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Self–Sensing Magnetic Bearings Driven by a Switching Power Amplifier Myounggyu D. Noh Eric H. Maslen January 1996
Self–Sensing Magnetic Bearings Driven
by a Switching Power Amplifier
Myounggyu D. NohEric H. Maslen
January 1996
Abstract
Active magnetic bearings require some form of control, based on feedback of theposition of the suspended object, to overcome open–loop instability and to achievetargeted system performance by modifying the bearing dynamics. In many applica-tions of magnetic bearings, a need to eliminate discrete position sensors may ariseeither from economic or reliability considerations. Magnetic bearings which estimatethe position from the information available in the electromagnet signals are referredto as “self–sensing”.
Previously, there have been two mainstream approaches for developing self–sensingmagnetic bearings. One approach is to use a Luenberger observer designed from alinearized state–space representation of voltage–controlled magnetic bearings. Due tothe nonlinearities involved with the physics of the bearing, this approach has limitedapplicability. The other approach considers the air gap as a parameter of the systemrather than a state. Previous attempts using parameter estimation were hampered byforce feed–through (the infiltration of force information into the position estimates).
In this dissertation, a nonlinear parameter estimation technique is presented bywhich the position of a rotor supported in magnetic bearings may be deduced fromthe bearing current waveform. The bearing currents are presumed to be developed bya bi–state switching amplifier which produces a substantial high frequency switchingripple. The amplitude of this ripple is a function of power supply voltage, switchingduty cycle, and bearing inductance. Inductance is predominantly a function of thebearing air gap or, equivalently, the rotor position, while the duty cycle is funda-mentally dependent upon the developed bearing force. Ideally, the estimator shouldexactly 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 voltageand duty cycle are both easily measured, the relationships can be inverted with anon–linear parameter estimator to extract the rotor position. The estimator embedsa model of the bearing inductance parameterized by the air gap. This simulationis subject to the same switching voltage as the actual magnetic bearing coils. Afeedback loop compares the simulated current waveform with the actual current andadjusts the gap length parameter until the two waveforms match.
The performance of the estimator is evaluated both by computer simulation andexperiment. The technique is demonstrated to produce a fairly wide bandwidth sensor(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 effects on the performance of the estimator. The embedded inductancemodel can easily incorporate these nonidealities without affecting the estimation per-formance.
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Contents
Abstract i
Nomenclature ix
1 Introduction 1
1.1 Prior research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Summary of present work . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Literature Review 6
2.1 Self–Sensing Magnetic Bearing . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 62.1.2 State–Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.3 Other Approaches . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Sensorless Electric Motor Control . . . . . . . . . . . . . . . . . . . . 72.3 Magnetic Bearing System . . . . . . . . . . . . . . . . . . . . . . . . 92.4 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . 102.5 Magnetization Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Magnetic Bearing System 12
3.1 Actuator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.3 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.4 Position Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4 Modeling 19
4.1 Coil Inductor Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Switching Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5 Experimental Setup 28
5.1 Design of the Test Rig . . . . . . . . . . . . . . . . . . . . . . . . . . 285.2 Specifications of the Test Bearing . . . . . . . . . . . . . . . . . . . . 295.3 Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.4 Position Sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.5 Dynamic Analysis of the Test Rig . . . . . . . . . . . . . . . . . . . . 30
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6 Switching Noise Demodulation 34
6.1 Idealized Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346.2 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
6.2.1 High Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 386.2.2 Full Wave Rectifier . . . . . . . . . . . . . . . . . . . . . . . . 406.2.3 Low Pass Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.3 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . 43
7 Parameter Estimation 51
7.1 General Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517.2 Idealization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 527.3 Stability of the Estimator . . . . . . . . . . . . . . . . . . . . . . . . 557.4 Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7.4.1 Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577.4.2 Inductor Simulation Model . . . . . . . . . . . . . . . . . . . . 58
7.5 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 587.5.1 Static Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587.5.2 Dynamic Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 597.5.3 Force Feed–through Effect . . . . . . . . . . . . . . . . . . . . 607.5.4 Bandwidth and Signal to Noise Ratio . . . . . . . . . . . . . . 63
8 Nonidealities 69
8.1 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 698.2 Hysteresis and Eddy Currents . . . . . . . . . . . . . . . . . . . . . . 748.3 Cross–Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.4 Back EMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 848.5 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 specifications of the test bearing . . . . . . . 30
6.1 Component values used in high–pass filter stage . . . . . . . . . . . . 406.2 Component values used in the low–pass filter . . . . . . . . . . . . . . 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 . . . . . . . . . . . . 133.2 A monopolar linear transconductance amplifier . . . . . . . . . . . . . 143.3 H-Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.1 8–pole magnetic bearing . . . . . . . . . . . . . . . . . . . . . . . . . 204.2 Block diagram of the inductor model . . . . . . . . . . . . . . . . . . 224.3 Gap displacement modulates the switching waveform . . . . . . . . . 234.4 Switching waveform with time–varying duty cycle . . . . . . . . . . . 244.5 Votage waveform during one switching cycle . . . . . . . . . . . . . . 254.6 Duty cycle variation for Im = 1 . . . . . . . . . . . . . . . . . . . . . 264.7 Harmonic contents of switching waveform . . . . . . . . . . . . . . . . 27
5.1 Sketch of the experimental setup . . . . . . . . . . . . . . . . . . . . 295.2 Position sensor static test result . . . . . . . . . . . . . . . . . . . . . 315.3 Mode shapes and natural frequencies of the test beam . . . . . . . . . 325.4 Relative locations of toggle clamp, sensor, and bearing . . . . . . . . 335.5 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 . . . . . . . . . . . 356.2 Representation of signal at each processing point . . . . . . . . . . . . 376.3 UAF42 configured as a high pass filter . . . . . . . . . . . . . . . . . 396.4 Frequency response of the high–pass filter stage . . . . . . . . . . . . 406.5 Circuit schematic of full wave rectifier . . . . . . . . . . . . . . . . . . 416.6 UAF42 configured as a low–pass filter . . . . . . . . . . . . . . . . . . 426.7 Frequency response of the low–pass filter stage . . . . . . . . . . . . . 436.8 Forward path filter response when ω = 2π · 120 . . . . . . . . . . . . 446.9 Frequency response of forward path filter when the duty cycle is fixed 466.10 Frequency response of forward path filter when the duty cycle is time–
varying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476.11 Ratio of offset to nominal gap . . . . . . . . . . . . . . . . . . . . . . 486.12 Forward path filter response of free vibration . . . . . . . . . . . . . . 496.13 Forward path filter response of forced vibration . . . . . . . . . . . . 496.14 Experimentally obtained frequency response of the forward path filter 50
7.1 Overall schematic of the estimator . . . . . . . . . . . . . . . . . . . . 52
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7.2 Frequency response of the forward path filter (simulation and approx-imation) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.3 Variation of the time–varying gain K . . . . . . . . . . . . . . . . . . 547.4 Idealized block diagram of the parameter estimator . . . . . . . . . . 557.5 Block diagram for absolute stability test . . . . . . . . . . . . . . . . 567.6 Disk D(Kmin, Kmax) and trajectory G(jω)F (jω) . . . . . . . . . . . . 567.7 Stability of the parameter estimator. The filled cicle indicates the gains
used in the experiments . . . . . . . . . . . . . . . . . . . . . . . . . 577.8 Analog realization of the controller . . . . . . . . . . . . . . . . . . . 587.9 Implementation of inductor simulation . . . . . . . . . . . . . . . . . 597.10 Linearity of the parameter estimator obtained from a static test. Max-
imum error from linearity is 1.3 % . . . . . . . . . . . . . . . . . . . . 607.11 The output of the estimator when ω = 2π · 120 . . . . . . . . . . . . . 617.12 Estimator output when the test beam is vibrating freely (constant duty
ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 627.13 Estimator output when the test beam undergoes a forced vibration
(time–varying duty cycle) . . . . . . . . . . . . . . . . . . . . . . . . 627.14 Experimentally measured force feed–through . . . . . . . . . . . . . . 647.15 Revised block diagram of parameter estimator . . . . . . . . . . . . . 657.16 Frequency response of parameter estimator and disturbance rejection 667.17 Power spectral density of the estimator output at steady state. The
root mean squared error is 0.9 µm (0.04 mil). . . . . . . . . . . . . . 667.18 Frequency response of parameter estimator (analytical vs. simulation) 677.19 Frequency response of parameter estimator (experimental) . . . . . . 68
8.1 Estimation error due to saturation . . . . . . . . . . . . . . . . . . . 708.2 Current rate vs. air gap length and bias current . . . . . . . . . . . . 718.3 Current rate when the bias current is 5 Amps. . . . . . . . . . . . . . 728.4 Slope versus displacement . . . . . . . . . . . . . . . . . . . . . . . . 738.5 Estimator response when the core material is saturated . . . . . . . . 738.6 Current waveform acquired by a digital oscilloscope. Sampling rate of
25 MHz was used. Additional filtering was applied to eliminate noise. 748.7 Eddy current effects modeled with a fictitious one–loop coil . . . . . . 768.8 Straight line hysteresis model . . . . . . . . . . . . . . . . . . . . . . 778.9 Current waveform generated by straight hysteresis model . . . . . . . 778.10 Estimation error due to jump discontinuity . . . . . . . . . . . . . . . 788.11 Modified parameter estimator accommodating jump discontinuity . . 798.12 Frequency dependent relative permeability . . . . . . . . . . . . . . . 808.13 Definition of θi and θp . . . . . . . . . . . . . . . . . . . . . . . . . . 818.14 Three–pole bearing showing sign and numbering convention used to
set up flux equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 828.15 Estimation error when back EMF is considered in the computer simu-
lation. (ω = 2π · 120 [rad/sec]) . . . . . . . . . . . . . . . . . . . . . . 878.16 Estimation error when back EMF is considered in the computer simu-
lation (ω = 2π · 1200 [rad/sec]) . . . . . . . . . . . . . . . . . . . . . 88
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8.17 Actual voltage waveform from switching amplifier . . . . . . . . . . . 89
B.1 Voltage waveform at 50% duty cycle . . . . . . . . . . . . . . . . . . 102
C.1 Nonlinear magnetization curve obtained by (C.4) . . . . . . . . . . . 106C.2 Minor hysteresis loops . . . . . . . . . . . . . . . . . . . . . . . . . . 107C.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
Ag Cross-sectional area of flux path at the air gap.
B Magnetic flux density.
d Lamination thickness.
g Nominal air gap of a bearing.
H Magnetic field intensity.
Hr Remnant magnetic field intensity.
i Coil current.
j√−1.
lc Length of flux path section.
L Inductance.
L0 Nominal inductance.
N Number of turns in coil.
R Electric resistance.
R Magnetic reluctance.
s Laplace variable.
tc Zero–crossing time
u Output signal of forward path filter.
V Voltage.
Vs Power supply voltage.
w Axial length of magnetic bearing.
x Gap displacement.
α Duty cycle.
δ Current jump due to hysteresis.
ζ Damping factor. Integration variable.
η Parameter describing nonlinear B–H curve.
σ Parameter describing nonlinear B–H curve.
µ Magnetic permeability.
µo Magnetic permeability of free space.
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µr Relative magnetic permeability.
µ0r Nominal relative magnetic permeability.
Φ Magnetic flux.
τ Switching interval
ω Excitation frequency.
Subscript g denotes air gap.
Subscript c denotes core material.
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Chapter 1
Introduction
Active magnetic bearings require some form of control based on feedback of the po-sition of the suspended object. This feedback is required, fundamentally, in order toachieve stability [21], but more importantly, in order to permit tailoring of the bear-ing dynamics to achieve targeted system performance [29]. Most magnetic bearingapplications use feedback of the position along each axis of magnetic control.
In many potential applications of magnetic bearings, the number of wires whichmust pass between the bearing controller and the components in the rotating ma-chine needs to be minimized. Such a requirement may arise either from economic orreliability considerations. An exaggerated example of this is provided by the appli-cation of magnetic bearings to heart pumps [2, 3, 12, 13, 66, 68] where wires musteither pass through the chest cavity (transcutaneous) or be inductively coupled to anexternal transmission/reception device. In either case, minimizing the wire count isa paramount design concern.
The wires feeding the electromagnets can, to some extent, be minimized by in-terconnection of the magnets. A more substantial reduction in wire count can beachieved by eliminating the discrete position sensing device and, instead, determiningthe position of the suspended object from information available in the electromag-net signals. Elimination of the discrete sensing device has such additional benefitsas lowering the cost of the system and removing the potential failure of the sensingelement, thus improving the reliability of the overall system. Other potential ad-vantages include elimination of sensor–actuator noncollocation [45] and reduction innoise infiltration because the bearing switching signal is now information rather thannoise (as it is when a discrete sensor is used).
Magnetic bearings which estimate the rotor position from information in the elec-tromagnet signals are referred to as “self–sensing”. When a bi–state switching ampli-fier is employed to drive the bearing coil, the bearing can be considered as a modulatedvariable reluctance sensor [56]. Modulation frequency, which is essentially switchingfrequency, is generally high enough for the modulated signal to retain good band-width, but low enough that the induced eddy current is negligible.
One of the objectives of this dissertation research is to develop a mechanism thatestimates the gap position by processing the current switching waveform. A real–time simulation of the bearing coil is introduced, and a feedback controller is used
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to force the simulated current to track actual current. The proposed method placesan emphasis on reducing the force dependence of the position estimator. Previousresearch efforts using parameter estimation techniques have been unable to overcomethis problem of force dependence. Other objectives are to identify nonidealities thatare normally neglected in modeling the magnetic bearing and to investigate the effectsof these nonidealities on the performance of the estimator.
Before summarizing previous research on self–sensing magnetic bearings, it seemsnecessary to point out the fundamental drawback to any self–sensing approach. Ca-pacity and thermal considerations usually lead to an actuator structure with magneticpaths whose reluctance substantially exceeds that of the air gaps at high frequencies(20 kHz or more). In contrast, high sensitivity and good rejection of magnetic nonlin-earities are achieved in variable reluctance sensors by ensuring that the iron reluctanceis on par with or substantially less than that of the air gaps. Thus, the overall sensingperformance of a self–sensing magnetic bearing can be expected to be inferior to thatof a discrete variable reluctance sensor when evaluated solely in terms of sensitiv-ity, bandwidth, and linearity. In deciding whether such an approach is appropriateto a given application, the system–level advantages of self–sensing bearings must beweighed against expected performance shortcomings
1.1 Prior research
There have been two mainstream approaches to date in research on self–sensing mag-netic bearings. One approach is to use the parameter estimation technique where thebearing air gap is considered as a parameter of the magnetic system. The presentwork takes this approach. The other approach is to treat the magnetic bearings andthe supported object as a whole system, thus considering the position as a staterather than a parameter. Chapter 2 provides a comprehensive survey on these twoapproaches as well as several other techniques. The current section focuses on the mo-tivation of this dissertation research inspired by the limitations of the prior researchefforts in realizing self–sensing magnetic bearings.
