Self–Sensing Magnetic Bearings Driven by a Switching Power Amplifier

120 Pages·2003·0.86 MB·English

Self{Sensing Magnetic Bearings Driven
by a Switching Power Ampli(cid:12)er
Myounggyu D. Noh
Eric H. Maslen
January 1996
Abstract
Active magnetic bearings require some form of control, based on feedback of the
position of the suspended object, to overcome open{loop instability and to achieve
targeted system performance by modifying the bearing dynamics. In many applica-
tions of magnetic bearings, a need to eliminate discrete position sensors may arise
either from economic or reliability considerations. Magnetic bearings which estimate
the position from the information available in the electromagnet signals are referred
to as \self{sensing".
Previously,therehavebeentwomainstreamapproachesfordevelopingself{sensing
magnetic bearings. One approach is to use a Luenberger observer designed from a
linearized state{space representation of voltage{controlled magnetic bearings. Due to
the nonlinearities involved with the physics of the bearing, this approach has limited
applicability. The other approach considers the air gap as a parameter of the system
rather than a state. Previous attempts using parameter estimation were hampered by
force feed{through (the in(cid:12)ltration of force information into the position estimates).
In this dissertation, a nonlinear parameter estimation technique is presented by
which the position of a rotor supported in magnetic bearings may be deduced from
the bearing current waveform. The bearing currents are presumed to be developed by
a bi{state switching ampli(cid:12)er which produces a substantial high frequency switching
ripple. The amplitude of this ripple is a function of power supply voltage, switching
duty cycle, and bearing inductance. Inductance is predominantly a function of the
bearing air gap or, equivalently, the rotor position, while the duty cycle is funda-
mentally dependent upon the developed bearing force. Ideally, the estimator should
exactly extract rotor position information while perfectly rejecting bearing force in-
formation.
When the bearing is a perfect inductor, the aforementioned functional relation-
ships are easily established and the gap dependence is monotonic. Since voltage
and duty cycle are both easily measured, the relationships can be inverted with a
non{linear parameter estimator to extract the rotor position. The estimator embeds
a model of the bearing inductance parameterized by the air gap. This simulation
is subject to the same switching voltage as the actual magnetic bearing coils. A
feedback loop compares the simulated current waveform with the actual current and
adjusts the gap length parameter until the two waveforms match.
The performance of the estimator is evaluated both by computer simulation and
experiment. Thetechniqueisdemonstratedtoproduceafairlywidebandwidthsensor
(at least 1 kHz) with acceptably low feed{through of the bearing force. The estimator
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also displays excellent linearity (less than 2 % deviation from linear).
Nonidealities such as saturation, hysteresis, and eddy currents are investigated to
assess their e(cid:11)ects on the performance of the estimator. The embedded inductance
model can easily incorporate these nonidealities without a(cid:11)ecting the estimation per-
formance.
ii
Contents
Abstract i
Nomenclature ix
1 Introduction 1
1.1 Prior research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Summary of present work . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Literature Review 6
2.1 Self{Sensing Magnetic Bearing . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 6
2.1.2 State{Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1.3 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Sensorless Electric Motor Control . . . . . . . . . . . . . . . . . . . . 7
2.3 Magnetic Bearing System . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Magnetization Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Magnetic Bearing System 12
3.1 Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.2 Ampli(cid:12)er. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.3 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.4 Position Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Modeling 19
4.1 Coil Inductor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
4.2 Switching Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Experimental Setup 28
5.1 Design of the Test Rig . . . . . . . . . . . . . . . . . . . . . . . . . . 28
5.2 Speci(cid:12)cations of the Test Bearing . . . . . . . . . . . . . . . . . . . . 29
5.3 Ampli(cid:12)er. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.4 Position Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.5 Dynamic Analysis of the Test Rig . . . . . . . . . . . . . . . . . . . . 30
