive
the FIR fil¬
the notch filter. Nevertheless it has the fol¬
of the open-loop
structure, low
Simple
Only
similar
high adaptation frequencies
sine necessary,
amplitude
and
vibration control
Adaptive
open loop
controller
77
P
Position
Amplifiers/
N-
controller
Bearings
+
JL
Sensors
Figure
The
adaptive
6.5: Structure
vibration control
shown that this method
AVC
was
sented. In
Rotor
can
implemented
on
of adaptive
(AVC)
was
vibration control
presented
in
[HoKn/94].
It
was
be used for current and vibration reduction. The
a
[TaKn/96] additional
rig
and
experimental results were pre¬
investigations on the stability and robust-
test
78
Chapter
ness
of the AVC
currents or
vibrations.
The
will be
theory
a more
of Synchronous
Currents
performed.
were
noted this method allows
already
As
6: Elimination
Fig.
eliminating
6.5 shows the
explained
either
configuration
speed-synchronous
for vibration control.
with the vibration reduction, because it allows
general approach.
6.3.1 Mode
of operation
x( t)
=
a0
and
+
a,j
definitions
cos
( Q. t)
+
ß ; sin ( Q t)
(6.3)
i
Eqn.
=
6.3 shows the Fourier
synchronous
Fourier
2
decomposition
coefficients
of
a
signal x\t\. olj
and
ß/
are
the
a.
of x(t). The vector
contains the
x
ß7
information
amplitude
on
and
phase
of the first harmonic of
x.
The AVC
works in this Fourier domain.
(x0)
x(t)
7f
U
Mt
Generation of
synchronous
7>
Fourier
Inverse
Adaptive
transform to
control law
time
coefficients
domain
np/
Position
+
t.
controller
Sensors
Figure 6.6:
Fig.
Amplifiers/ [
Rotor
Functional block
6.6 shows the functional block
diagram
Beanngs
diagram of AVC
of the
adaptive vibration com¬
pensation. x(t)
np performance signals, since the
objective of the AVC is to minimize this vector by applying u(t), the vector
of nc compensation signals. Here, in the configuration of vibration control
is called the vector of
nc
corresponds
to the
number of actuators, and
79
n
to
the number of
sensors.
Chapter
6: Elimination
Therefore
denotes the 2n
x
when
no
synchronous
compensation
adaptation
frequency.
rate
considering
and x0 the 2n
7
performance signals
characteristic,
the controller
can
be
designed
feed¬
unaf¬
without
vibration control
describes
the
loop
Tu+
=
quasi-static
Eqn.
for the
(6.4)
x0
model
of the
rotor's
synchronous
6.4 describes the transfer function of the
speed-synchronous parts.
It is the basis for the
vibration control. The transfer matrix T has the form 2n„
p
quadratic performance
function
J
This
quality
Eqn.
6.5. The derivation of the
u
x
the AVC.
closed control
function
compensation signals,
synchro¬
assumption the AVC can be considered to be of a
and the stability of the closed control loop remains
response. In other words:
A
7 vector of the
x
applied.
x
adaptive
2nc
Fourier coefficients of the
Theory of adaptive
6.4
the
Fourier coeffi¬
of AVC is assumed to be much lower than the controller
fected. Because of this
6.3.2
is
u
synchronous
With this
forward structure
Eqn.
1 vector of the
Fourier coefficients of the nc
vector of the
The
x
Currents
np performance signals,
cients of the
nous
of Synchronous
=
serves as
quality
x
2n
c
,.
criterion:
xTx
(6.5)
criterion must be minimized. For that
performance
Eqn.
6.4 is inserted into
function J for the
performance
leads to:
dJ
=
Setting Eqn.
6.6
equal
uTTTT + xnT
(6.6)
u
du
to zero and
solving
it for
u
results in the
global
con¬
trol law:
u
=
-(r7r)"" r7*,,
80
(6.7)
Chapter 6: Elimination of Synchronous
The
global
control law
depends
x0, the uncontrolled
on
which is undesirable. A recursive form of
Eqn. 6.7
Currents
performance signal,
must be
found. Therefore
the index i is introduced. It stands for the Fourier coefficients determined
from several
(10-20)
recent
between x;+; and X; the
*/
The
quasi-static
mance
quasi-static
+
/-*/
model is
revolutions.
rotor
=
model
can
By taking
the difference
be rewritten:
T(ui I-ut)
(6.8)
+
now
independent
of x0.
Minimizing
the
function / results in the local control law which has the
perfor¬
required
recursive form;
uj
with A the
optimal gain
+
=
1
Ui+Ax.,
(6.9)
matrix:
-1
t
A
A
=
=
-T
,
for
r
(6.10)
T
-(TT)
(6.11)
n,
n
Adaptive
open loop
controller
/.
n,-
n,
Position
Amplifiers/
controller
Bearings
+
Sensors
Figure
Eqn.
6.7:
Rotor
<-
Configuration of AVC for
6.10 describes the
general
uration of current elimination
case.
In most
(Fig. 6.7)
81
current
reduction
applications
the number
n
of
and in the
config¬
performance sig-
Chapter
6: Elimination
nals is
equal
Currents
the number nc of
to
transfer matrix T is
Eqn.
of Synchronous
quadratic
and
Therefore the
compensation signals.
6.10
Eqn.
can
be rewritten which leads to
6.11.
sin(Llt)
Position
controller
Bearing
Figure
Fig. 6.8
6.8: Detailed structure
of synchronous
current
compensation
shows the
complete AVC for current reduction. The synchronous
Fourier coefficients of the compensation signals are updated with a much
lower frequency (fu) than the control frequency. The sample-hold elements
and the discrete
integration imply
The structure of AVC is
quite
this.
similar to the structure of the
generalized
notch filter in
Fig. 6.3, except that the AVC shows an additional discrete
integrator (a) arising from the control law. Like the generalized notch filter
the AVC
uses a
speed-synchronous
convolution with the
rier coefficients. The
ing
Fourier
6.3.3
Reducing
this
and
the number
is denoted
are
used for the
the
subsequent
additions
lead
to
the
in the time domain.
The transfer matrix T is
speed £\
They
performance signals to generate the synchronous Fou¬
multiplications of sine and cosine with the correspond¬
coefficients
compensation signals
sine and cosine.
a
of gain
matrices
function of the
by 1\,
and hence
speed.
82
A;
operating speed.
indicates the
gain
T for
operating
matrix used at
Chapter 6:
A sufficient condition for stable
Eqn.
adaptation
Elimination
is
Currents
6.12. It is derived from
Eqn.
6.8 and 6.9.
a(I
+
ALTk)<l
(6.12)
ü(...) designates the maximum singular value
It
of Synchronous
shown in the
was
matrix
gain
speeds
the
Ak
previous
sections that for discrete
is calculated and saved in
optimal gain
much memory space,
.
matrices
especially
a
look-up
interpolated.
are
for systems
already mentioned,
the
operating
[Kno2/97] presents
adaptation. However,
optimal gain matrix.
several
for
strategies
gain
optimal gain
matrix that allows stable
be slower than with the
table. Between these
This
range. Therefore methods to reduce the number of
examined. As
speeds Qk the optimal
the
procedure requires
over a
wide
speed
matrices have been
matrix is not the
speed
of
only
adaptation
will
the memory usage of the
reducing
AVC:
-
A
single gain
the
-
-
speed
Prefilter
matrix A which
satisfies Eqn. 6.12 for all frequencies
in
range
in combination with
Speed-dependent gain
a
matrix
single A
matrix.
