Unformatted text preview: DESIGN OF HORIZONTAL SHAFT ACTIVE MAGNETIC
BEARING SYSTEM
Bostjan Polajzer , Drago Dolinar , Gorazd Stumberger ,
Joze Ritonja , Bojan Grcar , Kay Hameyer]
University of Maribor, Faculty of Electrical Engineering and Computer Science,
Smetanova 17, SI{2000 Maribor, Slovenia, bos[email protected] KU Leuven, Dept. E.E., Div. ESAT/ELEN, Leuven, Belgium 1 INTRODUCTION
Several principles of magnetic bearings operation are known 1], but the principle based on the use
of electromagnets to provide the force necessary for the levitation of a rigid body is the most widely
used. The magnetic eld has to be adjusted continuously to attain stable levitation and the required dynamics of the levitating body. This can be done only with controlled electromagnets. One
application are Active Magnetic Bearings (AMBs) where two pairs of radial bearings controlling
four DOFs are placed at each rotor end. The fth DOF is controlled by a pair of axial bearings.
Rotation, i.e. the sixth DOF, is controlled by an independent driving motor. AMBs o er signi cant
advantages due to their noncontact operation. Higher speeds, no friction, no lubrication, weight
reduction, precise position control and active vibration damping make them far superior to the
conventional bearings. AMBs are therefore a typical mechatronic product and are particularly appropriate for highspeed rotating machines. Commercial applications include pumps, compressors,
ywheels, milling and grinding spindles, turbine engines, centrifuges, etc.
The laboratory prototype of an AMB is presented in the paper. The mathematical model of
an AMB is determined separately for the mechanical and for the electrical part. The modeling is
restricted to the y{ axis. The dynamic model with lumped parameters is expressed in the time
domain. The model is coupled and nonlinear. The di erential driving mode is introduced to
avoid the redundancy of input variables. Also, the linearized equilibrium point deviation model is
given. Its parameters are determined by the numerical analysis of the magnetic eld 2] and by
measurements. The obtained model is used for control design 3]. Because of the decentralized
control, the same controller parameters are used for the y{ axis and for the x{ axis. A comparison
of experimental and simulation results for the control in y{ axis is shown along with experimental
results of high speed rotation. Basic mechatronic components of the experimental system are
brie y described as well. At the very end, some important ndings, di culties and suggestions
with respect to the problem of active vibration damping are summarized. 2 MAGNETIC BEARING SYSTEM MODELING
2.1 Laboratory prototype In this subsection the mathematical model of the laboratory prototype is presented. The system
consists of two axially allocated pairs of electromagnets, i.e. the vertical and the horizontal subsystem. In Fig. 1 a) we can see the schematic presentation of the horizontal{shaft magnetic bearing
system with its geometry, and in Fig. 1 b) y{ axis of the AMB. The four input variables of the
system are voltages on each electromagnet winding. If rotation and elasticity of the shaft are neglected, then the system has two DOFs. The two output variables of the system are shaft positions
in the x{ and in the y{ axis. The determination of the mathematical dynamic model is separated
into three steps. First, we write the equations of motion where several geometric relations occur
due to the axial allocation of actuators, sensors and weight. The next step of modeling deals with
the electromagnets. On the assumption that the iron core and coil windings are idealized we can
write the voltage equation for each electromagnet. In the last step we write the equations for the y fy
sy l lsy lay
lsx lax
fx
sx i1 000
111 11
111
000
111
u1 000 00
111
000
000
111
f x x y mg
A
i2 000 00
111 11
111 11
u2 000 00 z
a) y
z mg b) Figure 1: a) Schematic presentation of the horizontal{shaft magnetic bearing system, b) y{ axis of
the active magnetic bearings
electromagnetic force generated by each particular electromagnet excited by the coil current. Their
sum is the resultant electromagnetic force.
The system has only two independent DOFs, so we will restrict our further discussion to the
subsystem describing the y{ axis. It consists of two electromagnets with a serially connected pair
of coils. The subsystem shown in Fig. 1 b) is described by the mechanical equation of motion (1),
two voltage equations (2) where sign of the last term depends on the index (positive sign for h = 1,
negative sign for h = 2) and the equation of the resultant electromagnetic force (3). R is the resistance
of one electromagnet, L is the inductance of one electromagnet when the axis of the shaft is in the
center. k Nm2 =A2 ] is the force coe cient and ku Vs/m] the coe cient of the back{EMF. The
nominal air gap and equivalent shaft mass are denoted with and m respectively.