The most prominent research on self–sensing magnetic bearings was started by agroup of people at ETH ([79] for example). They used a linear state–space observer toestimate the gap displacement. Treating the magnetic bearing as a two–port system,the linearized state–space equation describing the system (magnetic bearing and sus-pending object) is observable as well as controllable. Therefore, using such techniquesas pole placement and linear quadratic regulator (LQR), a stable observer can be de-signed. As the dynamics of the suspending object is coupled with the dynamics of thebearing through back EMF (electro–motive–force), the estimator is unable to detectthe deflection due to static loads [9]. Another drawback of this approach is that theresulting controller/observer pair usually has poor robustness [20]. It is likely thatsmall parameter variations in the physical system will produce instability. Finally,as the dynamics of the bearing are combined with that of the rotor, it is difficult toemploy this approach when a switching amplifier is used. High frequency switchingnoise will degrade the estimator performance.
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Several researchers have taken parameter estimation approaches–treating the airgap as a parameter of the system. Okada et al. [64] developed a self–sensing mag-netic suspension using a demodulation technique developed on the same idea as theforward path filter described in Chapter 6. The Fourier coefficients of the voltageand current signals at the switching frequency give an indication of the position, asthe gap displacement modulates the amplitude of the switching waveform. A seriousdrawback of the approach is, however, that the estimation depends not only on thegap displacement but also on the duty cycle of the PWM switching amplifier. Thisforce feed–through will degrade the sensor performance. This dissertation researchspecifically addresses this issue. Okada suggests using a nonlinear compensator forthe duty cycle variation, even though it was not used in actual implementation. Thecompensator may be called a scaling factor since it removes the duty cycle dependencyby measuring the duty cycle directly. However, the scaling can cause several otherproblems. Chapter 6 elaborates on the scaling approach. The fundamental problemof this approach is that the performance of the estimator can be only as good as thedemodulation filter.
Utilizing the fact that the switching rate of a hysteresis amplifier depends upon theload impedance, Mizuno [59] proposes a method for developing a self–sensing bearing.Frequency demodulation is used to extract the position signal from the switchingwaveform. This approach faces the same problem as the one Okada proposed; theresult of the demodulation is dependent on the force information as well as the gapposition.
Gurumoorthy et al. [26] used a technique that originated from the research effortson the sensorless control of a switched reluctance machine [49]. They used a pair ofbearing coils as a sensor and injected a test signal which is a series of rectangularpulses. The amplitude of the resulting triangular waveform is a direct indication ofthe gap displacement. An obvious drawback of this approach is that the bandwidthof the estimator may be limited, as one part of the bearing needs to be switched off inorder to be used as a sensor. Moreover, the estimator can become quite complicated.Hence, this approach is inappropriate for low–cost applications.
A promising approach for self–sensing bearings is proposed by Iannello [31]. Forthe configuration where two opposing coils are wired in series, he derives an alge-braic equation relating the gap and the rectified voltage of each coil. One obviousdisadvantage of this approach is that it only applies to a specific wiring configurationand places restrictions on the controller design. This limitation can be overcome byadding a reference coil to an individual bearing coil. The addition of the referencecoil eliminates the need to wire the two opposing coils in series [30]. However, as theoverall inductance is increased, the bandwidth of the actuator has to be sacrificed.
1.2 Summary of present work
In this work, a nonlinear parameter estimation technique is developed to estimate theposition. A signal processing filter is designed to demodulate the switching waveform.The output of the filter is essentially a function of gap displacement, power supply
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voltage, and duty cycle. The rejection of force feed–through (duty cycle change) isachieved by adding a parallel simulation of an inductor model in which the outputpasses through the same filter as the actual current waveform. The error betweenactual and simulated inductance is used to correct the model, in order to ensure thedisplacement tracking. The estimator which is internally complicated and nonlinear isapproximated as a linear system with a time–varying gain from an input/output pointof view. Using circle criterion, a stability boundary with respect to the controllergains is obtained. A stability boundary obtained by computer simulations verifiesthe conservativeness of the circle criterion. The performance of the estimator isevaluated in terms of linearity, bandwidth, and signal to noise ratio. The effects ofthe nonidealities on the performance of the estimator are also presented.
1.3 Dissertation Outline
The motivation and objectives of the dissertation research and a summary of priorresearch efforts on self–sensing magnetic bearings appear in the first chapter.
Chapter 2 contains a comprehensive survey of literature on self–sensing magneticbearings as well as the magnetic bearing system in general. The chapter also includesa survey of the existing methods of sensorless motor drive control and nonlinearmagnetization models.
A summary of magnetic bearing systems follows in Chapter 3, where essentialcomponents comprising the system are described. Normally, a magnetic bearingsystem includes feedback controllers, amplifiers, magnetic actuators, and sensors. Therole of each component is presented in this chapter. The effect of replacing discretesensors with position estimators is also discussed.
The model which will be used by the parameter estimator is derived in Chapter 4.The model is constructed from fundamental equations describing the inductor aftermaking reasonable assumptions. The model retains the inherent nonlinearity of aninductor: the inductance is inversely proportional to the air gap length. Also pre-sented in this chapter is a harmonic analysis of the switching waveform generated byapplying bi–state voltage to an inductor. The analysis results in an exact frequencyspectrum of the switching noise.
Chapter 5 provides a description of the test rig used for verifying the proposedapproach. The objectives in designing the test rig are stated. Modal analysis of thetest rig provides a justification for using a eddy current probe as a reference againstwhich the estimator is evaluated.
A detailed explanation of the forward path filter which demodulates the switch-ing noise is presented in Chapter 6. Design process and issues involving the actualimplementation of the filter are presented in this chapter. This includes discussionson the problems of the demodulation filter and the scaling approach. Results fromcomputer simulations and experiments conclude this chapter.
The main results of this dissertation are presented in Chapter 7. The designand implementation of the parameter estimator are explained. The performanceevaluation of the estimator is done in terms of linearity, bandwidth, and signal to noise
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ratio. The term force feed–through is defined, and the performance of the estimatoris characterized with respect to it. Results obtained by computer simulations andexperiments are presented, which validates the approach taken.
Chapter 8 deals with some issues neglected during modeling stage. Magnetic satu-ration, which is the one of the nonlinearities, limits the performance of the estimator.This limitation due to saturation is fundamental to any self–sensing approach. Alsoin this chapter, suggestions are made, which remedy the errors introduced by thenonidealities.
Chapter 9 concludes the dissertation with conclusions and suggestions for futureresearch.
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Chapter 2
Literature Review
Due to its interdisciplinary nature, this dissertation requires a vast amount of liter-ature for its background. Needless to say, many contributions to different aspects ofthis research have been disregarded. This chapter provides a brief review of the lit-erature which has been consulted for the dissertation. A survey of the prior researchon self–sensing magnetic bearings begins the chapter. As the operating principle issimilar to magnetic bearings, a survey of sensorless motor drives, while somewhattangential, is worth examining as well. Following is a very brief review of the litera-ture on magnetic bearings which include actuators, controllers, and sensors. A surveyof the parameter estimation and magnetization models concludes this chapter.
2.1 Self–Sensing Magnetic Bearing
Self–sensing magnetic bearings have been investigated by various researchers in thepast ten years. Their efforts can be grouped into two broad categories which differ-entiate the underlying theoretical approach. One category is parameter estimationwhere the bearing air gap is treated as a time–varying parameter of an isolated dy-namic magnetic system – the bearing and amplifier combination. The present workfalls into this category. The other approach is to treat the magnetic bearings and thesupported object as a whole. In this case, the position of the supported object is astate rather than a parameter.
2.1.1 Parameter Estimation
Okada et al. [64] reported a realization of a self–sensing magnetic bearing using ademodulation technique similar to the forward path filter described in Chapter 6.Since the gap displacement modulates the amplitude of the switching waveform, theswitching frequency component of the voltage and the current is a direct indicationof displacement. Gurumoorthyet al. [26] used a technique evolved from sensorlesscontrol of a switched reluctance machine [49]. By switching off a pair of coils fora short period of time and injecting a test signal to that pair, they estimated theposition of the bearing. The test signal is a series of rectangular pulses which generate
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triangular waveform. The amplitude of the triangular waveform is a direct functionof the gap displacement.
Mizuno [62, 59] used phase–locked loops to estimate the gap displacement of amagnetic bearing when a hysteresis switching amplifier powers the bearing. Whena hysteresis amplifier is used, the rate of the switching is dependent on the loadimpedance. Thus, the gap displacement is frequency–modulated with the switchingsignal. A phase–locked loop circuit is used to demodulate the switching signal. Similarto Okada’s approach, the result of the demodulation is a function of displacement andforce change. Thus the problem of the force feed–through still exists in this approach.Iannello [31] proposed a configuration where two opposing coils are wired in series.Gap position is estimated from the measurements of the voltage drops in two coils.
2.1.2 State–Space
Vischer et al. [10, 8, 9, 60, 61, 78, 79] used a linear state-space observer to estimatethe gap displacement. The magnetic bearings are treated as a two–port system. Thelinearized state–space equation describing the system is observable as well as control-lable with the voltage as input and the current as output. Therefore, it is alwayspossible to design a stable observer for the given system. In their paper, Mizuno andVischer claimed that the sensorless bearing equipped with a state observer carrieshigh static load capacity [60, 79]. Mizuno [61] modified the state–space equation toaccommodate the eddy current effect and verified the model with experiments.
2.1.3 Other Approaches
Using LC tuned circuit, it is possible to stabilize magnetic suspension systems [23, 35].This technology uses the variation in inductance of the electromagnet. The inductanceis determined by the gap between the electromagnet and the suspended object. Thecontroller and the sensing mechanism are integrated in the tuned circuit. Generally,the LC tuned circuit provides little damping. Thus, external damping is necessary.
2.2 Sensorless Electric Motor Control
In contrast to the scarcity of the research on sensorless magnetic bearings, many ef-forts have been described to eliminate the separate displacement sensor in electricmotors. Since the basic operating principles of magnetic bearings are quite similarto those of the motor, it seems worthwhile to survey the literature on the sensorelimination techniques in motors. However, it should be pointed out that there arefundamental differences between the self–sensing magnetic bearing and the electricmotor counterpart. First of all, electric motors need angular position for commutationand control, whereas bearings require linear position information. In electric motors,one can assume that the inductance profile around one cycle is known a priori – theinformation which is not available with magnetic bearings. While it is not desirableto switch off a part of a magnetic bearing, electric motors do have nonactuating coils
7
which have been used as sensing elements. In spite of these differences, techniquesdeveloped for sensorless motors might be useful to magnetic bearings. Moreover, itneeds to be emphasized that sensorless motor drives are already commercially avail-able. This commercial availability provides the prospect of commercial realization ofself–sensing magnetic bearings.
Acarnley et al. [1] devised a method to extract position information from the riseand fall time of current waveforms in a switched reluctance motor. The rise and falltime is a function of incremental inductance which, in turn, is related to the position.However, they assumed that back EMF in motoring operation is insignificant, whichis not the case at high speed. Panda et al. [67] improved this technique by consideringthe back EMF. With computer simulation, they showed the effect of back EMF onthe chopping current rise and fall times. They used an analytical expression of flux-linkage which was obtained from least-square curve-fitting. As a result, applying theirtechnique requires experimental flux-linkage data as a function of rotor position.
Ehsani et al. [22] took the advantage of the large inductance variation in switchedreluctance motor drives and subsequently used phase inductance to detect position.The inverse of inductance is directly proportional to the frequency with some as-sumptions. Consequently, they injected pulse signals to the nonconducting phaseand encoded output frequency modulation. The FM signal can then be convertedto a signal which is a function of the position. Obviously, this technique cannotbe applied to magnetic bearing applications as there is no idle phase. MacMinn et
al. [51] started from the same idea as Ehsani [22] but differed in how to measurethe inductance of a motor. An impedance sensing method was developed in which asensing current pulse is compared to prescribed threshold values in order to detectthe commutation angle.
Kulkarni et al. [43] developed a technique to eliminate the discrete position sensorin an interior permanent-magnet synchronous motor drive. In their prior work [22],they used the wave form detection technique to search for the commutation instantsin switched reluctance motors. However, in this research, they derived analyticalexpressions for phase inductance in the internal permanent magnet motor. Theythen used the expressions to estimate the rotor position with the help of a look-uptable. They performed computer simulations to show the validity of their method.
Sepe et al. [72] developed full observers and showed that adaptive control shouldbe used to compensate for parameter mismatch. They also performed real timesimulation. Their method is too complex to implement; therefore, its practicality isdoubtful. Furuhashi et al [24] noticed that sliding mode controllers are insensitiveto parameter variations and disturbances, and applied the technique to develop anadaptive sliding mode observer for brushless DC motors. With computer simulation,they showed the validity of their method. However, this technique is not suitable forlow speeds. Schauder [73] used a model-reference adaptive system (MRAS) for theestimation of induction motor speed from measured terminal voltages and currents.
Matsui et al. [55] developed an algorithm to calculate position and speed fromdetected currents and calculated voltages from DC voltage, PWM pattern, and thedead time information. They verified their method with computer simulation andexperiment. The basic idea of this method is that the angular difference between
8
the actual and hypothetical axes can be estimated by the voltage difference betweenthe actual and hypothetical axes. They investigated how much the hypothetical axescould be displaced from actual ones without instability.
2.3 Magnetic Bearing System
For an overview of magnetic bearing systems, the Handbook of Lubrication and Tri-
bology written by Allaire et al. [5] and Active Magnetic Bearings by Schweitzer [71]provide comprehensive material on the topic. Generally speaking, a magnetic bearingsystem consists of magnetic actuators, controllers, amplifiers, and sensors. Ph. D. dis-sertations by Maslen [53] and by Keith [38] offer extensive surveys on the design andanalysis of magnetic actuators. More recently, Lin and Jou [46] suggested a nonlin-ear model of force and current relationship within a magnetic suspension system andidentified parameters of the model by curve fitting the experimental data. Allaire et
al [4] investigated the flux/force relationship of a solid magnetic thrust bearing de-signed for a laboratory pump. They used the bearing to support the shaft as well asto measure the thrust forces on the impeller. Since the bearing is to be used as a loadcell, an exact relationship between the current and the magnetic force is necessary.Since the phase lag due to eddy currents is relatively large at low frequency, theyconcluded that the solid thrust bearing is inappropriate for the dynamic test.
Active magnetic bearing systems are equipped with feedback controllers to over-come open–loop instability. Many different types of controller algorithms have beenconsidered for magnetic bearing applications. Of these algorithms, Proportional, In-
tegral, and Derivative (PID) controllers appear to be the most widely used [29]. Thereason for the popularity can be attributed to the simplicity and robustness. Othercontrol schemes include LQG [79], time delay [81], sliding mode [76], and H∞ [63] toname a few. Kanemitsu [37] compared some of these control laws.