A(Q)
The
prefilter approach, while advantageous for imposing little
computational burden, is a very difficult synthesis problem and is
additional
not exam¬
ined further here.
Single gain
matrix
The solution with
a
single gain
ory space is necessary and
dependence
no
matrix has the
additional
advantage that minimal mem¬
computation associated with speed
is needed.
For the further calculation the inverse of the convergence rate y is intro¬
duced. It is a measure for the adaptation speed. The matrix A has to satisfy
1
A'
is a
the
real matnx ol
si/e n x m
eigenvalues (unequal 0)
oi
The
XXr
smgulat
foi
n
\
=
83
allies
,((
ls
aie
defined
a(X)
=
as
the positive squaie
ma\(\ci<>(X)\)
tools
of
Chapter
the
6: Elimination
following
of Synchronous
condition for each
o(7
+
Currents
Qk [Knol/97]:
Aï\)<y<
(6.13)
/
Using Schur's complement, each of these conditions
linear matrix inequality (EMI) which is a convex
matrix A
can
be transformed to
constraint
on
the
I
AT
+
(6.14)
>0
T
yi
(i+ATky
Speed dependent gain
a
Tk
matrix
strongly
values vary
single gain
range. In this
For on-line
speed-dependent.
impossible
from each other it is may be
matrix A that satisfies
operating speed
the
gain
:l
y7
If the
a
Eqn.
case a
6.14 for all k
matrix
Ak
=
to
find
1,2,...,N covering
has to be found which is
the affine matrix
computational simplicity,
function is chosen here.
A,
Condition 6.14 is
now
A0
+
Qk Aj
extended to:
7+
(J
The task of
+
(A0
gain
+
Eqn.
6.12 holds for all
rate)
this
is
a
Qk.
1.
"X>0" stands for X
+
QkAj) Tk
(6.16)
>0
y7
synthesis
then is to determine
Aq
and
A;
such that
If y is minimized (i.e. maximized convergence
generalized eigenvalue problem (GEVP).
optimization problem
MATLAB [Gahi/95].
vex
(A0
O/,A7)-rA.)
matrix
synthesis
(6.15)
and
being positiv
e
can
be solved
definite, all
84
eigenvalues
using
of X
must
It is
a con¬
the LMI toolbox of
be positive.
6: Elimination
Chapter
6.3.4 Determination
The transfer matrix
on
Tk
of transfer matrix
the real system. In the
following figure
Currents
T
be calculated with
can
of Synchronous
a
model
or
it
can
be identified
the structure for identification is
shown:
sin(Qt)
S/H
—
S/H
"*l
J
<
f
'\*
fu
cos(Qt)
i+
Sensors
Amplifiers/1
Rotor
-
Bearings
i
Figure 6.9:
The structure for
Structure for
identifying Tk
identifying
the
is similar to the
Position
'
controller
transfer
adaptive
matrix
vibration compen¬
sation, except that the adaptation part is missing (compare Fig. 6.9 with
6.8).
This
tem
algorithm sends
with the
rotor
these test vectors
a
set
levitated. In order to
are
chosen to be the
matrix. These vectors
cosine
signals
manner as
to
2np feedforward
of
generate
simplify
2np
multiplied by
are
a
test Fourier vectors to the sys¬
the
required calculations,
columns of
the
2n
2n
identity
speed-synchronous
sine and
series of excitation
a
signals
in
x
exactly
position signals with the reference sine
produces 2np performance Fourier vectors in response to the
These vectors
are
Empirically good
20
and cosine
excitation.
the column vectors of the desired transfer matrix. To miti¬
gate the effects of noise the
over
same
AVC does.
The convolution of the
puted
the
results
test vectors are
are
applied during
a
longer period.
obtained if the Fourier coefficients
cycles.
85
are com¬
Chapter
6: Elimination
The identification
of Synchronous
can
that any
interest,
so
pendent
of the actual
Currents
be effected
gyroscopic
during operation
effects
are
at each
speed Qk
of
taken into account. To be inde¬
unbalance, the response of the system without excita¬
tion must also be measured at each
speed.
The
resulting
vector
from each column vector of the obtained response matrix to
86
is subtracted
yield Tk.
7. Realization and Results
This
chapter
describes the realization of the
complete
active
magnetic spin¬
dle system. The first section deals with the mechanical structure of the sys¬
tem. It
rotor
consists
mainly
of the
description
of the
magnetic bearings
design.
Figure 7.1: Components of the high-speed spindle
tronic
The
spindle
was
supported
with
a
system with both elec¬
s
first electronic system, which is described
in the next section. With this combination, first
experience
the elimination of the bias current and the cancellation of
nous
currents. A
speed
of
currents. That is illustrated
The
and the
knowledge
made with
speed-synchro¬
bearing
reached with very low
90'OOOrpm
by the measurements presented.
was
was
obtained from the system with the first electronics
was
incorporated in the further development. It was possible with a complete
digital control of all bearing axes (positions and currents) and a new printed
circuit technology to reduce the electronics size by 70% compared with the
first system (Fig. 7.1). The miniaturized electronics has the same functional¬
ity as the first prototype. The spindle can be operated with this electronics
up to a speed of 60'OOOrpm.
87
Chapter
7 Realization and Results
Figure
7.2:
Spindle
with miniatw ized control electronics
7.1
Magnetomechanical system
Fig.
7.2 illusüates the assembled
sectional
view
-
-
-
-
-
-
with the
of the magnetomechanical
structure consists
-
spindle
mainly
Housing with
ot the
water
jacket
Combined radial and axial
Elec
bearing
bearing
Displacement
trie
stiuctuie is
sensors
al drive
Auxiliary bearings
88
electronics The
shown
following components
Rotor
Radial
new
m
cross
Fig 7 3 The
Chapter
Hall
7: Realization and Results
sensors
Housing
Auxiliary bearing
Radial
displacement
sensors
Combined
bearing
Permanent magnet
synchronous
motor
Radial bearing
Radial
Water
jacket
/
Axial
displacement
Figure 7.3: Sectional
sensors
Auxiliary bearing
sensor
view
of the magnetomechanical
The rotation axis is vertical to minimize the radial
bined radial and axial
displacement
bearing
is
placed
bearing
system
force. The
com¬
in the upper section of the system.
The electric motor takes up the middle section and the radial
bearing
is
jacket, which surrounds the hous¬
ing. The interior can be evacuated to eliminate aerodynamic losses. Because
of the high speeds an additional protection cover was developed, which is
placed
below. The motor requires
not shown in this
a
water
figure.
89
Chapter 7: Realization
7.1,1
and Results
Bearing system
Figure
Fig.
7.4 shows the
The radial
are
turned
Magnetic bearings
permanent-magnet-biased bearings
displacement
by an angle
advantageous
7.4:
sensors
of 45°
since it is
rents are reduced. The
are
directly
attached
with the
compared
on
expense for control is
bearing
only slightly
The axial
are
end of
tables describe the
placed
for
reasons
Maximum
35
N
Force-displacement factor
kX!
100
kN/m
Force-current factor
Ki
14
N/A
Inductance
Ah
2.3
mH
8,7
250
\im
®iotoi
23
mm
"outei
78
mm
24
mm
air gap
Rotor diameter
Outer diameter
Length
'beat
90
m
%
this.
the measured
F
Magnetic
by
cur¬
electrical, magnetic and
bearing
force
This is
bearing
increased
bearings. Fig. 7.5 illustrates
force-displacement characteristics.
Radial
axes.