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6 Switching Noise Demodulation 34
6.1 Idealized Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
6.2 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2.1 High Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2.2 Full Wave Recti(cid:12)er . . . . . . . . . . . . . . . . . . . . . . . . 40
6.2.3 Low Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . 43
7 Parameter Estimation 51
7.1 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
7.2 Idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7.3 Stability of the Estimator . . . . . . . . . . . . . . . . . . . . . . . . 55
7.4 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.4.1 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.4.2 Inductor Simulation Model . . . . . . . . . . . . . . . . . . . . 58
7.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.5.1 Static Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.5.2 Dynamic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
7.5.3 Force Feed{through E(cid:11)ect . . . . . . . . . . . . . . . . . . . . 60
7.5.4 Bandwidth and Signal to Noise Ratio . . . . . . . . . . . . . . 63
8 Nonidealities 69
8.1 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
8.2 Hysteresis and Eddy Currents . . . . . . . . . . . . . . . . . . . . . . 74
8.3 Cross{Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
8.4 Back EMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
8.5 Other nonidealities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
9 Conclusion and Future Research 90
A Harmonic Analysis of Switching Waveform 99
B Harmonic Analysis of Demodulation 101
C Hysteresis Model 104
D Circuit Diagram of the Estimator 108
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List of Tables
5.1 Critical dimensions and speci(cid:12)cations of the test bearing . . . . . . . 30
6.1 Component values used in high{pass (cid:12)lter stage . . . . . . . . . . . . 40
6.2 Component values used in the low{pass (cid:12)lter . . . . . . . . . . . . . . 42
C.1 Parameter values used in simulation . . . . . . . . . . . . . . . . . . . 105
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List of Figures
3.1 Schematic of a typical magnetic bearing system . . . . . . . . . . . . 13
3.2 A monopolar linear transconductance ampli(cid:12)er . . . . . . . . . . . . . 14
3.3 H-Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 8{pole magnetic bearing . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.2 Block diagram of the inductor model . . . . . . . . . . . . . . . . . . 22
4.3 Gap displacement modulates the switching waveform . . . . . . . . . 23
4.4 Switching waveform with time{varying duty cycle . . . . . . . . . . . 24
4.5 Votage waveform during one switching cycle . . . . . . . . . . . . . . 25
4.6 Duty cycle variation for I = 1 . . . . . . . . . . . . . . . . . . . . . 26
m
4.7 Harmonic contents of switching waveform . . . . . . . . . . . . . . . . 27
5.1 Sketch of the experimental setup . . . . . . . . . . . . . . . . . . . . 29
5.2 Position sensor static test result . . . . . . . . . . . . . . . . . . . . . 31
5.3 Mode shapes and natural frequencies of the test beam . . . . . . . . . 32
5.4 Relative locations of toggle clamp, sensor, and bearing . . . . . . . . 33
5.5 Mismeasurement of position sensor due to noncollocation and resonant
frequencies at various toggle clamp positions . . . . . . . . . . . . . . 33
6.1 Voltage applied to coil during one switching cycle . . . . . . . . . . . 35
6.2 Representation of signal at each processing point . . . . . . . . . . . . 37
6.3 UAF42 con(cid:12)gured as a high pass (cid:12)lter . . . . . . . . . . . . . . . . . 39
6.4 Frequency response of the high{pass (cid:12)lter stage . . . . . . . . . . . . 40
6.5 Circuit schematic of full wave recti(cid:12)er . . . . . . . . . . . . . . . . . . 41
6.6 UAF42 con(cid:12)gured as a low{pass (cid:12)lter . . . . . . . . . . . . . . . . . . 42
6.7 Frequency response of the low{pass (cid:12)lter stage . . . . . . . . . . . . . 43
6.8 Forward path (cid:12)lter response when ! = 2(cid:25) 120 . . . . . . . . . . . . 44
(cid:1)
6.9 Frequency response of forward path (cid:12)lter when the duty cycle is (cid:12)xed 46
6.10 Frequency response of forward path (cid:12)lter when the duty cycle is time{
varying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.11 Ratio of o(cid:11)set to nominal gap . . . . . . . . . . . . . . . . . . . . . . 48
6.12 Forward path (cid:12)lter response of free vibration . . . . . . . . . . . . . . 49
6.13 Forward path (cid:12)lter response of forced vibration . . . . . . . . . . . . 49
6.14 Experimentally obtained frequency response of the forward path (cid:12)lter 50
7.1 Overall schematic of the estimator . . . . . . . . . . . . . . . . . . . . 52
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7.2 Frequency response of the forward path (cid:12)lter (simulation and approx-