Amplifiers are necessary for generating appropriate currents with respect to thecommand signal from the controller. Maslen et al. [54] stated the requirements of anamplifier to achieve the targeted performance. Because of high efficiency, switchingamplifiers are preferred to linear amplifiers in many applications [6]. Keith et al. [40]proposed a new switching algorithm called Minimum Pulse Width (MPW) and com-pared it with existing switching methods such as pulse width modulation (PWM),hysteresis, and sample and hold. Recently, there have been some research effortsto improve linear power amplifier. Cerruti et al. [16] proposed a class D amplifierwhich has two different voltage sources. A low voltage source drives the resistiveloads, and a high voltage source supplies energy to reactive loads. Wassermann andSpringer [80] also developed an amplifier called Linear Power Amplifier with CurrentInjection (LACI) based on the similar idea by Cerruti. However, LACI incorporatesa linear amplifier with a switching amplifier in such a way that, during static oper-ation, the linear amplifier is activated, while the switching amplifier is reserved forhigh dynamic loads.
Sensors supply the information to the controllers. Either position or flux can bemeasured, but position sensing is more widely used. Keith [38] developed an implicit
9
flux feedback control using a Hall effect sensor. Position sensing techniques availableto date are well summarized in the master thesis of Maurer [56], where he designed avariable reluctance sensor for canned pump applications. Boehm et al. [11] comparedseveral different position sensors including the Hall sensor, capacitive sensor, andeddy current sensor. They concluded that the eddy current sensors have the bestcharacteristics in terms of bandwidth and phase shift.
2.4 Parameter Estimation
Parameter estimation and identification is a well developed branch of system theory(see, for example, the book by Ljung and Soderstrom [48]). Of the many differentparameter estimation techniques, approaches similar to this dissertation research canbe found in the early works of adaptive control. In his book [25], Gibson collectedseveral early adaptive control algorithms, and model–adaptive control in his defini-tion is conceptually similar to the approach taken in this research. Margolis andLeondes [52] took a similar approach, which they called a parameter tracking servo.
2.5 Magnetization Model
Magnetization model is a model that relates the magnetic field intensity H to theflux density B. This relationship is often modeled as a linear one. Fundamentally,however, the magnetization process is nonlinear in nature due to saturation andhysteresis. Since the hysteresis is observed in many different fields of science, there isa substantial literature devoted to this phenomenon. In this dissertation research, theliterature survey on the hysteresis models is confined to the ones applied to magnetichysteresis, although they can be used to describe other kinds of hysteresis.
Macki et al. [50] surveyed existing mathematical models of hysteresis, two of whichappear to be widely used. One model is called the Duhem hysteresis operator whichuses an integral operator or a differential equation to describe hysteresis. Colemanand Hodgdon [18, 19, 27, 28] presented a model for ferromagnetic hysteresis based onthe idea of the Duhem operator. The model is described by a differential equation,H = α|B|[f(B)−H]+ Bg(B, B), in which the constant α and the material functionsf and g can be obtained from experimental data. Mayergoyz [57] investigated thePreisach–Krasnoselskii model of hysteresis [69, 42] and provided necessary and suf-ficient conditions under which actual hysteresis nonlinearities can be represented byPreisach’s model. Domain wall motion is caused by pinning sites encountered by thedomain walls as they move. Jiles and Atherton [33, 34] derived a differential equationby considering impedances to this kind of motion. Taking the eddy current effectsinto account, Jiles [32] extended his model to include the frequency dependence of thehysteresis curves. The models of Hodgdon, Mayergoyz, and Jiles are all representedby either differential equations or integral equations. Hence, the actual B−H curvecan only be obtained by solving the equations analytically or numerically.
Some researchers have taken a practical approach and derived a model for thehysteresis. Chari [17] and Lin et al. [47] reported models for the major hysteresis
10
loops. Springer [75] considered minor loops as well as major loops expressed by aseries of discontinuous partial differential equations. O’Kelly [65] proposed a modelsimple in mathematical form which needs a set of parameters at each discrete fluxdensity value from experimental data. His model is markedly similar in mathematicalexpression to the one used in Appendix C, but the model used in this dissertationresearch requires only four parameters to represent the families of major and minorhysteresis loops.
11
Chapter 3
Magnetic Bearing System
In this chapter, it will be stated what constitutes a typical magnetic bearing system.Most of the material herein is adapted from [5]. Unlike conventional bearings suchas rolling element bearings and fluid film bearings, magnetic bearings require activecontrol as they are open–loop unstable. This instability also exists even when per-manent magnets are used, since it is not possible to passively suspend an object inall axes. For rotating machineries, at least one axis must be controlled actively outof five degrees of freedom. Rotor position is fed back to the controller via positionsensors. The controller sends a command signal to the amplifier which produces nec-essary currents in the actuator coils. Magnetic forces are generated by the currentsin the coils. Employing an appropriate controller will stabilize the suspended rotor.Figure 3.1 shows the components of a magnetic bearing system.
3.1 Actuator
When electric current passes through a wire wrapped around a core of ferromagneticmaterial, magnetic flux is induced. This flux produces an attractive force on anotherferromagnetic material. For a given area S, the force can be computed using
~F =1
2µ0
∫∫
S
| ~B|2 da (3.1)
After several assumptions are made, the magnetic force produced by the magnetiza-tion of NI for the system shown in Figure 3.1 can be written as
F = η · µ0N2I2Ag
4g2(3.2)
The force is linearly proportional to the square of the magnetization NI and inverselyproportional to the square of the gap g. The derating factor η can account for theleakage and fringing effects as well as nonuniformity of the flux density across Sin (3.1). This factor is a function of nominal gap g and peak force Fmax [38]. When aswitching amplifier is used to drive the actuator coil, the duty cycle of the amplifieris directly related to the average value of the current I.
SuspendingObject
Actuator
Amplifier
Position Sensor
Controller -
6
Figure 3.1: Schematic of a typical magnetic bearing system
3.2 Amplifier
The amplifier converts low power controller output signals into high power actuatorinput signals (currents). It accomplishes this by regulating the flow of energy betweena power source and a load. As described in the previous section, magnetic force isdirectly determined from the current. Thus, transconductance operation is generallypreferred. In this case, the power source would be a current source. There are twotypes of amplifiers : linear amplifiers and switching amplifiers. The output current ofthe transconductance linear amplifier is proportional to the input voltage. In otherwords,
Iout = γVin (3.3)
Figure 3.2 shows the schematic of a linear amplifier. The output transistor is eitherpower FET or IGBT. The amplifier is operating in the linear range of the transistor;therefore, the efficiency of the amplifier is very low (usually, 5 ∼ 10 %). Moreover,linear amplifiers have lower capacity than that of switching amplifiers.
Because of high efficiency and large capacity, switching amplifiers are more widelyused than linear amplifiers. Switching amplifier switches fixed voltage of either +Vs or−Vs very fast so that the average current output is approximately proportional to theinput voltage. Output transistors are either “on” or “off”. Usually, four transistorsare used in one amplifier channel, and form an H-bridge as shown in Figure 3.3. The
13
Figure 3.2: A monopolar linear transconductance amplifier
duty cycle of the amplifier is the percent time spent in the “on” state compared tothe switching interval. Efficiencies of the switching amplifiers are generally very high(80 ∼ 90 %). Drawbacks of switching amplifiers are :
• Output contains harmonic distortion.
• Output is contaminated by the switching noise.
When a demodulation technique is used to estimate the displacement, the switchingnoise becomes a signal rather than a noise.
Pulse width modulation (PWM) is widely used as the switching algorithm. Thereare several other switching algorithms which do not treat the switching component asa linear amplifier. Some examples of those algorithms are : sample and hold, hystere-sis, and time delay (or minimum pulse width) [40]. The objectives of these algorithmsare to limit the switching rate so that the switching efficiency is preserved, and tomatch the average output current with the average input current. Asynchronous al-gorithms such as PWM, hysteresis, and time delay, have significantly lower distortionthan synchronous algorithms represented by sample and hold.
3.3 Controller
The primary tasks of the controller are to coordinate the transformation of sensorsignals when necessary, to collect any other parameters needed by the control algo-rithm, and to generate control current requests according to the control algorithm.The controller may also handle output coordinate transformation and biasing of theamplifier signal. In terms of the secondary tasks, the controller needs to permit mod-ification of the control algorithm and to implement diagnostic measurements. Thecontroller can be analog, digital, or hybrid.
14
uu Load
eVs
ee
eeActivate
for +Vs
Activatefor −Vs
Figure 3.3: H-Bridge
With either analog or digital hardware, a control algorithm can be implemented.Many different control laws have been applied to magnetic bearing applications, asreviewed in Chapter 2. Kanemitsu [37] compared some of these control laws. Pro-
portional, Integral, and Differential (PID) controllers are most widely used for thefeedback control of magnetic bearings because of simplicity, easy realization, androbustness [29].
3.4 Position Sensor
As Earnshaw’s theorem proves, a magnetic suspension is unstable unless supercon-ducting magnets are used. This is the reason that it is impossible to realize a purelypassive magnetic suspension system with ordinary permanent magnets. Utilizing themeasured information on the object, the feedback controller regulates the current inthe coil in such a way that the suspending object is in place. The controller may ex-ploit such information as position of the rotor, flux density, current, or combinationof these. There have been some efforts using flux measurement as the input to thefeedback controller [38]. However, since the controller design becomes easier with po-sition measurements than other measurements, the position of the object is the mostpopular parameter for feedback. For a linearized model, it can be shown that thesystem becomes a non–minimum phase with other measurements. The requirementsof a position sensor for magnetic bearing applications are :
• operation must be noncontacting
• bandwidth of the sensor should exceed that of the amplifier and the bearing
• sensors need to be durable, stable, and affordable.
15
In addition to these requirements, it would be desirable that the sensor see throughthe nonmagnetic canning material for canned pump applications [7]. Sensors arecommonly employed in differential pairs to eliminate effects of thermal or centrifugalstress growth. There are several types of position sensors that have been used, orhave potential to be used, in magnetic bearing applications [56, 11], such as :
• ultrasonic probes
• capacitance probes
• hall effect probes
• optical probes
• laser probes
• eddy current probes
• variable reluctance probes
Ultrasonic probes measure the acoustic impedance which depends on target po-sition. This sensor type has the advantage of a very long range and usually hasgood dynamic (bandwidth) and static (linearity) characteristics. A major drawbackof this type of sensor is that the resolution is too low to be used in magnetic bearingapplications.
Capacitance position sensors work on the principle of an ideal plate capacitorwhose reactance changes linearly with distance between the two capacitor plates, oneof which can be the levitated body. This type of sensor usually has good stabilityand linearity characteristics. Furthermore, capacitance transducers are not affectedby magnetic fields and have low temperature drift. The main disadvantage of thecapacitance type sensor is that it has very poor dynamic characteristics [11].
When a magnetic field penetrates a semiconductor plate perpendicular to the cur-rent, the Lorentz force produces a potential difference across the plate. The potentialdifference, called the Hall voltage, is proportional to the product of the current acrossthe Hall element and the magnetic field strength. The Hall effect sensors can be madevery small in size at low cost. Compared to other types of sensors, however, eventhe temperature compensated Hall effect sensors have high sensitivity to temperaturechanges. Moreover, the output of the sensor is highly susceptible to magnetic noise.The bandwidth of the sensor is usually low as well.
Optical sensors are widely used in many industrial applications including magneticsuspension systems [14]. This type of sensor comes in several variations, but usuallyconsists of a light emitting diode and a photo–sensitive element. The principle ofoperation could be based on interferometry, reflectance of the target material, orocclusion. Generally, optical sensors have high bandwidth and excellent linearity.One of the drawbacks of optical sensors is that the transmittance of the space betweenemitter and receiver must remain constant – a condition which precludes these sensorsfrom many applications.
16
Another optical sensor is the laser displacement sensor, based on the triangulationof a light beam. A laser beam is emitted from a laser diode, and this beam is reflecteddiffusely by the levitated object. A linear position sensor element is focused ontothis point and, depending on the location the reflected light hits this element ofthe sensor, a distance–dependent signal is obtained. This sensor type is sensitiveto the reflectivity of the levitated object. Surface changes of the object influenceits accuracy. The cost of laser sensors is relatively high compared to other types ofsensors. As with all probes based on optics, laser sensors may not be used in cannedpump applications unless the canning material is transparent.
Eddy current position sensors are the most commonly used sensors in magneticbearing systems. This sensor consists of a probe tip which contains either one ortwo coils. Temperature and noise compensation requires a two–coil arrangement,consisting of a sensing coil and a reference coil. The currents in these two coils oscillateat a high, constant frequency, typically between 0.5 – 2 MHz. The high frequencyprimary currents induce eddy currents on an electrically conductive material close tothese coils. The eddy currents on the target produce their own electromagnetic fieldwhich interacts with the field from the sensor and causes an inductance change in thesensing coil. The inductance change is then converted to a position signal througha balance circuit. Eddy current sensors exhibit excellent frequency response withvery little phase shift [11]. Linearity and sensitivity of this sensor type are also verygood. However, a few disadvantages of this type of sensor arise in magnetic bearingapplications. When a switching amplifier is used, the switching noise from magneticbearings can couple strongly into the probe signals. Thus the probes need to be placedat a distance from the bearings. This noncollocation of sensor with actuator can causeinstability due to unmodelled dynamics [45]. Also, small variations in permeability,due to machining, cause electrical runout on the shaft. The main drawback of eddycurrent probes is the cost of the sensors. They constitute a significant part of theoverall cost of a magnetic bearing system. Furthermore, a small mismatch in carrierfrequency of adjacent probes can cause a large beat frequency output signal in thelow frequency band.
A variable reluctance probe relies on the principle that the reluctance of the airgap between the probe and the target is linear to the distance between them. Theprobe senses the impedance change of the carrier signal due to reluctance variation.The frequency of the carrier should be lower than that of the eddy current sensor.The bandwidth of the sensor is limited by the eddy current effects but can be realizedup to 2 kHz. The cost of this sensor type is much lower than that of eddy currentprobes.
The position estimator described in this dissertation can be considered as a vari-able reluctance sensor, although a stand-alone variable reluctance sensor would bedesigned with different objectives. As stated briefly in Chapter 1, magnetic actua-tors are usually designed to maximize the capacity and the thermal dissipation. Thisrequirement leads to a structure with magnetic paths whose reluctance becomes sig-nificantly larger than that of air gaps at high frequency [58]. On the other hand,sensitivity and rejection of magnetic nonlinearities improve in variable reluctancesensors if the iron reluctance is less than that of the air gap. Therefore, in terms
17
of characteristics defining sensors (such as sensitivity, bandwidth, and linearity), adiscrete variable reluctance sensor would perform better than a self–sensing bear-ing. Another drawback of position estimation is that the estimator requires a tuningprocess, since the underlying principle is based on the model of the bearing. Anycommercially available sensor also needs some degree of tuning, although the tuningprocess may take less effort than achieving a self–sensing bearing. For a given specificapplication, a designer must decide whether the advantages of self–sensing bearingsoutweigh the expected drawbacks.