They
of space at the lower
mechanical dimensions of the
force-current and
the stators.
and disturbances of the
space-saving
displacement sensors
the rotor. The following
before installation.
Chapter
Combined
bearing
7: Realization and Results
Radial
Axial
Maximum fo rce
F
50
120
N
Force-displacement factor
K
120
300
kN/m
*/
20
50
N/A
2.3
3.8
mH
K,a
250
300
\xm
^rotor
30
38
Force-current
factor
Inductance
j
^wr,a
air gap
Magnetic
Rotor diameter
Outer diameter
"'outer
Length
33
t\
''bearing
mm
78
mm
24
mm
i
30-1
25 -i
20-
o
o
15
10
5
-
-|
J
0
*
0.5
Current
20
15
o
10
1.5
1
[AJ
n
J
-
o
I-l
Ph
5
20
40
60
80
Displacement [ umj
Figure
The measured forces
losses
were
7.5: Characteristics
are
lower than
underestimated,
7.3.4 show. The two radial
as
of all
expected.
three
bearings
In the calculations the iron
the power measurements
bearings
are
91
designed
with the
presented
same
in Sect.
parameters,
Chapter
7: Realization and Results
nevertheless the lower
bearing shows weaker forces. This difference arises
alloys used for the back iron and the axial bearing unit.
from the different
Auxiliary bearing
The rotor
tem
vent
be
can
suspended
by auxiliary bearings,
of
case
which
are
failure of the
a
magnetic bearing
conventional ball
bearings. They
sys¬
pre¬
destruction of the rotor. The
does not touch the
turn
in
with it. After
bearing clearance is 100\im. The rotor
auxiliary bearings during normal operation; they do not
an
emergency the ball
bearings
have to be
replaced.
7.7.2 Motor
A
synchronous
motor
with permanent-magnet excitation drives the
Figure
system. The
ported by
a
stator has
non-magnetic
carrier.
iron. The iron stack has the
losses caused
by
slots
are
large (10mm) compared to
ing forces can be ignored.
The motor has
50mm. It
a
length
consumes a
7.6: Electric motor
so-called
a
spindle
shape
air-gap winding. The winding is
This assembly is surrounded by the
of
a
shell and has
eliminated. Since the
the air gap of the
no
back
slots, therefore the
magnetic air
bearings,
sup¬
the
gap is very
magnetic
disturb¬
of 80mm and the outer diameter of the stator is
power of 2kW at
a
speed
of 120
'OOOrpm.
Chapter
7: Realization and Results
7.13 Rotor
Figure
7.7: Rotor
The assembled rotor is illustrated in
maximal diameter
lamination at the
rings
Fig.
(axial bearing disk)
bearing
7.7. Its
length
is
193mm, and the
is 38mm. The rotor has
a
high-speed
sections to minimize the iron losses. Two alumi¬
attached to the shaft
the
bearing
units. The
diametrically
magnetized permanent magnet shell of the motor takes the middle section. It
is enclosed by a carbon fibre bandage because of the enormous centrifugal
forces at high speeds. Carbon-fibre is light and has a large tangential tensile
num
are
strength (2968N/mm-').
near
A second
diametrically magnetized permanent mag¬
(in the figure left) of the rotor. Its field is
measured with two orthogonal Hall probes generating a sine and a cosine
signal. These speed-synchronous signals are used for the AVC.
net
is attached to the upper end
Shrink fits
As
already
mentioned the rotor
120'000rpm,
and the
most of the material
designed for a maximum speed of
rotor materials are subjected to severe loads. Therefore
is shrunk onto the shaft. Especially the design of the
was
shrink fits for the lamination is
difficult, because of the high specific weight
of the sheets and their relatively low tensile strength. Therefore the layout of
the shrink fits
was
done
using
calculations of stress and
the laminations
stress is not
a
finite element method.
displacement
((c), (d)). The task
higher
than the
was
yield point
93
of the aluminum
to
Fig.
7.8 shows the
ring ((a), (b))
define the diameters
so
and
that the
and that the inner diameter of the
Chapter
ring
7 Realization and Results
has still contact with the shaft at
von
S Mises
120'00rpm.
Mises Stress
(a)
[MPa]
Wve Ont 7^%)
D.
U
Displacement [mm]
Ma/itud^
-
+M04P 0
•
+13<Çe c
-
+1
-
+1 JMG 0"
j
Jp 02
-
+1 o<3e 02
Bl-
+1311e o2
jjll
+1 ^'p-Og
Bi +1 31 ^e 0°
§Hr~ +1 30?P 02
-
HHH"
+1 2S96-0'
+1 n77e 02
+1 2b4e-02
+1",1e02
von
Mises Strsss
Displacement |mm]
[MPa]
(C)
(d)l
a
O.
Figure
7.8: Stress and
displacement
calculation of
sheet
94
sensor
ring and rotor
Chapter
Modal
7: Realization and Results
analysis
(c)
10.9 kHz
5
9
m
m
7'
t
Figure 7.9: Measured modes of vibration (suspension free-free, positions of
radial bearing sections: 1 and 8)
with finite element method the rotor
Again
lated.
rotor
was
Unfortunately
showed
expected
a
large
at a
the modal
analysis
frequency
shell, which is glued
additional
pens in
points
mass,
reality.
of
1.7kHz, but it
onto
was
done
on
Fig.
7.9
(a) the
3 and 5, which is the
can
be
occurs
no
explained by
no
bending
mode
the permanent mag¬
was
modeled
as
Exactly this hap¬
large bending between the
of the permanent magnet shell.
95
simu¬
the assembled
stiffness to the rotor.
rotor shows
position
was
in the measurement at
the shaft. In the calculations it
which contributes
In
which
behavior
variation from the calculation. The first
2.4kHz (Fig. 7.9 (a)). The variation
net
dynamics
Chapter
7: Realization and Results
7.2 First control electronics
The first electronics, which is discussed in this
section, actually served
as
prototype. To understand the advantages and the disadvantage of the final
electronic system it shall still be described here.
7.2.7 Structure
The control system realized is shown below in
-^
7.10. The hardware
Fig.
con-
^
MCDs]
\
/.
j
,
[j
I
]
/
\
{
_
-
.\
A.
TMS
/'
,
/
i
—r-
7
320C50
A
D
Processor board
Analog
Sensor
current
evaluation
control
board
/1
\ /
H
H
Power electronics board
w.
xi<yi
/.
Wv2
Figure
7.10:
sists of three boards:
a
System
.ï->, Y ?
overview
sensor
power electronics board. In the
of first
electronic system
evaluation board,
following,
in detail
96
a
processor board and
these boards
are
discussed
a
more
Chapter
7 Realization and Results
Sensor evaluation board
This card includes the evaluation and the activation for 5
sors
for
and 2 Hall
displacement
probes
eddy-current
sen¬
for magnetic field measurements of
the permanent magnet attached to the upper end of the rotor. The excitation
frequency
a
of the
eddy-current
sensors is
312.5kHz. The electronics pioduces
displacement-proportional voltage signal
sensor
which
electronics derives from the
are
speed-dependent
Figure
with
orthogonal
a
bandwidth of 3.7kHz. The
Hall
and have the form of
probes voltage signals,
and
a sine
7.11: Sensor evaluation board (100
X
a cosine.