imation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.3 Variation of the time{varying gain K . . . . . . . . . . . . . . . . . . 54
7.4 Idealized block diagram of the parameter estimator . . . . . . . . . . 55
7.5 Block diagram for absolute stability test . . . . . . . . . . . . . . . . 56
7.6 Disk D(K ;K ) and trajectory G(j!)F(j!) . . . . . . . . . . . . 56
min max
7.7 Stabilityoftheparameterestimator. The(cid:12)lledcicleindicatesthegains
used in the experiments . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.8 Analog realization of the controller . . . . . . . . . . . . . . . . . . . 58
7.9 Implementation of inductor simulation . . . . . . . . . . . . . . . . . 59
7.10 Linearity of the parameter estimator obtained from a static test. Max-
imum error from linearity is 1.3 % . . . . . . . . . . . . . . . . . . . . 60
7.11 The output of the estimator when ! = 2(cid:25) 120 . . . . . . . . . . . . . 61
(cid:1)
7.12 Estimatoroutputwhenthetestbeamisvibratingfreely(constantduty
ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
7.13 Estimator output when the test beam undergoes a forced vibration
(time{varying duty cycle) . . . . . . . . . . . . . . . . . . . . . . . . 62
7.14 Experimentally measured force feed{through . . . . . . . . . . . . . . 64
7.15 Revised block diagram of parameter estimator . . . . . . . . . . . . . 65
7.16 Frequency response of parameter estimator and disturbance rejection 66
7.17 Power spectral density of the estimator output at steady state. The
root mean squared error is 0.9 (cid:22)m (0.04 mil). . . . . . . . . . . . . . 66
7.18 Frequency response of parameter estimator (analytical vs. simulation) 67
7.19 Frequency response of parameter estimator (experimental) . . . . . . 68
8.1 Estimation error due to saturation . . . . . . . . . . . . . . . . . . . 70
8.2 Current rate vs. air gap length and bias current . . . . . . . . . . . . 71
8.3 Current rate when the bias current is 5 Amps. . . . . . . . . . . . . . 72
8.4 Slope versus displacement . . . . . . . . . . . . . . . . . . . . . . . . 73
8.5 Estimator response when the core material is saturated . . . . . . . . 73
8.6 Current waveform acquired by a digital oscilloscope. Sampling rate of
25 MHz was used. Additional (cid:12)ltering was applied to eliminate noise. 74
8.7 Eddy current e(cid:11)ects modeled with a (cid:12)ctitious one{loop coil . . . . . . 76
8.8 Straight line hysteresis model . . . . . . . . . . . . . . . . . . . . . . 77
8.9 Current waveform generated by straight hysteresis model . . . . . . . 77
8.10 Estimation error due to jump discontinuity . . . . . . . . . . . . . . . 78
8.11 Modi(cid:12)ed parameter estimator accommodating jump discontinuity . . 79
8.12 Frequency dependent relative permeability . . . . . . . . . . . . . . . 80
8.13 De(cid:12)nition of (cid:18) and (cid:18) . . . . . . . . . . . . . . . . . . . . . . . . . . 81
i p
8.14 Three{pole bearing showing sign and numbering convention used to
set up (cid:13)ux equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
8.15 Estimation error when back EMF is considered in the computer simu-
lation. (! = 2(cid:25) 120 [rad/sec]) . . . . . . . . . . . . . . . . . . . . . . 87
(cid:1)
8.16 Estimation error when back EMF is considered in the computer simu-
lation (! = 2(cid:25) 1200 [rad/sec]) . . . . . . . . . . . . . . . . . . . . . 88
(cid:1)
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8.17 Actual voltage waveform from switching ampli(cid:12)er . . . . . . . . . . . 89
B.1 Voltage waveform at 50% duty cycle . . . . . . . . . . . . . . . . . . 102
C.1 Nonlinear magnetization curve obtained by (C.4) . . . . . . . . . . . 106
C.2 Minor hysteresis loops . . . . . . . . . . . . . . . . . . . . . . . . . . 107
C.3 Relative permeability based on major and minor hysteresis loops . . . 107
D.1 Circuit diagram of the position estimator . . . . . . . . . . . . . . . . 109
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Nomenclature
A Cross-sectional area of (cid:13)ux path at the air gap.
g
B Magnetic (cid:13)ux density.
d Lamination thickness.
g Nominal air gap of a bearing.
H Magnetic (cid:12)eld intensity.
H Remnant magnetic (cid:12)eld intensity.
r
i Coil current.
j p 1.
(cid:0)
l Length of (cid:13)ux path section.
c
L Inductance.
L Nominal inductance.
0
N Number of turns in coil.
R Electric resistance.
Magnetic reluctance.
R
s Laplace variable.
t Zero{crossing time
c
u Output signal of forward path (cid:12)lter.
V Voltage.
V Power supply voltage.
s
w Axial length of magnetic bearing.
x Gap displacement.
(cid:11) Duty cycle.
(cid:14) Current jump due to hysteresis.
(cid:16) Damping factor. Integration variable.
(cid:17) Parameter describing nonlinear B{H curve.
(cid:27) Parameter describing nonlinear B{H curve.
(cid:22) Magnetic permeability.
(cid:22) Magnetic permeability of free space.
o
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