18
Chapter 4
Modeling
In this chapter, a model of the bearing inductor is derived from the fundamentalequations governing the physics of the bearing. In order to obtain an idealized modelof the bearing inductor, the fundamental equations are simplified based on severalreasonable assumptions. These assumptions are also presented in this chapter. Themodel is used by the parameter estimator as a simulation of the actual coil. The modelalso shows how the displacement signal modulates the switching waveform. Harmonicanalysis of the switching waveform provides insights on the frequency spectrum of thewaveform and the ways in which demodulation can be achieved.
4.1 Coil Inductor Model
Illustrated in Figure 4.1 is a typical eight–pole magnetic bearing. Wires in each pairof adjacent poles are wired in series. The top pair and the bottom pair generate forcesvertically, while horizontal movement is restricted by the right pair and the left pair.Most of the flux generated by a current is confined within the path formed by thepair of the poles around which the current is flowing. As a result, this configurationhas very low cross–coupling between vertical and horizontal directions. If no mutualinductance is assumed, Faraday’s law of induction (for the pair of coils which arewound in series) can be written as
V = NdΦ
dt+ Ri. (4.1)
Another equation relating the flux Φ with the current i can be obtained by usingAmpere’s loop law which is stated as
∮
H · dl = Ni. (4.2)
Magnetization vector H is a function of the flux density. Assuming that the bearingis long enough that the flux in the axial direction is negligible, the loop law (4.2) canbe approximated as
Hclc + 2Hg(g ± x) = Ni. (4.3)
19
Figure 4.1: 8–pole magnetic bearing
The plus sign describes the loop law for the bottom pair of coils, while the minussign is related to the top pair of coils. Note that the positive displacement is definedas upward movement of the rotor from center position. The relationship betweenflux density and magnetic induction depends on the material. In the case of air,this relationship is linear and the proportionality constant is the permeability of freespace, µo:
Hg =Bg
µo
, (4.4)
where µo = 4π × 10−7 (T·m/A). Generally, the core material has nonlinear B−Hcharacteristics.
Hc = f(Bc, t). (4.5)
Nonlinearity includes hysteresis and saturation. Another magnetic nonideality is theeddy currents induced by the time–varying flux field B. As listed in Chapter 2, a widevariety of magnetization models exists. In Appendix C, a nonlinear model is proposedand compared with experimental data. Without the nonidealities, the B−H curve islinear, and can be described by
Hc =Bc
µoµr
, (4.6)
where µr is called the relative permeability of the material. This linear equationprovides a basis for the inductor model to be used in the real–time simulation of an
20
actual bearing coil. The aforementioned magnetic nonidealities would deteriorate theperformance of the estimator. Chapter 8 investigates in detail how the estimator isaffected by these nonidealities.
Assume that the cross–sectional area along the flux path remains constant andleakage and fringing fluxes are negligibly small. Then, using the conservation of flux,an approximation follows in which the flux density in the air gap is equal to the fluxdensity in the core.
Bc ≈ Bg. (4.7)
Substituting (4.4), (4.6), and (4.7) into (4.3), the magnetic flux in each flux path canbe obtained as
Φ =Ni
R , (4.8)
where the reluctance R is defined as
R =2(g ± x) + lc/µr
µoAg
. (4.9)
The total reluctance of the flux path R is the sum of the core reluctance and thereluctance of the air gap. The inductance of the coil can be obtained by differentiat-ing (4.8) with respect to current i and multiplying the number of coil turns N .
L = N∂Φ
∂i=
µoN2Ag
2(g ± x) + lc/µr
. (4.10)
The inductance is an inversely linear function of displacement x – a characteristicwhich is exploited in this work for the gap estimation. Nominal inductance L0 isdefined when the displacement is zero.
L0 =µoN
2Ag
2g + lc/µr
. (4.11)
Substituting (4.10) into (4.1), the simulation model of the bearing coil can be writtenas
di
dt=
2(g ± x) + lc/µr
µoN2Ag
(V − Ri − idL
dt). (4.12)
Voltage V in (4.12) is either −Vs or +Vs, since a bi–state switching amplifier isassumed to drive the bearing. The last term in the parenthesis of (4.12) is due toback electro–motive–force (EMF). Using chain rule, it can be shown that this backEMF is related to the velocity of the suspending object.
idL
dt=
∓µoN2Agxi
[2(g ± x) + lc/µr]2(4.13)
21
-2
µoN2Ag
- j?
--
l-
-∫
-
R
6
x
±
1
L0
+
V
i
+
−
Figure 4.2: Block diagram of the inductor model
In this dissertation, the gap is treated as a parameter describing the system (4.12). Ifback EMF term is included in the model, the estimation becomes difficult, since thegap is not only a parameter but also a state. Fortunately, the back EMF is usuallysmall compared to V − iR when a bi–state switching amplifier is used to generate V .In the sequel, the back EMF is ignored, and the inductor model is given as
di
dt=
2(g ± x) + lc/µr
µoN2Ag
(V − Ri). (4.14)
The effect of back EMF on the performance of the estimator is investigated in Chap-ter 8.
A block diagram representation of (4.14) is shown in Figure 4.2. The analogsimulation of the inductor used by the parameter estimation is based on this blockdiagram. A detailed description of the simulation model is presented in Chapter 7.
4.2 Switching Waveform
As mentioned in the previous section, the voltage across the bearing coil is suppliedfrom a bi–state switching amplifier. Since the resistance of the coil R is usually verysmall, the current waveform generated by the equation,
di
dt=
2(g ± x) + lc/µr
µoN2Ag
(V − Ri) (4.15)
22
1
sL(x) + R- -
?
6-
6-
6-
Voltage V Current i
Gap x
Figure 4.3: Gap displacement modulates the switching waveform
resembles triangular waveform. The structure of (4.15) also indicates that the currentwaveform can be considered as a carrier signal which the displacement signal modu-lates. AM radio is based on the same principle that low frequency signal modulatesthe amplitude of high frequency carrier wave. The modulation process is conceptuallyillustrated in Figure 4.3, where the voltage is assumed to be a square wave. In otherwords, the duty cycle of the amplifier is kept constant at 50 %.
The duty cycle of the amplifier is defined as the ratio of the time the amplifierapplies the positive voltage to the total switching interval. At steady state, whilemaintaining the constant average current, the duty cycle needs to remain constantideally at 50 %. In reality, however, the duty cycle needs to be slightly over 50 %, inorder to compensate the voltage drop due to the resistance of the coil. On the otherhand, if the bearing is required to generate dynamic forces, the duty cycle would betime varying. The resulting switching waveform would be similar to the one shownin Figure 4.4.
With constant duty cycle, gap position estimation is simply a matter of an ampli-tude demodulation of switching waveform. As the amplitude is a function of the dutycycle as well as the gap, a mechanism is needed to reject duty cycle dependence fromthe output of demodulation in order to estimate the displacement correctly. The firststep in building a position estimator would be to construct an amplitude demodula-tor and evaluate its performance subject to the variable duty cycle. In Chapter 6, anonlinear filter is constructed for this purpose.
Before moving into the design of the demodulation filter, harmonic analysis ofthe switching waveform seems appropriate. Such analysis provides some insights onhow to design a demodulator. When a switching amplifier is driving a bearing, it isgenerally assumed that the output of the amplifier (coil current) is tracking the input(requested current). The coil current is quadratically related to the magnetic force,
23
Figure 4.4: Switching waveform with time–varying duty cycle
but the relationship is often linearized.
Fmag ≈ kxx + kii. (4.16)
For a double–acting configuration (control currents of same magnitudes and oppo-site polarities are applied to two diametrically opposing coils), the magnetic forceis exactly linear to the control current. Quadratic or linear, the current is directlyrelated to the magnetic force. Since the switching amplifier is described by the dutycycle, one must draw a correlation between the duty cycle and the current. Once thiscorrelation is available, it is easy to specify the duty cycle variation in terms of forcefluctuation. The derivation of an equation relating the duty cycle variation to thecoil current is given below.
Assume that the requested current to the amplifier is given as
Ireq = Ib + Im sin ωt. (4.17)
This form of the requested current is very likely in cases where a rotor undergoessynchronous vibration. Assume that a natural sampling PWM switching amplifier [77]is used, and the switching frequency ωs is an integer multiple of ω. Then, the numberof switching cycles in one period of the requested current is
nc =ωs
ω
During one switching cycle in the time interval [tk, tk+1], the voltage waveform isshown in Figure 4.5. Two instants, t+ and t−, are defined as the moments at whichthe voltage changes from −Vs to +Vs and, conversely, in the opposite direction. Sincethe duty cycle for this time period is fixed at αk, these two timings are given as
t+ =τ
2(1 − αk)
24
-
+Vs
−Vsαkτ
tk t+ t− tk+1
Figure 4.5: Votage waveform during one switching cycle
t− =τ
2(1 + αk)
The transitions are assumed to occur instantaneously. The current waveform can beobtained by integrating the voltage signal over time. Assuming that the resistance ofthe coil is negligible, the current during the one cycle [tk, tk+1] is
ik(t) =
Ik −Vs
L(t − tk) , @, tk < t < tk + t+
Ik −Vs
L(1 − αk)τ +
Vs
L(t − tk) , @, tk + t+ < t < tk + t−
Ik +2Vs
L· αkτ − Vs
L(t − tk) , @, tk + t− < t < tk+1
(4.18)
From (4.18), the current Ik+1 at time tk+1 is
Ik+1 = Ik +Vsτ
L(2αk − 1). (4.19)
As assumed previously, the coil current is following the requested current. Therefore,
Ik = Ib + Im sin ωtk
Ik+1 = Ib + Im sin ωtk+1
The high frequency renders an approximation that
Ik+1 ≈ Ib + Im sin ωtk + Imωτ cos ωtk
= Ik + Imωτ cos ωtk(4.20)
If the duty cycle is given of the form
αk =1
2+ αm cos ωtk ,
then by comparing (4.19) with (4.20), the variation of the duty cycle from 50 % is
αm =ωLIm
2Vs
. (4.21)
Regarding (4.21), two observations can be made :
25
1002 5 101
2 5 1022 5 103
2 5 104
Frequency (Hz)
10-2
2
5
10-1
2
5
100
2
5
101
2
5
102
Dut
ycy
cle
varia
tion
m(%
)
Figure 4.6: Duty cycle variation for Im = 1.
1. Duty cycle variation is proportional to the driving frequency ω. In other words,for a fixed current fluctuation Im, the duty cycle variation at low frequencywould be much smaller than the variation at high frequency. In Figure 4.6, theduty cycle variation is less than 1 % until the frequency is 100 Hz, for the casewhere Im = 1 Ampere.
2. For a given current fluctuation, the bandwidth of the amplifier can be estimatedfrom (4.21), using the fact the αm should be less than 50 %. When the currentfluctuation is 1 Ampere, Figure 4.6 shows that the amplifier is unable to generatethe current tracking the requested current after 5 kHz. Since the resistance ofthe coil is ignored, this estimation is apparently conservative.
Harmonic contents of the switching waveform shown in Figure 4.4 can be identifiedby approximating the waveform with a Fourier series
i(t) = a0 +∞
∑
n=1
(an cos nωt + bn sin nωt) (4.22)
and evaluating the Fourier coefficients an and bn. Appendix A details the procedureof deriving the coefficients analytically. The magnitude of each harmonic is definedas
|In| =√
a2n + b2
n (4.23)
26
1022 5 103
2 5 1042 5
Frequency (Hz)
10-10
10-9
10-8
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
101
Mag
nitu
de
Figure 4.7: Harmonic contents of switching waveform
Figure 4.7 shows the normalized harmonics of the switching waveform, when theduty cycle is changing at the frequency of 500 Hz. Higher harmonics are very smallcompared to the fundamental except at the switching frequency at35 kHz.
27
Chapter 5
Experimental Setup
Based on the model presented in the previous chapter, an amplitude–demodulationfilter and a parameter estimator are implemented using analog circuit components.The performances of the filter and the estimator are evaluated through computersimulations and experiments. Before proceeding to the design of the demodulationfilter, this chapter explains the details of the experimental setup.
The setup consists of a slender beam, an 8–pole magnetic bearing, a switchingamplifier, and a position sensor used as a reference signal. The details of each com-ponent are presented in this chapter. First, the objectives in designing the test rigare stated, followed by the specifications of the test bearing. Brief descriptions of theswitching amplifier and the position sensor used in the rig are presented in Section 5.3and 5.4. Finally, results of the dynamic analysis of the test beam are given in Sec-tion 5.5, along with a discussion on how the sensor–actuator noncollocation affectsthe position sensor as a reference signal.
5.1 Design of the Test Rig
A test rig is designed to verify the proposed parameter estimator experimentally. Fig-ure 5.1 shows the sketch of the test setup. The journal is supported by a beam whichhas a greater compliance in the vertical direction than in the horizontal direction.This compliance prevents the instability caused by the negative stiffness of the mag-netic bearing. The bearing is an 8–pole design, which is described in Section 5.2 indetail. Two toggle clamps can be positioned at various points along the beam on eachside of the bearing. By controlling the clamping location, various structural resonantfrequencies can be obtained. Dynamic analysis of the beam confirms the range ofthe resonant frequenies as well as the sensor/actuator noncollocation. The analysisis described in Section 5.5. While injecting constant biases into the top and bottomcoils, a free vibration (therefore, constant duty cycle) can be generated by hitting thebeam with a hammer. A sinusoidal coil current can also be requested through thebearing amplifier for a forced vibration (time–varying duty cycle). The motion of thejournal is measured with a eddy current type position probe as a reference signal.
28
Figure 5.1: Sketch of the experimental setup
5.2 Specifications of the Test Bearing
The bearing used in the test is an 8–pole design, schematically similar to the oneshown in Figure 4.1. Table 5.1 lists the dimensions and specifications of the bearing.The load capacity of the bearing can be calculated from
Fmax = 2 × 1
2µo
B2sAg (5.1)
where Bs is the saturation flux density. The saturation flux density depends on thematerial. For silicon iron, Bs is usually around 1.2 Tesla, which defines the maximumforce that the bearing can generate. In this dissertation, however, the load capacityis defined as the amount of force that the bearing is able to generate in the linearoperating range. Thus, Bs is lower than the actual saturation flux density (0.7 Teslain this case), which results in the load capacity of 38.9 N (8.74 lbf) using (5.1).Assuming the reluctance due to core material is negligibly small, the flux density canbe obtained from the loop law :
B =µoNI
2g(5.2)
Thus, the current which causes the material to saturate is
Is =2gBs
µoN= 4.1 (Amp) (5.3)
29
Number of poles 8Diameter of rotor 38.1 mm (1.5 in.)Inside diameter of stator 39.1 mm (1.54 in.)Outside diameter of stator 81.3 mm (3.2 in.)Axial length of stator 12.7 mm (0.5 in.)Stator pole height 13.7 mm (0.54 in.)Stator pole width 7.9 mm (0.31 in.)Radial gap, g0 0.508 mm (0.020 in.)Number of coil turns per leg, N/2 69Pole face area, Ag 99.68 mm2 (0.1545 in.2)Length of iron core length, lc 79 mm (3.12 in.)Nominal inductance , L0 2.35 mH
Table 5.1: Critical dimensions and specifications of the test bearing
5.3 Amplifier
A Minimum Pulse Width (MPW) switching amplifier [40, 39] is used to drive thetest bearing. The output device of the amplifier is H–bridge, which is shown inFigure 3.3. The switching rate of the amplifier is 70 kHz. Thus, at constant dutycycle, the switching frequency is 35 kHz. A direct current power supply supplies afixed voltage of 70 Volts to the amplifier.