100mm)
Processor board
The
pled
sensor
and
signals
digitized by
the TMS320C50,
fed to the piocessor board
aie
a
7
12) They
are sam¬
fast A-D converter. Heart of the whole electronics is
a
digital signal
piocessoi
clocked with 40MHz and woiks with
width of 16bits. Since
(Fig.
a
large
of Texas
fixed-pomt
volatile memory
is
Instruments.
arithmetic with
integrated,
a
It
is
data
the board exter¬
nally needs only an EEPROM (electrically erasable and programmable read
only meraoiy). A serial interlace allows communication with a host com¬
puter. The processor board geneiates the CPU and PWM clock, which
synchiomzed
to minimize
desired currents with
a
are
chstoitions. The piocessoi boaid generates the
fast D-A converter.
97
Chaptet
7 Realization and Results
Figure
7.12
Processor card
(J 00
x
100mm)
Power board
The power boaid shown
Fig
7 13
responsible tot generating the bear¬
ing currents Therefore 5 current amphfieis are leahzed on it. Each ampli¬
fier has a full bridge, which is driven by a PWM signal with a frequency of
65kHz The d c link voltage is 48V and the lull bridge can geneiate a maxi¬
mum
log
in
continuous anient of ±2 5A
PI
controller, which
is
is
The actual current
synchronized
with the PWM
The power board geneiates all necessaiy
analog
for the
sensor
Figure
is
fed back to
generation.
supply voltages
±12V and 5V
evaluation and 5V foi the digital electronics.
7 13
Pern
et
elec tionic
98
s
board ( 100
x
an ana¬
160mm )
Chapter
7: Realization and Results
Software
7.2.2
Software
on
host computer
The serial interface allows communication between the control electronics
and
host PC. The
a
(MCDS)" is
a
called
so
software
on
"Magnetic
the PC for the
Control and
development
Development
Studio
of the DSP software
and for the controller
design. The controller can be composed of a PID con¬
troller, low-pass and high-pass filters, notch filters etc. The contents of vari¬
ables can be monitored and changed via this software.
Realization ofposition controllers and AVC
The DSP software is
programmed in assembler to prevent unnecessary
overhead due to non-optimal compilers. As in the previous section men¬
tioned, the DSP is a 16bit fixed-point processor. Unfortunately fixed-point
means some more programming effort, for
example for scaling or divisions.
The
position
sist of
a
controllers
are
PID controller with
called with
lead-lag
frequency
a
elements
of 10.4kHz-
reproducing
They
con¬
the D part and
a
low pass filter.
The
adaptive vibration control was realized according to Sect. 6.3. It is
applied to all radial axes. Therefore np nc 4 (number of performance sig¬
=
nals
=
size 8
number of
x
8. The
compensation signals
multiplication
=
=
4) and the gain matrix A is of the
of A with the
performance
vector
x
consists
therefore of 64 additions and
multiplications. Although the DSP executes an
addition and a multiplication in a single clock cycle, this step is still quite
time-consuming. The determination of the Fourier coefficients of the perfor¬
mance
signals
and the calculation of the
every controller
cients of the
Forty
out
of the
adaptive
was
is
speed.
updated
compensation.
measurements
noted that the
variation with
executed
with
only 40Hz.
measured between 10Hz and 2kHz for the
vibration
Therefore the
levitated. It
were
are
whereas m, the vector of the Fourier coeffi¬
compensation signals,
transfer matrices
scopic.
est
cycle (10.4kHz)
compensation signals
The rotor is not very gyro¬
done at standstill with the rotor
were
diagonal elements
Therefore
Ak
Ak=A0
+
was
of
Tk
underwent the great¬
chosen to be:
i\a,l
99
lay¬
(7.1)
Chapter
7: Realization and Results
This form has the
ments,
only
a
advantage
single
as
the two
(A0
=
&fAi)
+
4n2
tions:
Additions:
A0'XL
+
4n"-2n
2n
Multiplica¬
4n2
tions:
The GEVP
Cljl
(£>k
was
Total:
uk
8n2
2n
8n2
4n2
+
Xk
Total:
uk
4n2+2n
2n
1
optimization
+
xk
4n"-2n
Multiplica¬
=
with 64 ele¬
tables show:
following
4n2
Additions:
uk+i
A;
number has to be saved. Besides this it reduces the
computing expenditure
uk+l
that instead of the 8x8 matrix
4rt*+2n+l
2n
carried out in three
speed
ranges: 50-300Hz,
300-600Hz and 600-2000Hz. Three linear matrix functions
(An
and
aj)
were
found, which allow stable adaptation between 50Hz and 2kHz.
7.2.3 First results
Current controller
The measured transfer function of the
illustrated in
Fig.
7.14. It has
low-pass
analog
current controller
characteristics with
a
(i/ij)
is
bandwidth of
3.5kHz,
Closed-loop
behavior
7.15 shows the transfer function of the
Fig.
excited
first
on
the
bearing
rigid bending
not
bending
show
a
(parallel mode,
second rigid bending
mode is
distinct
position
mode
about 50Hz, and the
first
current and the
now at
1800Hz, It
peak.
100
closed-loop system.
was
taken
as
It
was
response. The
App. A.2) is at a frequency of
mode (tilting mode) at 150Hz. The
is well damped, as the curve does
see
Chapter
10
i
r
7 Realization and Results
-i
r
„fv.y4f'iw*
_j
-10
500
1000
1500
,
2000
i
i
2500
3000
3500
4000
4500
5000
3000
3500
4000
4500
5000
i_
Frequency [Hz]
500
1000
1500
2000
2500
Frequency [Hz]
Figure 7.14: Transfer function (icixj/iiX]) of the power amplifier (first elec¬
tronics)
500
1000
1500
2000
Frequency [Hz]
Figure 715: Measured transfer function ( W/^J °ttne closed loop of the
spindle s\ stem with the first electionics
101
Chapter
7: Realization and Results
Run-up
Fig.
7.16
(top)
shows the
speed-synchronous
control
current
of the upper
1.4
1.2
Ift
•
1
Ph
<
Ci
Without AVC
With
0.8
0.6
in
3
U
0.4
0.2
-
^flh-M—*-»"*•
500
1000
Rotation
500
speed [Hz]
pi
*-
Without AVC
—
With AVC
o
d
Ph
t—r-r-T^^*
Q
500
0
1000
Rotation
Figure
radial
speed [Hz ]
7.16: First electronics: Synchronous control
during
ment
bearing during
run-up
current
run-up with AVC turned off and
speed
about 10Hz. Above and up to 1.3kHz it is turned
is
and
displace¬
(upper bearing)
turned off below 30Hz since the resolution of the
only
.500
on
on.
The AVC is
determination is
and the
synchro¬
less eliminated. Above 1300Hz
(marked with (a))
the adaptation is frozen because of the bending critical (see Sect. 6.1). The
nous current
more or
compensation signals
are
still
applied
but their Fourier coefficients
are
not
updated.
Fig.
7.16 below shows the
displacement during
102
run-up with and without
Chapter
AVC turned
turned
cally
on.
7: Realization and Results
Note that the rotor turns around its axis of inertia with AVC
on.
Therefore the
position
is minimized since the rotor is mechani¬
well balanced. The reduction of current is thus also
accompanied by
a
reduction in rotor vibration.
7.3 Miniaturized control electronics
The
used
processor
in
the
miniaturized
TMS320F240. The structure of this
ferent to the
ware
design.
developed
new
control
DSP controller is
previously presented TMS320C50,
and it
Because of this it is first discussed
hardware is
electronics
completely
requires
more
is
a new
the
dif¬
hard¬
in detail before the
explained.
7.3.1 The TMS320F240
Flash
3
fa
EEPROM
bu1
U
Oh
O
,F?