2 y
m d 2 = f ; m g fl
dt
uh = ih R + L dih ku dy
dt
dt
i2
f = k ( ;1y)2 ; ( (1) h=1 2
i2
2 (2) ! (3)
+ y )2
The AMB model given in the form of equations (1), (2) and (3) is multivariable, coupled and
nonlinear. Let us assume constant model parameters. Voltages u1 and u2 are system inputs, the
position y is the output, and the common disturbance consists of the equivalent gravity force mg
and the load force fl . If we bear in mind that the system is totally controllable the redundancy
of input variables becomes evident. For its elimination we introduce the di erential driving mode.
The bias current i0 is forced through the coils of both electromagnets. Considering the given
assumptions the resultant electromagnetic force f0 is zero. But as we know there always exist the
load force fl and the disturbances as well as the gravity force mg, we have to add the socalled
control current (i
i0) in the upper coil and subtract it in the lower coil (4). In this way a
SISO system is obtained where the input variable is the socalled control voltage u or the control
current i in case of a current fed system. i1 := i0 + i i2 := i0 ; i (4) 2.2 Linearized equilibrium point deviation model Among the model equations (1), (2) and (3), only equation (3) is nonlinear. It is linearized in the
equilibrium point (i0 y0 ) where i0 is an arbitrary bias current and y0 the position of the shaft's axis in the center (y = 0). Taking into account the rst term of the Taylor expansion of equation (3)
about the equilibrium point the electromagnetic force is given by the wellknown linear equation
(5). The current gain coe cient ki N/A] and the position sti ness coe cient ky N/m] are de ned
as (6). After rearranging equations (1), (2) and (5) the linearized AMB model is obtained (7). The
equivalent model in the inputoutput domain is de ned by the transfer function (8). f = ki i + ky y (5) 2
ky := @@f
= 4k i0
3
y (i0 y0 )
2
ki i + ky y ; m ddty2 = 0
Ri + L dit + ki dyt ; u = 0
d
d
Y
G(s) = U (s) = mL s3 + mR s2 +kik2 ; k L) s ; k R
(s)
(i y
y = 4k i0
ki := @@f
2
i (i0 y0 ) Y (s)
I (s) (6)
(7)
(8) ms y
I
= m ski;ky Gel (s) = U ((ss)) = mL s3 +mR s2 +(;i2k;ky L) s;ky R
(9)
2
k
The parameters of the linearized equilibrium point deviation model of the laboratory prototype of
active magnetic bearings are shown in Table 1. Current gain and position sti ness are calculated
by the numerical analysis of the magnetic eld 2] using the nite element method (FEM). The
calculated results have been compared with the measured values. Gmech(s) = 2 Table 1: Parameters of the linearized model in equilibrium point (i0 y0 )
data
resistance
inductance
equivalent mass
current gain
position sti ness
bias current
nominal air gap parameter
value
determination
R]
0.6
measured
L H]
0.0048
measured
m kg]
5
measured
ki N/A]
32.7
FEM
ky N/m]
108350
FEM
i0 A]
2.5
free parameter
;3 construction data
m]
0:6 10 3 CONTROL
y r ef GconY i r ef GconI u Gel i Gmech y Figure 2: Control structure of the system
This AMB represents unstable system, therefore we need a closedloop control system to stabilize
it. The cascade structure has been used (Fig. 2). In the inner loop the current controller GconI (s)
is responsible for the best possible reference current tracking. Let us assume that the latter is
perfect. Then it is enough to use only the mechanical transfer function Gmech(s) (9), de ned by
two real poles (s 1 2 = 164:58), for the further position controller design. A PID controller (10) has
been used for stabilization purposes. Its parameters are de ned by increasing the amplitude of the frequency characteristic at low frequencies, and by trying to attain an adequate phase margin for
the chosen cross frequency in case of high frequencies. As the system consists of two independent
DOFs decentralized control has been implemented. Therefore the same controller parameters has
been used also for the x{ axis. The position controller was set up as follows: controller gain
Kcon = 10000, integral time constant Ti = 0:03 s, derivative time constant Td = 0:003 s and
parasitic derivative time constant Td0 = Td =10.