5.4 Position Sensor
As a reference signal to the estimator, an eddy current probe is attached to the bear-ing. For calibration, a static test is done using the journal as a target. Figure 5.2shows that the output voltage of the sensor monotonically decreases as the gap be-tween sensor head and target increases. A linear regression over the range between13.4 mm and 14.0 mm results in a sensor gain of 4.28 V/mm (0.109 V/mil).
5.5 Dynamic Analysis of the Test Rig
A dynamic analysis of the beam used in the test rig is necessary to check the resonantfrequencies of the beam and to assess the effect of sensor noncollocation with thebearing. Figure 5.3 shows the mode shapes of the beam without toggle clamps. Thefirst bending mode occurs at 131 Hz which is close to the experimentally measuredfrequency, 122 Hz.
When the toggle clamps are secured at various positions along the beam, the beamhas different resonant frequencies. Using this behaviour, the frequency spectrum ofthe estimator can be evaluated. The dotted line in Figure 5.5 shows the variation ofthe resonant frequency according to the different locations of the toggle clamps. Therelative position of a toggle clamp, d, is defined in Figure 5.4.
30
13.2 13.3 13.4 13.5 13.6 13.7 13.8 13.9 14.0Gap (mm)
-6.0
-5.5
-5.0
-4.5
-4.0
-3.5
-3.0
-2.5
-2.0
-1.5
-1.0
Sen
sor
outp
ut(V
)
Figure 5.2: Position sensor static test result
Since the eddy current probe is not located at the same position as the bearing,the probe do not directly measure the movement at the bearing location. This inac-curacy becomes severe when the clamps are close to the bearing. Given the relativepositions of the clamp, the sensor, and the bearing shown in Figure 5.4, the firstbending mode of the beam with appropriate boundary conditions gives an indicationof noncollocation effect. The solid line in Figure 5.5 shows the ratio of sensor mea-surement to the actual displacement xs/xa as a function of the position of the clampsalong the beam. When d/L is over 0.8, the sensor substantially underestimates theactual displacement at the bearing.
This noncollocation effect places an upper limit on the frequency below whichthe estimator can be tested. As demonstrated in Figure 5.5, the maximum resonantfrequency which can be obtained from the test rig is around 900 Hz. The maximumoccurs when the ratio of the toggle clamp position to the length of the beam is0.8. Another limitation on the frequency is that the vibration level becomes verysmall when the toggle clamps are close to the bearing. Since the static deflection isproportional to the cubic of the beam length, the magnitude of vibration at xs/xa =0.8 would be 0.8 % of the level obtained without toggle clamps. The forces with thesame magnitude are assumed in both cases. If the vibration level is smaller thanthe resolution of the sensor or of the estimator, the dynamic test data would bemeaningless. Due to these two limitations, experimental evaluation of the estimatoris restricted. From repeated experiments, it is determined that the dynamic testbeyond 500 Hz is unreliable.
31
First Mode (131 Hz)
Second Mode (493 Hz)
Third Mode (1622 Hz)
Figure 5.3: Mode shapes and natural frequencies of the test beam
32
ClampSensor
-
-d
L
6 6
xs xa
Bearing
-
Figure 5.4: Relative locations of toggle clamp, sensor, and bearing
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0d/L
0.8
0.85
0.9
0.95
1.0
1.05
1.1
x s/x
a
0
200
400
600
800
1000
1200
1400
1600
Firs
tmod
e(H
z)
First modexs/xa
Figure 5.5: Mismeasurement of position sensor due to noncollocation and resonantfrequencies at various toggle clamp positions
33
Chapter 6
Switching Noise Demodulation
The forward path filter demodulates the switching waveform and filters out the signalwhich is a function of duty cycle, power supply voltage, and gap length. In thischapter, an idealized analysis of the forward path filter is presented to show how itdemodulates the amplitudes of switching noise. The forward path filter consists ofa high pass filter, a full wave rectifier, and a low pass filter. The latter part of thischapter is devoted to explaining how these components are implemented with actualdevices. The results of computer simulations and experiments conclude the chapter.
6.1 Idealized Analysis
The demodulation process using the forward path filter can be illustrated in two ways.Assuming a sinusoidal displacement and fixed duty ratio, a harmonic analysis of theprocess reveals that the output of the filter is indeed a function of the displacement.The relationship between the output of the filter and the displacement can also beshown by an analysis of the filter action during one switching cycle. The analysisis idealized since it assumes that the filters are perfect. This second analysis is veryuseful in showing the effect of duty cycle on the output of the filter.
When the duty cycle is fixed at 50 %, the voltage waveform becomes a square wave,and the current waveform is just a triangular wave whose amplitudes are an indicationof the gap displacement, as illustrated in Figure 4.2. Expanding the voltage waveformwith a Fourier series, it is possible to analytically show the amplitude–demodulationprocess through harmonic analysis. The analysis assumes that the displacement issinusoidal. Demodulation is achieved by using a full-wave rectifier and a low–passfilter. Given the model of the inductor derived in Chapter 4 as
di
dt=
2(g ± x) + lc/µr
µoN2Ag
(V − Ri) , (6.1)
and assuming that the displacement is represented by a sinusoid x = xm sin ωt, theaction of the rectifier and the low–pass filter can be shown analytically. The detailsof the harmonic analysis are presented in Appendix B. The analysis concludes that
34
-
+Vs
−Vsαkτ
tk t+ t− tk+1
Figure 6.1: Voltage applied to coil during one switching cycle
the output of the filter can be expressed as
u =8
π3
(
Vsτ
µoN2Ag
)(
g ± x +lc
2µr
)
. (6.2)
Indeed, the result of demodulation is a monotonic function of the displacement x.The preceding analysis was based on such assumptions as:
• A much higher switching frequency than the bandwidth of the dynamic system.
• Fixed duty cycle of the amplifier at 50%.
• Ideal low–pass filter.
The first assumption is generally true, since most of the dynamic systems have abandwidth not higher than 2 or 3 kHz. Therefore, the ratio between the frequency ωand ωs is less than 0.1.
(
ω
ωs
)
max
< 0.1 (6.3)
An imperfect low–pass filter will have some effect on the performance of the forwardpath filter, although the effect should be small if the first assumption holds. Thesecond assumption is the most significant for degrading the performance of the filter.Investigation of the filter action over one switching cycle makes it clear how the filteris affected by the time–varying duty cycle.
Assuming that the voltage waveform applied to the bearing coil during one switch-ing cycle is as shown in Figure 6.1, the current waveform can be obtained by inte-grating the input voltage signal. Rewriting (4.18) with k = 0 and α = αk for the sakeof convenience,
i(t) =
I0 −Vs
L(x)t , @, 0 < t < t+
I0 −Vs
L(x)(1 − α)τ +
Vs
L(x)t , @, t+ < t < t−
I0 +2Vs
L(x)· ατ − Vs
L(x)t , @, t− < t < τ
(6.4)
35
where t+ and t− are the moments at which the polarity of the voltage changes from−Vs to +Vs and from +Vs to −Vs, respectively. The duty cycle α is defined as
α ≡ t− − t+
τ(6.5)
Then t+ and t− can be given in terms of the duty ratio α.
t+ =τ
2(1 − α)
t− =τ
2(1 + α)
Since the switching occurs at high frequency, it is reasonable to assume that thelow frequency component of the signal varies linearly with time during one switchingcycle. Therefore, the high–pass filtered switching waveform can be approximated as
iHP (t) = i(t) −[
I0 +Vs
L(x)(2α − 1)t
]
(6.6)
Physically, the high–pass filter removes the low frequency component from the currentsignal which is generated due to duty cycle variation. Thus, (6.6) has an average valueof zero. As before, the demodulation of the filtered signal is achieved by a rectifier anda low–pass filter combination. Eq. (6.7) shows the rectifier output of the high–passfiltered signal.
|iHP (t)| =
Vs
L(x)· 2αt , @, 0 < t < t+
Vs
L(x)(1 − α)(τ − 2t) , @, t+ < t <
τ
2Vs
L(x)(1 − α)(2t − τ) , @,
τ
2< t < t−
Vs
L(x)· 2α(τ − t) , @, t− < t < τ
(6.7)
Figure 6.2 illustrates how the signal is processed after each pass. The application ofan ideal low–pass filter to the output of the rectifier results in the average value ofthe output over one switching interval.
u =1
τ
∫ τ
0
|iHP (t)|dt (6.8)
= α(1 − α)
(
Vsτ
µoN2Ag
)(
g ± x +lc
2µr
)
(6.9)
The duty cycle represents the average voltage applied to the bearing coil. Hence thefiltered output is not only dependent on the gap but also on the time rate of thebearing force. This result is quite similar to the one in [64], where the estimator isbased on the same principle as the forward path filter, but differs in implementation.
36
Rectifier
High Pass Filter
0 τ/2 τ
Figure 6.2: Representation of signal at each processing point
37
An interesting observation can be made from (6.9). Obviously, if the duty cycleis measured, then obtaining the gap displacement is simply a matter of scaling thefilter output. There are, however, several problems with this approach. A directmeasurement of the duty cycle can be obtained only by measuring the average outputvoltage. Such measurements can introduce additional delay and filtering. Even if themeasured α is available, the division process can be complicated and time consuming.The most serious drawback of the scaling approach is that the end result is only asgood as the forward path filter performance. As discussed later in this chapter andin the next chapter, the performance of the forward path filter is unsatisfactory fora position estimator. The parameter estimator described in Chapter 7 improves theestimation considerably.
6.2 Realization
There are several aspects of implementing the forward path filter which deserve someattention. The filter is composed of a high–pass filter, a full–wave rectifier, and alow–pass filter. All of these components are realized with analog integrated circuits.
6.2.1 High Pass Filter
The high pass filter is necessary to remove any low frequency component in theswitching signal. The low frequency components are generated due to duty cyclevariation. These components are directly related to the electromagnetic forces inthe bearing. The actual filter is implemented using a Burr-Brown UAF42 universalactive filter chip [15]. The filter is a state variable filter [44] and can be configured asa low–pass, a high–pass, or a band–pass filter. Figure 6.3 shows the circuit diagram.Four resistors R1 and two capacitors C1 are supplied with the chip as well as fouroperational amplifiers. The resistance of R1 is 50K 0.5% and the capacitance of C1
is 1000pF with the accuracy of 0.5% as well. A state–space equation can be used toexplain the basic principle of the filter. Let x1 be the output of the first operationalamplifier from the left and x2 be the output of the second operational amplifier. Then,two first–order differential equations can be derived.
[
x1
x2
]
=
[
−2ζhωc1 ωc1
−ωc1 0
] [
x1
x2
]
+
[
KHP ωc1
0
]
Vin (6.10)
The cut–off frequency ωc1 is determined by the external resistor RF1.
ωc1 =1
RF1C1
.
The purpose of the high–pass filter is to extract only the switching noise from the cur-rent waveform. Therefore, the cut-off frequency of the filter should be selected belowthe switching frequency but high enough to remove any low frequency componentswhich might be present in the bandwidth of the actuator. In the experiments, the
38
Figure 6.3: UAF42 configured as a high pass filter
frequency is chosen at 4.46 kHz which is about 0.13 of 35 kHz switching frequency.The damping factor ζh is set by two other external resistors RG and RQ.
ζh =1
2
(
2 +R1
RG
)(
RQ
RQ + R1
)
.
To obtain a maximally flat response in the pass band, the damping factor should be0.707 (Butterworth filter). Due to the discreteness of the component values, a ζh of0.713 is used in the experiments. The forward gain KHP is defined as
KHP =R1
RG
The last operational amplifier is used as a low–pass filter whose cut–off frequency ismuch higher than the switching frequency. One purpose of the filter is to reduce anyunwanted high frequency noise. The filter also amplifies the signal so that the rectifierfollowing the filter operates properly. The transfer function of the overall high–passfilter stage (more accurately band–pass filter) is
Vout
Vin
= K1
(
s2
s2 + 2ζhωh1s + ω2
h1
) (
ωh2
s + ωh2
)
(6.11)
where the forward gain K1 is defined as
K1 = KHP
(
RF2
R2
)
39
Component Values AccuracyRG 20 kΩ 1 % externalRQ 23.2 kΩ 1 % externalR1 50 kΩ 0.5 % internalRF1
35.7 kΩ 1 % externalR2 10 kΩ 1 % externalRF2
49.9 kΩ 1 % externalC1 1000 pF 0.5 % internalC2 20 pF 10 % external
Table 6.1: Component values used in high–pass filter stage
1022 5 103
2 5 1042 5 105
Frequency (Hz)
10-3
2
5
10-2
2
5
10-1
2
5
100
2
5
101
2
Mag
nitu
de
MeasuredPredicted
1022 5 103
2 5 1042 5 105
Frequency (Hz)
0
50
100
150P
hase
(deg
.)
MeasuredPredicted
Figure 6.4: Frequency response of the high–pass filter stage
and ωc2 is determined by
ωc2 =1
RF2C2
Table 6.1 lists the actual component values used in the experiments. In Figure 6.4, itis shown that the Bode diagram of the transfer function (6.11) agrees well with themeasurements.
6.2.2 Full Wave Rectifier
As described in Section 6.1, a simple way to achieve an amplitude demodulationis through the use of a rectifier followed by a low–pass filter. Figure 6.5 shows arealization of an active full wave rectifier [36].
During the design stage of the rectifier, one needs to be cautious that the inputsignal to the rectifier is within a bound. The lower end of the bound is determinedby the threshold value below which the signal to noise ratio of the rectifier outputis unacceptably low. The other end of the bound is decided by either the slew rate
40
Figure 6.5: Circuit schematic of full wave rectifier
limitation or the output limitation. All op amps have slew rate limitation, usuallygiven as volts per microseconds. If the input voltage swing is too large, the op ampmay not be able to amplify signals without significant phase delay. Output limitationmeans that there is a limit for the magnitude of the rectifier output. The outputof the rectifier passes through a low–filter, and then is added by a constant voltagewhich accounts for the nominal gap. If the output of this summing junction is greaterthan the supply voltage, the sensitivity of the filter to the displacement would becomezero. These upper and lower bounds can be determined during the tuning process.