G
i-,
16
e.s
w
16-'
S-H
Ü
g
'
<&
JS
<+-i
<D
G
*i'G Oh il
00
16
i
G
Digital
signal
16
%£%,
12
processor
ÖfJ
Interrupt
A-D
A~
'
Start'conversion
D
conversion
module
Figure
-j-^-^-^-j
7.17: Block
diagram of FMS320F240
Fig.
7.17 sketches the structure of the TMS320F240. This device is spe¬
cially designed for mechatronical applications. Core of the chip is a 16bit
fixed-point DSP,
which is driven
by
a
20MHz clock.
103
Chapter 7: Realization and Results
In addition to the DSP
the
ory)
following
can
integrated
the
same
chip:
independent. Power
directly without any glue logic and with no com¬
for the
burden
on
are
DSP.
2 fast lObitA-D converters
They
can
log inputs
by
DSP
A
by
be started
the DSP
loading
-
are
module
be driven
can
putational
-
(Static Random Access Mem¬
SRAM
an
generate 12 PWM signals of which 9
devices
-
with
controller functions
generation
-A PWM
It
core
an
the PWM
core). 2 multiplexers
each. The termination
can
of the
select 1
signal
conversion is
without
out
of 8
signalled
ana¬
to
the
interrupt.
non-volatile memory
large
The EEPROM has
a
read and executed
during object
A serial
short
time
access
(1 cycle). Program data
be
can
time directly from the EEPROM.
interface
Communication with host is
7.3.2 Structure
trollers for all five
significantly. Fig.
without
axes.
glue logic.
electronics
implementing digitally position
Therefore the
outlay
on
and current
hardware
can
area
con¬
be reduced
7.18 shows the system overview of the miniaturized
trol electronics. Since the board
reasons
possible
of the miniaturized control
The TMS320F240 allows
mized for
generation module (again
con¬
of the power electronics must be mini¬
of costs, the current measurement
implemented
was
at
a
separate print. Therefore the system consists of four boards. All PCBs have
the size 60
I.
x
80mm and
are
presented
in the
following
sections.
Module", in this
The official
designation
exclusively
lor the PWM generation Theietoic the author intioduccd the
loi this unit
is
"Event
Managei
Gcneiation Module"
104
application it
designation
is
used
'PWM
Chapter
7: Realization and Results
Processor board
TMS320F240
s
MCDS
.2
PWM control
signals
Central
4}
3
S T3
<U
©
processing
j
unit
I
10
T
Current sensing
Sensor
board
evaluation
_Â
board
>
k
L
k
H V#v
i
03
O
A/-A/
w;
O
S
7.
o
'
-1
o
A2-A2
xi>yi
i
O
Figure
«
7.18:
System
overview
<
of miniaturized magnetic bearing
control
electronics
Sensor evaluation board
The
sensor
evaluation board
(Fig. 7.19)
evaluation of the first electronics.
evaluation for 5
the
sensor
eddy-current
signals
result in
processor. It is driven
first
by
electronics, which is
ware
and cost
Again
sensors
fulfils the
same
tasks
as
the
sensor
it consists of the excitation and the
and 2 Hall
voltage signals,
which
probes.
are
The evaluation of
applied directly
to the
single supply voltage of 5V (compared with the
driven by 5V7 and ±12V) to reduce again the hard¬
a
expenditure.
105
Chapter
7: Realization and Results
Figure 7.19:
Sensor evaluation
(60
x
80mm)
Processor board
Fig.
7.20 shows the processor board with the TMS320F240. All 12
signals (5 positions,
5 currents and 2 field measurements)
corresponding inputs
of the processor and
are
are
conditioned
by
analog
applied
to
the
it. The serial
interface of the processor allows communication with the host PC. An addi¬
tional external SRAM is available to allow flexible
ports and
a
four channel D-A converter
Figure
can
7.20: Processor board
106
programming. Digital
monitoring.
be used for
(60
x
80mm)
Chapter
7. Realization and Results
Power electronics board
The heat
dissipation
Figure
in
power electronic system is
a
7.21:
Principle of the Thermagon
usually
very space
con-
PCB system
suming. Large cooling attachments are necessary if the power devices are
not cooled actively by ventilators or water. Therefore a new printed circuit
board (PCB) technology was used, which allows forgoing the cooling body.
Fig.
In
7.21 the
principle
Under the circuit
for
high voltages
15 times
copper
higher
plate,
layer (copper)
is
than the conventional
that
serves
directly
guaranteed.
is
Figure
1
lies the
Thermagon system is sketched.
T-preg dielectric. Besides insulation
it has the property of very
system requires the
heat sink
of the so-called
use
high
FR41.
simultaneously
thermal
conductivity,
about
This structure is
as carrier
and
as
pressed onto a
cooling body. This
of surface-mounted power devices (SMD). Their
soldered with the PCB
so
that
7.22: Power electronics boaid
The thernial conductmtv ol FR4
is
0 3\V/m
Tcompaied
107
to
optimal
(60
x
power
80mm)
T pies with ^W/m°C
dissipation
Chapter
Fig.
7. Realization and Results
7.22 shows the realization of the power board. It is
supplied
and consists of 5 full
bridges. They
of the DSP and
able to generate currents of ±5A at
are
are
driven
directly by
with 40V
the PWM outputs
a
frequency
of
36.4kHz,
Current measurement board
The current measurements
on
this board generates
applied
a
current-proportional voltage signal,
to the processor. All necessary
tal) of the electronic system
Figure
7.3.3
done with shunt resistances. The evaluation
are
are
7.23: Current
supply voltages (5V analog
generated on this board.
measurement
board
(60
x
which
and
digi¬
80mm)
Software concept
already mentioned in the description of the hardware, the system
only two lObit A-D converters. 5 actual current values, 5 actual position
As
nals and 2 Hall
generation
sensor
signals
module starts the
The end of the A-D
Fig.
is
must
be
digitized during
conversion
conversion is
troller software consists of
two
sig¬
operation. The PWM
without
signalled
7.24 the software concept and the
has
to
influencing the DSP core.
the DSP by an interrupt. In
rough timing are presented. The con¬
parts: a main loop and an interrupt routine.
108
Chapter
A-t>
7: Realization and Results
Digital signal
conversion
module
Interrupt
0
processor-
Main
routine
X?J?
_
loop
K
PC
*2
Tp
PWM
hphl
v_-v--
2T,PWM-
_
G
,_*.
PC
72
A2-A;
O
a
çc
G
O
CC
-t—>
a
3T
PWM
Com
*-H
o>
/-,z
G
tu
ÖD
CC
PC
Oh
4T
PWM
z
>>
*/,\/
X>
PC
1-4
C3
x}
G
O
H
^7>7v7
OD
i
/-1/-1
>
G
O
o
Q
PC
ft
i
A2'A2 \
CC
AVC
L,H
m
ce
Inter
]
,
—
».
8TPWM
AVC
\->,v->
L
t
».
_
I
Figure
7.24:
Software concept
(CC: Current Control PC: Position Control
munication, AVC:
Adaptive
109
I:
Interpolation,
Vibration
Control)
Corn: Com¬
Chapter
7: Realization and Results
Interrupt routine
The
interrupt
routine is started every
A-D conversion is
an
Depending
executed
on
or
control
rent
position
loop
is
Hall
cur¬
simple PI controller within the interrupt
controller frequency of 9.1kHz. The position
as a
cycle (2.3kHz).
rate of the
The
synchronized.
synchronized every 8th interrupt. This results
frequency of 4.6kHz. The Hall probes are digitized
every 16
sampling
the
valid data of
programmed as a state machine.
appropriate current controllers are
control in the main
current
a
the
as soon as
axis is
an
tion controller
polated
signals
implemented
was
routine. This allows
control of
available. It is
the converted
the
274\is (36.4Hz)
in
a
posi¬
and inter¬
In the later sections it will be shown that
signals
is too low for
a
stable AVC
over
the
whole
operating speed range. But it is impossible to raise this frequency
without lowering the frequency of the current control and therefore the cur¬
rent
quality.