+
GconY (s) = Kcon s TiT 1 s Td0 + 1
(10)
s i s Td + 1 4 SIMULATION AND EXPERIMENT
In simulations (MATLAB { Simulink) all parts of the experimental system shown in Fig. 3 a) were
considered in addition to the nonlinear actuator model (Fig. 3 b). Let us describe some parts
of the system. First, the inductive position sensor with a sensitivity of 7.7 mV/ m and 20 kHz
cuto frequency was chosen very carefully. An additional lter for sensor crosstalk elimination
was implemented afterwards at 5 kHz. Next, the analogue current controller and a 20 kHZ PWM
switching ampli er were used with a 300 VA per channel. Finally, the digital PID position controller
with antiwindup was implemented into the power PC environment with a sampling time of 100 s. a) b) Figure 3: a) Experimental system and b) laboratory prototype AMB
Fig. 4 shows the comparison of calculations and measurements on the laboratory prototype. Only
the comparison of control in the y{ axis is shown. It is obvious that the agreement of results is very
good { excellent damping agreement and acceptable sti ness disagreement. However, the result
for the stability test was quite di erent. The upper stability limit of the experimental system was
much lower than the one we established theoretically. This means that we can not achieve a very
high sti ness. The reason for this are the actuator limitations. As a result, insu cient forces are
generated for a wide range of shaft positions. This conclusion is con rmed by FEM calculations
and measurements of the f (i y) relation. In Fig. 5 the results of the rotation test, where the
position error does not exceed 15 m, are also presented. In Fig. 5 b) the shaft elasticity problem
turns up in addition to the rotor unbalance problem which is more evident in Fig. 5 c). 5 CONCLUSION
The paper deals with the modeling and the analysis of AMB laboratory prototype. If we take into
account that the system has two independent DOFs, then the analysis of the model is restricted
to the y{ axis. The di erential driving mode is introduced and the linearized equilibrium point
deviation model is written. Its parameters are measured and calculated by the numerical analysis of
the magnetic eld. The PID controller design of the y{ axis is included. Because of the decentralized
control, the same control design is used also for the x{ axis. 2.5
2 simulation 2 position [m] position [m] 3 experiment 1 1.5 simulation
experiment 1
0.5
0 0
0.3 a) 0.35
time [s] 0.4 0.45 −0.5 b) 0.3 0.35
time [s] 0.4 0.45 Figure 4: The position response of y{ axis from the equilibrium point: a) to the reference
step function (0.2 mm) and b) to the load step function (60 N)
2
y−position [m] y−position [m] −5 x 10 1
0
−1
−2
−2 a) −1
0
1
2
x−position [m] x 10−5 −5 x 10 2
y−position [m] −5 2 1
0
−1
−2
−2 b) −1
0
1
2
x−position [m] x 10−5 x 10 1
0
−1
−2
−2 c) −1
0
1
2
x−position [m] x 10−5 Figure 5: The experimental position response: a) at 3890 rpm, b) at 6600 rpm and c) at 8040 rmp
The presented work represents one of the rst steps in the research of modeling, analysis and
control design of AMB at our institution. Although we used one of the most simple control methods
we came to the following important conclusions:
the rigidity of the system is increased by a higher bias current i0 and controller gain Kcon
the system dynamics is improved by an appropriate derivative time constant Td
the in uence of disturbances is reduced by an appropriate integral time constant Ti
the system sti ness insu ciency is caused by actuator limitations, therefore an appropriate
actuator should be carefully chosen with respect to the expected load forces and available
power supply or vice versa
the nonmodeled dynamics of the shaft elasticity and unbalanced rotor becomes evident at
the socalled critical speeds and this is why the control design for active vibration damping
should also employ the nonmodeled rotordynamics. Acknowledgments
The authors are grateful to the Belgian Federal o ce for scienti c, technical and cultural a airs for promoting
the W&T cooperation with Central and East Europe by giving the DWTC grant to D. Dolinar. Thanks
are due to the Ministry of Science and Technology of Slovenia for the nancial support. References
1] G. Schweitzer, H. Bleuer, and A. Traxler, Active Magnetic Bearings { Basics, Properties and Applications
of Active Magnetic Bearings. Zurich: VDF, 1994.
2] Olympos  Finite Element and Optimization Package, see http://www.esat.kuleuven.ac.be/elen/elen.html,
online help.
3] C. Knospe, \PID control for magnetic bearings," in Short Course on Magnetic Bearings, Lecture 7,
(Alexandria, Virginia), 1997. ...
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 Electromagnet, Equilibrium point, magnetic bearing, laboratory prototype, active magnetic bearings

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