6.2.3 Low Pass Filter
A low–pass filter is implemented with the same state–variable filter as the high–passfilter. The cut–off frequency directly determines the bandwidth of the forward pathfilter and will be limited by the high–pass filter. In the experiments, the cut–offfrequency is set at 1.75 kHz. Figure 6.6 illustrates the realization of the low–passfilter stage using UAF42. Table 6.2 lists the component values selected to implementthe filter stage. The transfer function of the filter stage is
Vout
Vin
= K2
(
ω2l
s2 + 2ζlωls + ω2l
)
(6.12)
The damping factor ζl is set at 0.651, which makes the filter stage slightly under–damped. In Figure 6.7, the measured frequency response of the low–pass filter is
41
Figure 6.6: UAF42 configured as a low–pass filter
Component Values AccuracyRG 50 kΩ 1 % externalRQ 38.3 kΩ 1 % externalR1 50 kΩ 0.5 % internalRF1
90.9 kΩ 1 % externalR2 10 kΩ 1 % externalC1 1000 pF 0.5 % internal
Table 6.2: Component values used in the low–pass filter
42
1012 5 102
2 5 1032 5 104
Frequency (Hz)
10-2
2
5
10-1
2
5
100
2
Mag
nitu
de
MeasuredPredicted
1012 5 102
2 5 1032 5 104
Frequency (Hz)
-180
-160
-140
-120
-100
-80
-60
-40
-20
0
Pha
se(d
eg.)
MeasuredPredicted
Figure 6.7: Frequency response of the low–pass filter stage
compared with the Bode diagram of the transfer function (6.12). Again, the analysispredicts the measurements with little error.
6.3 Simulation and Experimental Results
The performance of the forward path filter is evaluated through computer simula-tions and experiments. Simulations are designed to emulate the actual implementa-tion as closely as possible. First, models of the high–pass filter (6.11) and low–passfilter (6.12) are discretized suitable for a digital simulation. Then, a model of theamplifier (either PWM or time delay) is digitally simulated. With the digital rep-resentation for each component of the filter, the following steps are taken for thesimulations.
1. Assume that the movement of the suspending object is sinusoidal and can berepresented as g + xm sin ωt.
2. To emulate the time–varying duty ratio, a sinusoidal current Ireq = Ib+Im sin ωtis requested to the amplifier. As discussed in Section 4.2, the duty cycle varia-tion is directly related to the current fluctuation Im and the driving frequencyω.
3. The model of the switching amplifier generates a pulse train of voltage accordingto the requested current.
4. The inductor model (6.1) is integrated over time with the gap displacement andthe voltage input.
5. The current waveform passes through the digital simulation of the high–passfilter, the rectifier, and the low pass filter.
43
20 22 24 26 28 30 32 34 36 38 40time (msec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
x,x
(m
)
EstimatedActual
20 22 24 26 28 30 32 34 36 38 40time (msec)
-50
-40
-30
-20
-10
0
10
20
30
40
50
erro
r(
m)
Figure 6.8: Forward path filter response when ω = 2π · 120
In Figure 6.8, one can see that the output of the filter is following the actual dis-placement fairly well, but with noticeable phase lag to the actual displacement whenω = 2π · 120 (rad/sec).
The phase lag (or bandwidth) determines if the filter is viable as a replacementof the discrete position sensor. One can evaluate the bandwidth of the filter by itsfrequency response. As the filter is a nonlinear device, a sine sweep test needs to bedone in order to obtain a frequency response. Thus, the aforementioned steps areperformed for various frequencies, and the harmonic components are computed by
44
numerically calculating the Fourier integral,
an =2
T
∫ T
0
x sin nωtdt (6.13)
bn =2
T
∫ T
0
x cos nωtdt (6.14)
The magnitudes of the harmonic components are
|Xn| =√
a2n + b2
n (6.15)
and the phase angles of the harmonic components are
6 Xn = tan−1
(
an
bn
)
(6.16)
Figure 6.9 shows the frequency response of the forward path filter obtained throughsimulated sine sweep test. The current request is maintained at Ib = 2.5 [Amp], whichwill hold the duty cycle constant. Magnitudes of the first two harmonic componentsX1, X2 are shown as ratios to xm. Higher order components are not drawn simplybecause their magnitudes are negligible small. In the figure, the primary harmoniccomponent X1 varies little with the frequency, while the second harmonic componentX2 increases appreciably at high frequencies. Any harmonics Xn other than thefundamental, X1, act as noises. Therefore, the signal to noise ratio of the filter athigh frequency is unsatisfactory. Besides the superharmonic noise, the filter outputhas a significant phase lag to the actual displacement. To be precise, the phaselag is over 30 degrees after 500 Hz. As the estimator will be eventually used as areplacement of the discrete position sensor, the phase lag should be kept as small aspossible. A large phase lag will destabilize the bearing controller.
The frequency response of the forward path filter is shown in Figure 6.10, inwhich the request current is sinusoidal. As discussed before, this sinusoidal requestedcurrent forces the duty cycle to be time–varying. The magnitude of X1 becomesslightly larger than the fixed duty cycle case in high frequency region. Since theoutput of the filter is quadratically related to the duty cycle, the time–varying dutycycle has a greater effect on the second harmonic than the first. Compared to thefixed duty cycle case, X2 has increased by about an order of magnitude in the lowfrequency region. Therefore, the time–varying duty cycle considerably affects thesignal to noise ratio of the filter. A variable duty cycle causes little change on thephase of the first harmonic component.
As well as the increase of the second harmonic, the time–varying duty cycle hasa detrimental effect on the estimation of the nominal gap. When the duty cycleis time–varying, Figure 6.11 shows that the DC component of the filter drasticallyincreases with the driving frequency. If the duty cycle is fixed, the DC component isunchanged.
Experiments verify the results of computer simulation in which the time–varyingduty cycle degrades the performance of the forward path filter as a position estimator.
45
2 5 1022 5 103
2
Frequency (Hz)
10-5
10-4
10-3
10-2
10-1
100
101
Mag
nitu
de
2X1X
2 5 1022 5 103
2
Frequency (Hz)
-200
-150
-100
-50
0
50
Pha
se(d
eg.)
2X1X
Figure 6.9: Frequency response of forward path filter when the duty cycle is fixed
46
2 5 1022 5 103
2
Frequency (Hz)
10-5
10-4
10-3
10-2
10-1
100
101
Mag
nitu
de
2X1X
2 5 1022 5 103
2
Frequency (Hz)
-200
-150
-100
-50
0
50
Pha
se(d
eg.)
2X1X
Figure 6.10: Frequency response of forward path filter when the duty cycle is time–varying
47
2 5 1022 5 103
2
Frequency (Hz)
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
g/g
variable duty cyclefixed duty cycle
Figure 6.11: Ratio of offset to nominal gap
For a test with fixed duty cycle, constant bias currents are maintained in the top andbottom pairs of coils in the bearing described in Chapter 5. The beam is hit by ahammer to induce a free vibration, which is measured by an eddy current probe.Figure 6.12 shows that the filter output is following the actual displacement aftergain and offset adjustment.
Variable duty cycle is obtained by requesting sinusoidal current in the coil to theswitching amplifier, thus forcing the beam to move continuously. Figure 6.13 showsthe filter output along with the actual displacement when the beam is experiencing aforced vibration. One can see that the output is noisier than the fixed duty ratio case.This is because the demodulation filter does not employ a mechanism to increase thesignal to noise ratio.
Shown in Figure 6.14 is the experimentally obtained frequency response of theforward path filter. Due to the lack of data and restrictions of the test rig discussedin Chapter 5, the quality of the plot is not satisfactory. Nonetheless, within thecapacity of the test rig, one can make a relevant point regarding the experimentallymeasured spectrum. Notice that there is a significant variation in the magnitudeand phase of the filter output. This variation reveals that the output is severelycontaminated by the noise. As discussed previously, the signal to noise ratio of theforward path filter is affected by the superhamonics of the filter output. Moreover, thefilter has no mechanism to enhance its signal to noise ratio. The parameter estimatorproposed in this dissertation employs a feedback control which increases the signal tonoise ratio significantly.
48
0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1time [sec]
-0.15
-0.1
-0.05
0.0
0.05
0.1
0.15
x,x
(mm
)
FP filter outputActual motion
Figure 6.12: Forward path filter response of free vibration
0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1time [msec]
-0.1
-0.08
-0.06
-0.04
-0.02
0.0
0.02
0.04
0.06
0.08
0.1
x,x
(mm
)
FP filter outputActual motion
Figure 6.13: Forward path filter response of forced vibration
49
1012 5 102
2 5 103
Frequency (Hz)
10-1
2
5
100
2
5
101
Mag
nitu
de
1012 5 102
2 5 103
Frequency (Hz)
-150
-100
-50
0
50
100
150
Pha
se(d
eg.)
Figure 6.14: Experimentally obtained frequency response of the forward path filter
50
Chapter 7
Parameter Estimation
In the previous chapter, the limitations of the forward path filter as a position esti-mator were described. A parameter estimation technique parameterizing the air gapis used to enhance the performance of the estimator. The current chapter starts witha general description of the approach taken in the dissertation research, followed by asection explaining the principle of the estimator. The material that follows is a proofconcerning the stability of the estimator. Issues involving the actual implementationof the estimator are discussed. Finally, the performance of the parameter estimatoris evaluated.
7.1 General Approach
The proposed method using a parameter estimation technique estimates the air gaplength in the following manner. As before, the measured current waveform is filtered(using the forward path filter) to extract the amplitude of the switching waveform.At the same time, a real time analog simulation of the magnetic bearing inductanceis supplied with the actual amplifier output voltage V and an estimate x (assumedvalue) of the air gap.
di
dt=
2(g ± x) + lc/µr
µoN2Ag
(V − Ri). (7.1)
The resultant simulated switching waveform is filtered through the same forwardpath filter as the measured waveform. The amplitude of the simulated current isthen compared with that of the actual current, and the error is used to update theestimated air gap:
x∗ = arg minx
[
|i| − |i(x)|.]
(7.2)
This approach is called parameter estimation because the position estimation updatesthe air gap length which is a parameter of the model. This approach may be called aprimitive adaptive system [25] or a parameter tracking servo system [52]. Figure 7.1shows the overall structure of the parameter estimator.
51
ForwardPath Filter
ForwardPath Filter
j PI?
6
+
−
-
-
-i
i
BearingModel
-
?
-
x
V
Figure 7.1: Overall schematic of the estimator
7.2 Idealization
The output of the forward path filter as a function of the duty cycle and the gapdisplacement has been derived in the previous chapter.
u = K(g ± x). (7.3)
The reluctance of the stator is ignored for the convenience of analysis. The gain Kis a function of the duty cycle α, and thus is time–varying.
K =1
2α(1 − α)
Vsτ
gL0
(7.4)
where the nominal inductance L0 is defined as
L0 =µoN
2Ag
2g.
Regarding (7.3), one needs to realize that it is derived through an analysis in thetime domain, based on the assumptions that the high–pass and low–pass filters areperfect. In reality, these filters are not ideal and their transfer functions are givenby (6.11) and (6.12). From these transfer functions, one can deduce an approximatetransfer function of the forward path filter :
u = KF (s)(g ± x) (7.5)
where F (s) is given as
F (s) =
(
K1ωh2
s + ωh2
)(
K2ω2l
s2 + 2ζlωls + ω2l
)
. (7.6)
52
In (7.6), the transfer function for the high–pass filter is omitted on the grounds thatit is applied only to the switching noise and has no contribution to the input–outputrelationship. Figure 7.2 shows the bode diagram of (7.6) compared with the result ofthe computer simulation. The magnitude of the simulated filter response stays closeto the approximation up to the vicinity of the high–pass filter’s cut–off frequencywhere the unfiltered switching noise comes into play. The phase of the approximatedtransfer function (7.6) is in good agreement with the result of the computer simulation.
The importance of (7.5) is that the forward path filter which is nonlinear andinternally complicated is approximated as a linear system with a time–varying gainK. Utilizing this linearity, the analyses that follow become substantially easier thandealing with the nonlinear filters.
The time–varying gain K in (7.5) satisfies
Kmin ≤ K ≤ Kmax. (7.7)
The duty ratio of 50 % renders K equal to Kmax which is
Kmax =1
8
Vsτ
gL0
(7.8)
The gain becomes the minimum when the duty cycle varies the most from 50 %, asshown in Figure 7.2. Symbolically, the minimum gain Kmin is
Kmin =1
2α(1 − α)
Vsτ
gL0
(7.9)
where α is defined as
α = max(αmax, 1 − αmin)
Since the real–time simulation of the bearing inductance is based on the modeldescribed in Chapter 4, the equation governing the simulation can be written as (7.1).If the simulated current passes through the same forward path filter as the actualcurrent signal, the output of the filter is approximated as
u =1
2α(1 − α)
Vsτ
gL0
(g ± x) (7.10)
or in frequency domain,
u = KF (s)(g ± x) (7.11)
Subtracting (7.11) from (7.5), the error between the two outputs of the filters can bewritten as
e = KF (s)(x − x) (7.12)
To accomplish parameter tracking, a proportional and integral type controller isused. Generally, the proportional block is approximated to have infinite bandwidth.
53
1012 5 102
2 5 1032 5 104
Frequency (Hz)
10-5
10-4
10-3
10-2
10-1
100
101
Mag
nitu
de
Linearized ModelSimulation
1012 5 102
2 5 1032 5 104
Frequency (Hz)
-200
-150
-100
-50
0
50
Pha
se(d
eg.)
Linearized ModelSimulation
Figure 7.2: Frequency response of the forward path filter (simulation and approxima-tion)
-
6K
α
Kmax
0.50 1
Kmin
αmin αmax
Figure 7.3: Variation of the time–varying gain K
54
j- KF (s) j G(s)
KF (s) j
-
?
---?
6
g
x u
u
ex
g
+
±+
−±
+
Figure 7.4: Idealized block diagram of the parameter estimator
In reality, the bandwidth of the block is limited, as the op amps used to implementthe controller have finite gain bandwidth. Actual implementation of the PI controlleris a phase lag configuration with a pole and a zero, where the pole is closer to theorigin of the complex plane than the zero is. The transfer function of the controllercan be written as
G(s) =
(
Kp +Ki
s + ωi
)(
ωpi
s + ωpi
)
(7.13)
Summarizing the analysis thus far, the parameter estimator can be approximatedas a linear system with time–varying parameter K, through an idealization of thenonlinear forward path filter. Figure 7.2 shows the block diagram of the linearizedparameter estimator.
7.3 Stability of the Estimator
As the system is approximated as linear with a time–varying parameter, the abso-lute stability of the estimator can be shown using the circle criterion [41]. Absolutestability means that the system is globally uniformly asymptotically stable for anyvariation of K within the prescribed boundary. In other words, the estimator is sta-ble for any values of K satisfying Kmin < K < Kmax, when the system has arbitrarynonzero initial conditions. The block diagram illustrated in Figure 7.5 is modifiedfrom Figure 7.2 to be appropriate for the absolute stability test. D(Kmin, Kmax) isdefined to be a closed disk in the complex plane. The diameter of the disk is the linesegment connecting the points −1/Kmin + j0 and −1/Kmax + j0. Since F (s)G(s) isopen–loop stable, the criterion guarantees the absolute stability if the Nyquist plot ofF (jω)G(jω) does not enter the disk D(Kmin, Kmax) nor encircles it. Circle criterionis a conservative measure, since the circle includes any variation of the gain, whichmight be restricted in some way. A graphical illustration of the circle criterion isshown in Figure 7.6.