Before
A-D conversion
Main
returning
loop
the
input signals
of the next
selected.
are
controllers and the vibration
position
in the main
loop.
The
the MCDS software
programmed
so
layout
on
that
of the first electronics.
ous
the main
loop
All five
are
to
filters. The actual
of the
the host PC
They
position controllers can
(Sect. 7.2.2). Therefore
allow the
they
can
be
compensation
same
functionality
composed
implementation
of
consists of
a
as
were
executed
be done
using
the controllers
the controllers
PID controller and vari¬
only
a
single
PID control¬
ler.
Adaptive
Sixty
Vibration Control
transfer matrices have been measured between 5Hz and 1.2kHz. It
already indicated during
possible above a speed of
was
these measurements that
is
about 700Hz. The
sampling
rate
(2.3kHz) of the Hall
sensor
no
reason
signals. Again
stable
for this is the low
the GEVP
ried out and four
allow stable
adaptation
was car¬
speed-dependent gain matrices have been found. They
adaptation in the speed ranges: 40-220Hz, 220-320Hz, 320-
400Hz and 400-550Hz. Above 550Hz, the adaptation is frozen.
110
Chapter
7 Realization and Results
7.3.4 Measurements and results with the miniaturized control
electronics
!~V
1500
Frequency [Hz]
200
100
1000
1S00
Frequency [Hz]
Figure 7.25:
Measurement
of the transfer func
troller
twn
of the digital
current con¬
(icxl/idlI)
-70
80
C
90
100
110
120
200
600
Freqienoy
400
H7
600
1000
Frequency [Hzj
Figure 7.26:
Measured
transfer function of the closed control loop (^/ifjx2)
Chapter
7 Realization and Results
Current
amplifiers
digital control
mentioned in the
already
As
with
amplifiers have a control frequency
36.4kHz. Fig. 7.25 shows the measured
controller. It has
Closed
loop
a
cur¬
of
frequency
(icixilicxj)
of the
a
behavior
(X]licjxj)
of the closed control
Fig. 7.26. Both rigid body modes
magnetic bearing system with the first
mode is at 100Hz and the
vibration control
Fig. 7.27
shows the
bearing
are
loop
is pre¬
shifted
control
slightly compared to
electronics. The parallel
mode at about 300Hz.
tilting
Adaptive
of the upper radial
current
300Hz with the AVC turned off and
bearing
at
a
of
speed
on.
-
Ml
A
00ms
I
40 0mA
Figure
7.27: Measured
7.28 the run-up
tronics is
presented.
bearing
on
practically complete
current is reduced by
of
a
speed
m
with the miniaturized elec¬
a
similar result
ripple
synchronous
is within this range.
about 0.3A
112
speed-synchronous
the range of the first
Note that the current
current
unit
as
with the
of 50 and 550Hz the AVC results
than 90%
is
off (upper figure)
corresponds to 0.5A)
60'OOOrpm
elimination of the
more
2000
04 OS 41
with AVC turned
The measurements show
first electronics. Between
-10 omA
r
SJan
current
speed
A
n SO'b
(figure below) (1
to a
OOms
2000
04 05 06
and AVC turned
Fig.
Ml
*v
SJan
H S0\
In
PWM
transfer function
sented in
!
a
controlled
bandwidth of about 2.8kHz.
The measured transfer function
the
digitally
of 9.1kHz and
rent
current
the
previous section,
(Fig 7.27)
currents.
bending
in
a
The
mode.
and that the remaining
Chapter
7: Realization and Results
1.4
1.2
1
a
Ph
<
0.8
a
0.6
~
u
Rotation
1000
800
600
400
200
speed [Hz]
Without AVC
With AVC
Q
10
Rotation
Figure 7.28: Synchronous control
speed [fb]
tem
under different
power
operating conditions.
with controlled
switching
consumption
significant.
complete
sys¬
It is noticeable that the power
con¬
table shows the total power
following
indicates
run-up
consumption
The
sumption
displacement during
(upper radial bearing
and
current
with miniaturized control electronics
Total power
1000
800
600
400
200
bearing
consumption
equal
current
losses and iron losses of
for levitated rotor is
The total power
only
consumption
113
to
totally
of the
zero
is
14.4W.
This
8.8W. The additional
1.2W. The effect of the AVC is
at a
speed
of 300Hz with AVC
Chapter
7: Realization and Results
turned off is 20W. The AVC reduces the power
to
the value at
zero
Power
again
Consumed Power
of operation
[W]
3.9
amplifiers disabled
amplifiers enabled, bearing system
not
connected
Power
4.4W
speed.
State
Power
consumption by
amplifiers enabled,
current
Rotor levitated,
Rotor levitated,
controlled
=0
no
speed=300Hz,
114
JA
A
15.6
speed=0Hz
speed=300Hz,
Rotor levitated,
bearing
5.6
AVC
AVC
20
15.6
8. Conclusion and Outlook
This work derives the fundamentals for the
design of integrated and reason¬
shown by examination of already
magnetic bearing systems. It is
existing work that a further integration of
able active
the greatest cost reduction. For
bearing
a
an
active
magnetic bearing
control flux. In conventional
erated
current
technology,
the
magnetic
current must be minimized.
The flux in
and
this, with
the power electronics results in
by
electric currents. The
discussed in this work is
a
premagnetization
flux
both of these
gen¬
magnetic bearings
premagnetization
flux of the two
are
bearings
generated by permanent magnets. They
need any electric bias current.
homopolar.
consists of
The bias flux of both
It is shown that the iron losses in the rotor
are
do not
bearing types
is
smaller than with
heteropolar configuration. The first bearing type is a radial bearing. It has
a shorter length than conventional homopolar permanent-magnet-biased
bearings. The second type is a combination of a radial and an axial bearing.
The two units are excited by the same permanent-magnet flux. This is again
space-saving and allows a short rotor, which results in higher bending criticals. The experimentally determined bearing forces were smaller than those
a
calculated, since the iron losses
The
speed-synchronous
were
underestimated.
currents caused
by
unbalance represent the
parts of the control currents. Different methods to reduce the
currents are thus
examined.
of inertia instead of its
They
all result in the rotor
geometrical
turning
largest
synchronous
about its axis
axis. An
appropriate technique is the
well-known Adaptive Vibration Control (AVC). It has an open loop struc¬
ture and therefore does not influence the stability of the closed loop system.
Its theory is reasonable and the layout is very simple. Nevertheless its
implementation consumes much memory space, especially for systems
which have a large operating speed range. This work presents the imple¬
mentation of
an
nous currents
and
extended AVC, which allows
uses
only
a
reducing
the
speed-synchro¬
minimal amount of memory space.