The circle criterion determines if the system is stable depending upon the vari-ous values of proportional gain Kp and integral gain Ki of the phase lag controllerdescribed earlier. The results of the stability test are displayed in Figure 7.7, when
55
j G(s)
KF (s)
-
--
6
u = 0
u
ex
Figure 7.5: Block diagram for absolute stability test
-
6
G(jω)F (jω)
ab
a = − 1
Kmax
b = − 1
Kmin
M
Figure 7.6: Disk D(Kmin, Kmax) and trajectory G(jω)F (jω)
56
10-42 5 10-3
2 5 10-22 5 10-1
Proportional gain Kp
10-1
2
5
100
2
5
101
2
5
102
2
5
103
Inte
gral
gain
Ki
Simulation ( m=0.1)Simulation ( m=0.01)Circle criterion
STABLE
UNSTABLE
Figure 7.7: Stability of the parameter estimator. The filled cicle indicates the gainsused in the experiments
the duty cycle variation is assumed to be between 40 % and 60 %. The results arecompared with the stability boundary obtained from computer simulations. The out-most boundary is obtained when the variation of the duty ratio is 1 %. The boundarywhich is in the middle, is the result of the computer simulation when the variationof the duty cycle is 10 %, the same value as the circle criterion. As expected, thecircle criterion results in a more conservative boundary than the simulation. It shouldbe noted that the actual stability boundary lies between the boundary by the circlecriterion and the boundary by the simulation, since the circle criterion ignores anystructural variation of the gain, and thus results in the worst case boundary. Thefilled circle in Figure 7.7 indicates the gains actually used in the experiments.
7.4 Realization
7.4.1 Controller
The controller is implemented using analog circuit devices. Shown in Figure 7.8 is thecircuit diagram of the controller. According to experiments, it turned out that theproportional block has less influence on the performance of the estimator than theintegral block. Constant voltage is added to the output of the controller to accountfor the nominal gap of the bearing. This keeps the average output of the integratorat zero.
57
Figure 7.8: Analog realization of the controller
7.4.2 Inductor Simulation Model
Real time simulation of the inductor model (7.1) is implemented using analog circuitcomponents, as shown in Figure 7.9. The voltage from the switching amplifier isscaled down, so that the output of the differential receiver is in the operating range ofthe op amp. The estimation of the reciprocal of the inductance is multiplied with thescaled switching voltage using Burr–Brown MPY634 analog multiplier. The estimatedcurrent is obtained by integrating the output of the multiplier. The resistance of thecoil is simulated by adding a resistor in the feedback path of the integrator. All threecomponents are null–adjusted to minimize drift which may be accumulated in thesimulated output.
7.5 Performance Evaluation
7.5.1 Static Test
The linearity of the estimator can be evaluated by a static test. The test is donethrough a manual process by inserting shims of different thicknesses between thebeam and the support bars before clamping, thus achieving constant air gaps. Theoutput of the estimator is measured in reference to the eddy current probe. Figure 7.10shows the result of the static test. In this figure, the circles are the actual outputof the estimator, compared to a straight line obtained from a linear regression. Themaximum error from linearity is 1.3 %. One problem of the estimator encounteredduring the static test is that the estimation increases suddenly at large gap length(over 0.65 mm). This sudden increase can be explained as follows. As the air gap
58
Figure 7.9: Implementation of inductor simulation
length becomes large, the amplitude of the switching noise increases. The op ampsused to realize a forward path filter have an output limitation which is limited bythe supply voltage. Therefore, if the signal at any point in the forward path filterbecomes larger than this limit, the signal remains at the limit. A wider linear regionthan presently obtained is possible if the gain of the forward path filter is lowered.However, the sensitivity of the estimator has to be compromised.
7.5.2 Dynamic Test
Dynamic performance of the position estimator can be evaluated from the bandwidthand the signal to noise ratio. The bandwidth indicates how fast the estimator canrespond to a transient input. On the other hand, the signal to noise ratio repre-sents the disturbance rejection of the estimator. Prior to the investigation of thesecharacteristics, it is worth examining the output of the estimator when the actualdisplacement is changing at a specific frequency. The results can be well comparedwith those of the same tests for the forward path filter, which were described in Sec-tion 6.3. The result of a computer simulation obtained through the procedure laidout in Section 6.3 is presented in Figure 7.11. Comparing the output of the forwardpath filter in Figure 6.8, the error between the true displacement and the estimationdecreased significantly. Since identical forcing functions are assumed in both cases,the comparison also illustrates that the parameter estimator decreases the force feed–through. The force feed–through is defined and discussed in the next section. Oneshould also notice from this comparison that the parameter estimator decreases thephase lag. This means that the estimator has a wider bandwidth than the forwardpath filter. Section 7.5.4 deals with this subject in detail.
Improvements in position estimation can also be seen in experimental results. Fig-ure 7.12 shows the output of the estimator as compared with the output of the eddycurrent probe, when the beam is undergoing a free vibration. To see the effect of a
59
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65Gap length (mm)
4.5
5.0
5.5
6.0
6.5
7.0
7.5
8.0
8.5
9.0
Est
imat
orou
tput
(V)
Linear FitEstimator
Figure 7.10: Linearity of the parameter estimator obtained from a static test. Maxi-mum error from linearity is 1.3 %
time–varying force, a sinusoidal current is imposed on the sensing coil to excite thebeam. The result of the test is shown in Figure 7.13. It is obvious from a comparisonwith the output of the forward path filter shown in Figure 6.13, that position sensingbased on parameter estimation is superior to the demodulation filter in terms of band-width and rejection of force. These two characteristics are investigated extensively inthe following sections.
7.5.3 Force Feed–through Effect
The magnetic flux generates attractive forces. The flux is induced by electric cur-rents in the coil wrapped around the actuator. This force acts on objects made fromferromagnetic material. Electromagnets can produce only attractive forces which areproportional to the square of applied coil currents. Since a switching amplifier isdriving the coil, the current is a high frequency triangular wave whose low frequencycomponents are determined by the command signal from the bearing controller (as-suming that the amplifier can be modeled as linear). The high frequency componentsof the current do not affect the dynamics of the suspended object, since the band-width of the mechanical systems would be much lower than the switching frequency.Hence, only the low frequency components of the current signal need to be consideredas far as the force is concerned. The duty cycle of the amplifier which was definedin Section 6.1 determines the low frequency components. Chapter 6 shows that the
60
0 2 4 6 8 10 12 14 16 18 20time (msec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
x,x
(m
)
0 2 4 6 8 10 12 14 16 18 20time (msec)
-50
-40
-30
-20
-10
0
10
20
30
40
50
erro
r(
m)
Figure 7.11: The output of the estimator when ω = 2π · 120
61
0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1time [sec]
-0.1
-0.08
-0.06
-0.04
-0.02
0.0
0.02
0.04
0.06
0.08
0.1x,
x(m
m)
Estimator outputActual motion
Figure 7.12: Estimator output when the test beam is vibrating freely (constant dutyratio
0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1time [msec]
-0.1
-0.08
-0.06
-0.04
-0.02
0.0
0.02
0.04
0.06
0.08
0.1
x,x
(mm
)
Estimator outputActual motion
Figure 7.13: Estimator output when the test beam undergoes a forced vibration(time–varying duty cycle)
62
output of the amplitude demodulation filter has quadratic dependence on the dutycycle. The parameter estimator removes this dependence by introducing a simulationmodel driven by the actual voltage signal which contains the duty cycle information.Therefore, theoretically, the position estimation should not be affected by the time–varying duty cycle. In actuality, due to imperfectness of the filters, the duty cycledoes affect the performance of the estimator.
The term force feed–through is introduced to measure the estimation error due tothe time–varying duty cycle. Since the direct measurement of duty cycle is difficult,the force feed–through S x
F is defined as the sensitivity of the estimator output to theforce variation.
SxF =
x
4F(7.14)
Nondimensionalization of (7.14) results in a more general representation of S xF .
SxF =
x/g
4F/Fmax
=
(
x
g
)
·(
Fmax
4F
) (7.15)
The sensitivity function S xF can be measured experimentally. The test procedure is
as follows. First, the beam described in Chapter 5 is secured with two toggle clampslocated as closely as possible to the bearing, thus ensuring no movement of the beamat the bearing location. Then a sinusoidal current of known frequency is requested tothe amplifier. The bias and the amplitude of the current are chosen in such a way asto obtain the desired force fluctuation. The output of the estimator is the estimationerror due to the applied force. Figure 7.14 shows the result of the force feed–throughtest. The estimation error is presented as a sensitivity to the force fluctuation. Thesensitivity is approximately constant up to 1 kHz, which means that the estimationerror is nearly linear to the force fluctuation. The peak at 1.6 kHz appears to becaused by the structural resonance of the test rig.
7.5.4 Bandwidth and Signal to Noise Ratio
Bandwidth determines how fast the estimator is able to track actual displacement,whereas signal to noise ratio (S/N ratio) indicates how accurate the estimation is. Ifthe estimator is to be used in magnetic bearing applications, its bandwidth shouldexceed those of the controller, actuator, and amplifier. The main reason that eddycurrent probes are typically used in magnetic bearing applications is that these probeshave excellent dynamic characteristics. The bandwidth can be decided by eithermagnitude or phase of the frequency response of the estimator. As a replacement ofthe discrete position sensor, the magnitude of the estimator should be close to unity,and the phase lag needs to be as small as possible. Excessive phase lag will destabilizethe bearing controller, and the exact threshold value can be obtained by a stabilityanalysis of the controller and estimator combined.
63
1012 5 102
2 5 1032
Frequency (Hz)
10-2
2
5
10-1
2
5
100
(x/g
)(F
max
/F
)F/Fmax = 14.87 %F/Fmax = 11.90 %F/Fmax = 8.92 %F/Fmax = 5.95 %
Figure 7.14: Experimentally measured force feed–through
Signal to noise ratio is the ratio between the desired signal (estimation) and thenoise (estimation error). This ratio depends on how well the estimator can reject thenoise disturbances to which it may be subjected. Two possible sources of noise areforce feed–through and switching noise. As discussed in Section 7.5.3, a large changeof the duty cycle results in an estimation error. Also, the output of the forward pathfilter contains the residual of switching noise which is not filtered out completely.Figure 7.15 is a block diagram of the parameter estimator revised from Figure 7.2.Nominal gap g is not shown in this block diagram, since it is added to the outputas well as the input. From the block diagram, the transfer function from the actualdisplacement x to the estimation x is
H(s) =KF (s)G(s)
1 + KF (s)G(s)(7.16)
where the transfer function of the forward path filter F (s) is given as (7.6). Eq. (7.13)defines the transfer function of controller. Similarly, the transfer function from noiseto estimation can be derived as
N(s) =G(s)
1 + KF (s)G(s)(7.17)
Frequency responses of H(s) and N(s) are given in Figure 7.16. The magnitudeof H(s) remains flat at unity to around 2 kHz, but the noise increases rapidly after 2
64
KF (s) j G(s)
KF (s)
-
---
6
x ex
?
n
Figure 7.15: Revised block diagram of parameter estimator
kHz. This means that the accuracy of the estimation becomes poor at high frequency(> 2 kHz) although the estimator is still tracking the actual displacement. It shouldbe noted, however, that the noise level throughout the spectrum is subtantially small.Experimentally obtained noise spectrum of the estimator ouput also supports theanalysis. Shown in Figure 7.17, the power spectral density of the estimator outputhas a peak at around 2 kHz, but the peak is very small in magnitude. The root–mean–squared error is 0.9 µm (0.04 mil).
The frequency response of the idealized transfer function H(s) is verified withcomputer simulation, as demonstrated in Figure 7.18. The magnitude and phase ofH(s) matches extremely well with the result of the simulation up to around 3 kHzwhere the assumption of a perfect high–pass filter no longer holds. Comparing thefrequency response of the parameter estimator with that of the forward path filtershown in Figure 6.9 and Figure 6.10, one can immediately see that the bandwidthhas been improved significantly by the employment of parameter estimation.
The experimentally obtained frequency response of the estimator is presentedin Figure 7.19. As for the experimental spectrum of the forward path filter, thequality of the spectrum is poor due to the lack of data. As discussed in Chapter 5,a dynamic test beyond 500 Hz is not possible with the existing test rig. Consideringthis limitation, the magnitude and phase remains constant to 500 Hz except withseveral spikes. Hence, one can conclude that the experimentally measured frequencyresponse agrees with the analysis and simulation within the limitation of the testsetup.
65
1012 5 102
2 5 1032 5 104
Frequency (Hz)
10-4
10-3
10-2
10-1
100
101
102
H(s
),N
(s)
N(s)H(s)
Figure 7.16: Frequency response of parameter estimator and disturbance rejection
5 1022 5 103
2 5 1042 5
Frequency [Hz]
0.0
0.02
0.04
0.06
0.08
0.1
0.12
Pow
ersp
ectr
alde
nsity
(m
/Hz)
Figure 7.17: Power spectral density of the estimator output at steady state. The rootmean squared error is 0.9 µm (0.04 mil).
66
1012 5 102
2 5 1032 5 104
Frequency (Hz)
10-1
2
5
100
2
5
101
Mag
nitu
de
IdealizedSimulation
1012 5 102
2 5 1032 5 104
Frequency (Hz)
-40
-30
-20
-10
0
10
20
30
40
Pha
se(d
eg.)
IdealizedSimulation
Figure 7.18: Frequency response of parameter estimator (analytical vs. simulation)
67
1012 5 102
2 5 103
Frequency (Hz)
10-1
2
5
100
2
5
101
Mag
nitu
de
1012 5 102
2 5 103
Frequency (Hz)
-150
-100
-50
0
50
100
150
Pha
se(d
eg.)
Figure 7.19: Frequency response of parameter estimator (experimental)
68
Chapter 8
Nonidealities
In the previous chapter, the simulation model used by the estimator is based on theidealized model derived in Chapter 4. In the modeling process, several nonidealitiesare ignored. These nonidealities include saturation, hysteresis, eddy currents, backEMF, and cross-coupling. This chapter discusses effects of these nonidealities on theperformance of the estimator. Where possible, suggestions are made to incorporatethe nonidealities into the embedded simulation model. This modification of the modelwill presumably improve the estimation.