These elements have been tested
high-speed spindle with a first control
electronics. Its current control is analog. With this system a speed of
120 'OOOrpm was reached with speed-synchronous control currents, which
on a
775
Chapter
always
were
The
8: Conclusion and Outlook
smaller than the current
knowledge, gained
from this system led to
tronics. The current control
currents and a new
bearing
power electronics
was
ripple.
was
realized
a
a
electronics
The
the reduced
digitally. Through
mounting technology
realized.
miniaturized control elec¬
further
volume is
about 70%
smaller than the volume of the first electronics. With this electronics
tion
speed
only
up to
of the
integration
rota¬
a
60'000rpm was reached. Unfortunately the AVC functions
42 'OOOrpm, since the sampling rate of the reference signals is too
of
small.
Measurements with the miniaturized control electronics showed that the
consumption during rotation with AVC remains at
0Hz. Another fact emerging from these measurements
the
total power
level
at
as
iron losses
are
the
of the total power
largest part
material has to be chosen very
These
findings
may
be
consumption.
same
is that the
Therefore the
carefully.
embodied
in
magnetic bearing system
further
designs:
-
Because
of the AVC
the
small. The maximum
This
-
can
bearing forces during operation
force
be taken into
occurs
employed,
axes can
very
start-up and is only
during
account for the winding layout.
With the DSP controller
position of all bearing
are
a
digital
brief
control
be realized. This is
of the current and
a first step towards
the
complete integration of the signal processing electronics. The
required sampling rates were not reached, but fortunately it is only a
question of (a short) time till faster devices with more functionality are
available.
-
By reducing
integrated.
ceivably
the
bearing
currents
the power electronics
can
For small systems the whole control electronics
be accommodated in
a
multichip
116
module
(MCM).
be further
might
con¬
Appendix: Modelling
This
chapter
discusses the
modelling
of
a
magnetic bearing system
the rotor. In the first section the model of
ing
includ¬
bearing for a single axis is
derived. It is shown that large bearings especially have a similar transfer
function to spring-damper systems. The next sections extend the rotor
model step by step. It is first considered to be rigid, then gyroscopic cou¬
pling is introduced and in the last section the necessary theory for modelling
an elastic rotor is presented.
a
Single-axis magnetic bearing
A.l
8.1 shows the structure of
Fig.
xd
^\
-Amplifier— k,
Controller
A
-f.
which is
'c
Position
_
a mass m,
+
->
supported by
w
f
\
x
A—(//mj-
a
single
axis
•
\fd
Figure 8.1:
magnetic bearing.
grator and
a
Structure
The
of a single-axis magnetic bearing
subsystem
feedback with the
actuator-mass
gain Av
This
the system. A PD
tee
a
stable
(proportional differential)
suspension at least:
consists of
negative
a
double inte¬
stiffness destabilizes
controller is
required
to guaran¬
(*./;
To eliminate static control
forces, the controller
can
errors,
be extended with
is considered to be ideal. It has
as
long
as
which arise for
a
gain
an
example from static
integrating part. The amplifier
of 1. This
the mechanical time constants
777
are
assumption
is
permissible
much larser than the electrical
Appendix : Modelling
time constants. The disturbance transfer function
Gn
=
can
thus be calculated:
7
£
ms2
fd
Kdpkts
+
Kppki
+
-
(8.2)
fcx
Eqn.
8.2 is similar to the disturbance transfer function of
by
spring-damper system [Gemp/97]:
a
(8.3)
=
ms"
damping
is calculated
+
ds
+
the
proportional part Kpp
d
In small
bearings
=
k
by multiplying
controller with the force-current factor
adjusted by
suspended
1
Gj
The
a mass
k
Kdp *,.
the differential part
The
kj.
bearing
Kjp
stiffness
of the
can
be
of the controller:
~
Kpp kj
'
-
kx
(8.4)
with small mechanical time constants the electrical time
constants must be taken into consideration. The
amplifier
can
thus
no
longer
be assumed to be ideal.
Current controller
Bearing
'
-^O+
>
K,
pc
+
Kic
^h
Figure 8.2:
Fig.
8.2 shows the model of
ler and the
bearing
coil
a
\hlA\
Structure
real
are
model has two
1
erf amplifier
amplifier.
coil. PI controllers
,
It consists of
a
current control¬
often used for current control. The
integrators and thus shows a behavior similar to a secondorder-low pass filter. Especially the phase loss of the amplifier causes stabil¬
ity problems in high-speed spindle systems.
775
Appendix
A.2 Mechanical model
A.2.1 The
rigid
of the
:
Modelling
rotor
rotor
A
A"
70,Y,n
Jh<pt
x~z
plane
AX]
y-z
plane
Figure
8.3:
Rigid
rotor in
The coordinate system shown in
following
figure
figure
x-z
and y-z
parallel
shows the model of the
below in the y-z
planes
8.3 contains the definitions used in the
considerations. The c-axis is
rotor. The middle
and the
Fig.
the
plane.
to
the rotation axis of the
rigid
rotor
in the
x-z
plane
The model of the rotor consists of
119
a
Appendix
:
Modelling
shaft, which has
no mass.
All the
mass
attached to the rotor at the center of
allows
12
and
is concentrated in
gravity.
This form
l2
denote the distances from the center of
magnetic bearing forces Fsj
and
Fs2
rotation axis.
The
Jj
position
gravity
describes the moments of
of the rotor is described
and the rotation
ered to be small. To
and
Fs2
ing
on
are now
t;
=
=
simplify
on
Fy
a
position (x,
angles \
and y. The
force F
Tx
=
are
and
(FV,FV)
a
y,
Jq
parallel to the
orthogonal to
z)
of the center
and (p
are
bearing
consid¬
forces
Fs}
torque T (TvTf)
act¬
gravity:
FSxl
+
-l]Fsx]
~
and is
inertia, which
the
bearing posi¬
the end of the shaft.
gravity
Fsx2
+
Fx
1
Fsyl
llFsrJ
+
l2Fsx2
_/
-
Fsy2
Fx
/
following equations
(Eqn. 8.7 and 8.8) and in
/
_;
equations
describe the
the y-z
l
i/cp
my
Jjl
=
equations
=
=
=
F\
/
F
1
7
-/
(8.5)
0
sx2
tsyl
2_
of motion
plane (Eqn.
mx
Fsx,
=
J2Fsx2
With these transformations the
1
=
_
The
chosen since it
the
to
the further calculations the
transformed to
the center of
Fx
angles £,,q>
by
gravity
act
is the moment of inertia related to the center of
of
was
explaining the gyroscopic coupling (A.2.2).
tions. The
Jq.
disk, which is
a
(8.6)
Fsy2
can
be
easily
of motion in the
derived.
x-z
plane
8.9 and 8.10)
(8.7)
rv
(8.8)
Fy
(8.9)
Tx
(8.10)
120
Appendix
With these
equations
of motion and the
$
1.
bearing
2.
mode
Figure
8.4:
Rigid body
rigid body mode,
the
tilting mode,
of the
i-
rigid body
mode
modes
rigid body modes can be estimated. They are shown in Fig. 8.4.
rigid body mode, the parallel mode, the rotor oscillates parallel.
ond
Modelling
frequencies
1
Y
rigid body
model the
:
the rotor shows
a
In the first
In the
sec¬
periodic tipping
movement.
A.2.2 The
rigid rotor with gyroscopic coupling
Figure
The model is extended
rotation
frequencies
now
the
8.5:
Gyroscopic coupling
with the rotation of the rotor.
gyroscopic coupling
shows the rotor disk which rotates with the
momentum
L0,
which is
parallel
cannot be
speed
to the rotation axis:
72/
Especially at high
neglected. Fig. 8.5
Q. It has the
angular
Appendix : Modelling
L0
The theorem of
angular
momentum has the
torque T
says that
a
changing angular
consequence:
as
dL(
=
(8.12)
Tt
8.5 illustrates this theorem. A small deflection of the rotation axis
results in
a
changing angular
momentum
AL
It is
(8.11)
[Kneu/90]
momentum
T
Fig.
J0Q
-
orthogonal
8.8 and 8.10)
to
can
Lq
~J0
«
and has
[Good/891:
(8.13)
Acp
•
negative sign.
a
be extended with the
The torque
equations (Eqn.
gyroscopic coupling:
Jjiç
=
Tv
J0Ql
(8.14)
Jjîz,
=
Tx -J0LÎ(p
(8.15)
This leads to the model of the
rigid
Ml/m\
F
"Ü
Q
•
+
rotor
with
gyroscopic coupling:
X
x
A j
.
i
7
-/,
-h h
7
/,
i
^T11^^////
FT
1
Jv
w2^
9
9.
9
A-K
!
-
^2
~JnQ.<-
'œ
JqQ.-4
F
1
7\
i
I
y+
Fs\2
1
1
1,
-/,
l+
L
F
~
>
—I
sxl\
8.6: Model
-,S
///_/h
Ç
i
—
Ç
J
>
>
7/m
x—
Figure
A(p
v
y
of the rigid
rotor
122
with
gyroscopic coupling
Appendix
:
Modelling
A.2.3 The flexible rotor
Fs2A
8.7: Flexible rotor
Figure
At
high frequencies
the rotor cannot be considered to be
which describes the first
bending
[Ferm/90]. Fig. 8.7 shows
is
replaced by
and
P;
P2.
and
which
dr2
The
and
dr2
bearing
are
forces
Fig. 8.8. dxj
Fsy] correspond
two
mass
nodes),
is the
Sj
and
S2
are
again
can
and
5;
dyj
to the
be
and
modeled with
and
S2.
The
denote the
P^
FsI
two
and
a
shaft
disk.
Pj
and
respectively S2
Fs2. They
act on
spring-damper systems
spring-damper
model of
orthogonal components
components of Fs}.
Figure 8.8: Spring-damper model of the flexible shaft
123
rigid
rotor
denote the ends of the rota¬
S;
replaced by
A model
Jeffcot
is still concentrated in
the vectors between
attached to the ends
sketched in
Fsxl
the two shaft ends.
The elastic shaft
P2.
are
and
(with
the extension to the flexible rotor. The
flexible shaft. All the
F2 designate
tion axis.
and
a
mode
rigid.
Sj
of
is
drp
Appendix
:
Modelling
Starting point
the
spring
is
Eqn.
8.16. It describes the balance of forces in
constant and
b}
the
FsI
Fsj
=
Fsyl
The
same
=
Fsl
k]dr1
=
cos(Qt)
is done with
=
=
dyj
Note that all components
=
bjdrl
+
and
Fsxj
(8.16)
Fsv]:
k1cos(Qt)drI
+
b1cos(Qt)dr1
(8.17)
k1sin(Qt)drI
+
bjsin(Qt)dr]
(8.18)
=
dr} cos(Qt)
(8.19)
drr sin(Qt)
(8.20)
depend
-LÏ
the
on
describe the derivations of
dx,
dxj
drjsin(Qt)
^JL_
rotating speed
and
+
£1 The next two
dyj.
dr,cos(Clt)
(8,21)
,
=
dvj
=
kj
drf.
dxj
equations
=
FsI sin(Qt)
with
damping.
is divided into its components
Fsxl
Pj,
Ü
dy,
dr,cos(Qt)
•
^1^
+
dr,sin(Qt)
(8.22)
,
=dx2
Now
and
Eqn.
8.17 and 8.18
can
be rewritten
so
that
they only depend
on
dxj
dyj.
Fsv/
=
Fsx]
=
kjdxj
+
bj (dxj
kjdyj
+
h
j
(dyj
124
+
-
Qdxf)
(8.23)
Qd.Xj)
(8.24)
Appendix : Modelling
Interestingly the two axes are coupled. The equations are similar
gyroscopic coupling (Eqn. 8.14 and 8.15). In [Ferm/90] this coupling
denoted as the quasi-gyroscopic coupling.
Now all components exist which
flexible
rotor.
loop system will
emerge, which
ear
control
close to
algorithms
or
model. This model
non-linear elements such
can
r*-
dx,
-x
1/b,)-
-+.
\QJ
d\,
dx,
d^
ci
.
.
i
«
,-—\ d\~.
-
-K5
d\->
-
:_
Jj/b}^^!
k-, <-
f\j
\xl
i
j
1/m
^-±* j -V /
-\^
F u2
h\ \J
1
-/j
A
tcV
Jj
9
-Jcß^
rxL
i
/]
_>>o
///a—•
+
i,-i,
F
i
r
tu'o
—
1/m
/
/
•
->
/
1
h
Aà
1-1,
-
i
k,
-
d\
F
->l/h,
À
-
V
V
—
+
-
->
A
—
—
il
<
a
<-
J+ j
dx,
d\j
dx~,
„
]/h}- _2J7
ki
dx,
-4-
Figure 8.9: Complete model of the Jeffcot
125
as
non-lin¬
be simulated very
reality.
-+-ï-
be
magnetic bearings a closed
easily simulated using for example
limitations. The system thus
*;
can
a
of
case
be
can
simulating
Simulink. Simulink allows
complete
the
In the
suspension.
extended with the
is thus
necessary to describe the behavior of
are
Fig. 8.9 below shows
to the
rotor
^
r
1
^j?
v^ss*'
726
«r
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tional
Gifu
[Schw/94]
Symposium
City, 1997
on
Gerhard Schweitzer; Hannes Bleuler;
Active
Magnetic Bearings,
vdf
Alfons
Traxler
Hochschulverlag
an
der
ETH Zürich, Zürich 1994
[TaKn/96]
SamirM. Tamer; Carl R.
Knospe
Robust
Adaptive Open Loop Control of Periodic Distur¬
bance: Analysis, Synthesis and Implementation, University
of Virginia, Report No. UVA/643092/MAE96/489, Jan.
1996
[ Uey a/94]
Ueyama
Hi roch ika
Magnetic bearing device, Euopean patent 0612928A1,
Europäisches Patentamt, 1994
[Yonn/81]
Jean-Paul Yonnet
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actions
on
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Magnetics, Vol. Mag-17, No 1, January 1981
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References
[Zhan/951
Jing Zhang
Power
Amplifier
for Active
Magnetic Bearings,
Nr.: 11406, ETH Zürich, 1995
737
Diss. ETH
A^i'ik
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Ésu.
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)
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.
(Val
732
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Curriculum vitae
Particulars:
Name:
Betschon
First
Felix Ernst
names:
Date of birth:
May 7th
Birthplace:
Wettingen
Citizen:
Laufenburg, Switzerland
Marital status:
unmarried
Father:
Franz Felix Betschon
Mother:
Ruth Alice Betschon née Schürch
1970
Education:
1977-1983
Elementary
School in Heiden
(Primarschule)
1983-1985
Junior
Fligh
in Heiden
(Sekundärschule)
1985-1990
High
School in natural sciences in
Trogen
(Kantonsschule Typus C)
1991-1996
Studies in electrical
Federal Institute of
engineering
at
the Swiss
Technology (ETH)
in Zur¬
ich
in
electrical
engineering
1996
Diploma
degree)
1996-2000
Doctoral thesis at the Electronic
and
Engineering
Design Laboratory (ETH Zurich)
733
(masters