8.1 Saturation
Saturation decreases the relative permeability of the ferromagnetic material. At anextremely high flux level, the permeability becomes equal to the permeability ofthe free space µ0. The reduction in the permeability affects the operation of theestimator. By examining the dynamics of switching, the effect of the reduction canbe made obvious. The time rate of the coil current is described by
di
dt=
2(g ± x) + lc/µr
µoN2At
(V − Ri). (8.1)
On the other end, the switching dynamics in the simulated bearing are governed by
di
dt=
2(g ± x) + lc/µ0r
µoN2At
(V − Ri), (8.2)
where µ0r is the nominal relative permeability used by the simulation model. If the
feedback is working to make i track i, the error between the actual displacement andthe estimation, due to the change in the permeability, is
x − x = ± lc2
(
1
µr
− 1
µ0r
)
. (8.3)
A proper model of saturation would make it possible to assess the effect of saturationon the performance of the estimator.
69
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0Flux density (T)
0.0
0.05
0.1
0.15
0.2
0.25
0.3
Est
imat
ion
erro
r(m
m)
Figure 8.1: Estimation error due to saturation
A magnetization model is given as a form of analytic functions in Appendix C.Using this model, one can evaluate the effect of the reduced permeability due tosaturation. When considering only the major hysteresis loop, the reciprocal of thepermeability, 1/µr, can be approximated as
1
µr
=1
µ0r
+ sgn(B)
(
1 − 1
µ0r
)
η · e(|B|−Bs)/σ
1 + η · e(|B|−Bs)/σ. (8.4)
Eq. (8.4) is obtained by differentiating the average of the two envelopes of the B−Hcurve with respect to B, as discussed in Appendix C. Since a switching amplifier isassumed to drive the bearing coil, however, the B−H loop forms minor hysteresisloops. The permeability of minor hysteresis loops would be substantially lower thanthe permeability obtained from (8.4) (see Figure C.3). Therefore, one would under-estimate the core reluctance if the permeability obtained from major hysteresis loopsis used.
Figure 8.1 shows the estimation error due to saturation. At low flux density, theerror, which remains constant, can be easily fixed in the actual implementation. Thissolution involves changing the DC voltage added to the output of the controller. Atthe high flux level, the constant error increases rather drastically. This increase isproblematic for the estimator. It would be difficult to eliminate the error on–line.
Besides the constant estimation error, the saturation causes a fundamental prob-lem. Excessive saturation forces the gain of the estimator to change its sign. In (8.1),the slope of the current waveform is a function of the gap displacement and the rela-
70
0.3
0.4
0.5
0.6
0
1
2
3
4
520
25
30
35
40
Air gap length (mm)Bias current (A)
Cur
rent
rat
e (A
/mse
c)
Figure 8.2: Current rate vs. air gap length and bias current
tive permeability. The relative permeability is determined from the minor hysteresisloops at a certain flux density. Ampere’s loop law for the magnetic flux path can bewritten as
2B
µo
(g ± x) + H(B)lc = Ni. (8.5)
Eq. (8.5) shows that the flux density B is not only a function of the current but alsoa function of the gap displacement x. Solving (8.5) for a fixed i, the slope (8.1) at agiven displacement can be obtained. In the magnetically linear region, the relationshipbetween the slope and the displacement is linear; however, at small gaps with highbias current, the relationship becomes nonlinear.
This argument is verified both by simulation and by experiment. Presented inFigure 8.2 is the current rate di/dt versus air gap length and bias current obtainedfrom solving (8.1) and (8.5). At high bias and small air gap, the current rate sig-nificantly deviates from the linear surface. In fact, one can see that a reversal ofsensitivity occurs at high bias and small air gap. Figure 8.3 shows the current rateas a function of the air gap length when the bias current is 5 Amp. As expected, thepermeability of the minor hysteresis loops cause more distortion.
In Figure 8.4, the result of the simulation is compared with experimental data. Inorder to obtain the test data, the following steps are taken. First, a digital oscilloscopecaptures the switching waveform at the sampling rate of 25 MHz. For the set of datain one switching interval (either positive or negative slope), a linear regression isperformed. The points in Figure 8.4 are obtained by taking the average of the tenseparate slopes (the results of the linear regressions). The experimental results also
71
0.3 0.35 0.4 0.45 0.5 0.55 0.6Air gap length (mm)
26
28
30
32
34
36
38
Cur
rent
rate
(A/m
sec)
Minor loopMajor loop
Figure 8.3: Current rate when the bias current is 5 Amps.
verify that the sensitivity of the current slew rate to the gap changes dramaticallywhen saturation occurs–in this case, at high bias current (4.5 amperes) and small airgap. By incorporating the saturation model into the simulation of the bearing, thegain error due to sensitivity change can easily be fixed as long as the relationship issingle–valued. If the relationship is double–valued, the estimation becomes unreliable.A possible solution for the problem is to utilize the time history of the estimates andthe measurements of the average current. By devising a simple model of the surface(Figure 8.2) and incorporating it into the embedded inductor model, the sensitivityreversal may be compensated. If this problem is corrected, self–sensing magneticbearings can be used in applications which require extreme performance (such as jetengines). The problem certainly calls for additional research efforts.
Figure 8.5 shows the result of computer simulation of the position estimator, whenthe core material is saturated. The bias current of 5.95 Ampere is used to ensure thesaturation. The simulation utilized the nonlinear magnetization model described inAppendix C. Since the bottom pair of the poles are used in the simulation, the neg-ative extremes correspond to the smallest gaps and the highest flux density. Aroundthese points, the estimated displacement is moving in the opposite direction to theactual displacement. The result also shows the offset error due to the decrease inpermeability.
72
0.35 0.4 0.45 0.5 0.55 0.6 0.65Air gap length (mm)
22.0
22.5
23.0
23.5
24.0
24.5
25.0
25.5
26.0
26.5
Slo
pe(A
/mse
c)
.
........ ......
....
.......
...
.
......
.....
...
...
...
...
...
..............
.
.
..
........
.
.
.....
........
...
...
........
..
.. .......
........
...
.........
. EXPERIMENTSIMULATION
Figure 8.4: Slope versus displacement
0 2 4 6 8 10 12 14 16 18 20time (msec)
-200
-150
-100
-50
0
50
100
150
200
x,x
(m
)
Figure 8.5: Estimator response when the core material is saturated
73
0.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08time (msec)
1.85
1.9
1.95
2.0
2.05
2.1
2.15
2.2
Cur
rent
(Am
p)
Figure 8.6: Current waveform acquired by a digital oscilloscope. Sampling rate of 25MHz was used. Additional filtering was applied to eliminate noise.
8.2 Hysteresis and Eddy Currents
Besides saturation, magnetic nonidealities which were neglected in the modeling pro-cess include hysteresis and eddy currents. Magnetic hysteresis makes the output (fluxdensity B) dependent not only on the input (field intensity H) but also on the timehistory of the input. Eddy currents are created by the time–varying magnetic fluxin a magnetic material. This dissertation neglects the eddy currents created by amoving conductor (known as Foucault currents [70]).
When a switching amplifier is driving the magnetic bearing, both of these effectsforce the switching waveform to have a much higher current rate at the start of aninterval than during the rest of the interval. This anomaly can be measured andis illustrated in Figure 8.6. Since the interval of higher current rate is considerablyshort compared to one switching cycle, one can idealize the switching waveform as atriangular wave with jump discontinuities.
Without eddy currents, the jump discontinuity induced by hysteresis will be de-pendent upon the separation of the loading and unloading portion of the minor hys-teresis loops, shown in Figure C.2. As the separation of the minor loops is muchsmaller than that of the major loops, the jump due to hysteresis would be negligibleat a high switching rate. It is thus reasonable to assume that the jump discontinuityis mostly due to eddy currents. The jump induced by eddy currents can be explainedas follows.
74
Consider two conductive loops which are essentially parallel, as illustrated in Fig-ure 8.7. One of the loops is driven by an applied voltage, V , while the other is drivenby linked magnetic flux Φ. Each of the loops has some finite resistivity. The drivenloop models the electromagnet coil, while the other loop models conduction pathsin the magnet iron which permit eddy currents. To analyze this linked circuit, firstapply Faraday’s law to each loop:
V
R− I =
N
R
dΦ
dt(8.6)
−Iec =1
Rec
dΦ
dt(8.7)
These two expressions can be added to obtain
V
R− (I + Iec) =
(
N
R+
1
Rec
)
dΦ
dt. (8.8)
By Ampere’s loop law for a simple linear magnetic circuit with static reluctance R,
ΦR = Iec + NI. (8.9)
Substituting (8.9) into (8.8) and solving for Φ produces
dΦ
dt= −R RecR
R + N 2Rec
Φ +Rec
R + N 2Rec
V.
For time scales significantly smaller than the time constant of this equation,
τ N2Rec + R
RecRR ,
this differential equation is adequately modeled by
dΦ
dt≈ Rec
R + N 2Rec
V. (8.10)
Finally, (8.7) and (8.9) lead to
I ≈ R Rec
R + N 2Rec
∫
V dt +1
R + N 2Rec
V. (8.11)
When a switching amplifier is driving the inductor, V is equal to either +Vs or−Vs, which means that the current waveform will be a series of ramps (integrals of thevoltage square wave) with discontinuities between each successive ramp. The jumpdiscontinuity is illustrated by Figure 8.9. The magnitude of these discontinuities willbe determined by the extent to which eddy currents are generated in the material. Inthe limit as Rec → ∞, the eddy currents go to zero and so do the discontinuities. In
75
Figure 8.7: Eddy current effects modeled with a fictitious one–loop coil
the limit as Rec → 0, the induced flux goes to zero (the material is a superconductor),and the current simply follows the applied voltage according to Ohm’s law.
To investigate the effect of this jump discontinuity on the performance of theestimator, a simple model of magnetization is utilized, instead of using the modelpresented in Appendix C. The model in this section is ideal in the sense that itonly results in the discontinuity in current waveform. The model is composed of twostraight lines described by
H =
1
µB + Hr if B > 0
1
µB − Hr if B < 0
(8.12)
where µ = µoµ0r. An example of the hysteresis loop is shown in Figure 8.8. In (8.12),
Hr represents the separation of the loading and unloading curves of a loop. For astatic case, this separation will be determined by the remnant magnetization. In acase where a switching amplifier is used, the amount of jump due to eddy currentswill determine Hr.
Applying the same analysis done in Section 6.1, the output of the forward pathfilter with the discontinuous current waveform as the input signal is
u =α
2
[
4
(
1 − tcτ
)(
tcτ
)
− α
]
Vs(g ± x)τ
gL0
+δ2
α· gL0
Vs(g ± x)τ, (8.13)
where the jump δ is obtained as
δ =2Hrlc
N.
76
6
- H
B
Hr
−Hr
B = µ(H − Hr)
B = µ(H + Hr)
-
Figure 8.8: Straight line hysteresis model
?
6δ
0 tc τ
6
-
i
t
Figure 8.9: Current waveform generated by straight hysteresis model
77
0 5 10 15 20 25 30 35 40Hr (A/m)
0.0
0.001
0.002
0.003
0.004
0.005E
stim
atio
ner
ror
(mm
)
Figure 8.10: Estimation error due to jump discontinuity
The zero–crossing time tc is a function of δ described by
tc =τ
2
[
1 − δ
α· gL0
Vs(g ± x)τ
]
. (8.14)
If δ = 0, then tc becomes 0.5τ as before. The simulated bearing model is based on thelinear magnetic material; therefore, the output of forward path filter with simulatedcurrent is unchanged :
u =1
2α(1 − α)
Vsτ
gL0
(g ± x). (8.15)
Since the feedback element forces u to track u, the estimation error due to the dis-continuity can be revealed by subtracting (8.15) from (8.13). The error is a functionof duty cycle α and the actual displacement x. Figure 8.10 shows the error due todiscontinuity as a function of the separation of loading and unloading curve Hr whenα = 0.5 and x = 0. It should be noted that the result is highly exaggerated, since theminor hysteresis loops formed by high frequency switching have much smaller sepa-ration of two envelopes of B−H curve than the major hysteresis loops. Excludingsome thrust bearings which have a long flux path, most radial bearing applicationsshould find little effect from the jump discontinuity generated by hysteresis and eddycurrents.
In cases when the estimation error due to discontinuous switching waveform iscritical, the simulation model can easily be modified to accommodate the discontinu-ity. The current in one switching interval can be approximated as the ideal waveform
78
ForwardPath Filter
ForwardPath Filter
j PI?
6
+
−
-
-
-i
i
1gL0
j- -?
∫-j --
R
6
jγ?
- -
-
1L0
V
u
u
x
Figure 8.11: Modified parameter estimator accommodating jump discontinuity
added by an offset which has the same polarity of the applied voltage. Therefore,the simulated current can be made to track the actual current with discontinuity byadding a fraction of actual voltage to the simulated current. The block diagram ofthe modified parameter estimator is presented in Figure 8.11.
Not only does the eddy current induce the jump discontinuity in switching wave-form, but it also decreases the permeability of the stator. In fact, the permeabilityµ is a function of driving frequency. By solving a simplified Maxwell’s equation, thefollowing expression for frequency dependent permeability can be derived [58].
µr(jω) = µor ·
tanh(√
jωσµo · d/2)√jωσµo · d/2
(8.16)
where σ is the resistivity of the core material, d is the lamination thickness, and µo =µoµ
or. Figure 8.12 is the plot of frequency dependent permeability (8.16). This figure
clearly shows that the relative permeability at switching frequency fs is substantiallysmaller than the nominal relative permeability µo
r. This decrease in permeabilitywill result in offset error as discussed in the previous section. However, as long asit does not change with time, the offset in estimation can easily be fixed by addingan appropriate constant to the nominal gap term in the simulation model. Thisadjustment is automatically made in the implementation by changing the added DCvoltage to the output of the controller.
79
1012 5 102
2 5 1032 5 104
2 5 105
Frequency (Hz)
2
5
103
2
5
104
2
5
105
Rel
ativ
epe
rmea
bilit
y
ro
r(f s)
Figure 8.12: Frequency dependent relative permeability
8.3 Cross–Coupling
If a cross–coupling exists between axes, the estimator in one axis is affected by dis-placements in other axes. For example, for an eight–pole magnetic bearing shownin Figure 8.13, any displacement in the y axis may produce estimation error in thex axis. There are two factors which may cause cross–coupling between axes. Sincethe geometry of the bearing is circular, any movement in one direction may intro-duce gap change in the perpendicular direction. Mutual inductance also couples thex and y axes, since the simulation model in the estimator is based solely on theself–inductance.
Geometric cross–coupling can be identified by investigating the gap change in eachpole with respect to transverse movements in the x and y directions. The air gap ofthe ith horseshoe magnet in Figure 8.13 can be written as
gi = 2g − 2x cos θi cos θp − 2y sin θi cos θp, (8.17)
assuming the average gap in each pole can be approximated by the gap at the linewhich starts from the center of the rotor and crosses the middle point of the pole.The current rate at the ith horseshoe is
Ii ≈gi + lc/µr
µoN2Ag
· Vs. (8.18)
80
Figure 8.13: Definition of θi and θp
The sensitivities of the current slope to the displacement x and y are
∂Ii
∂x= − 2Vs
µoN2Ag
· cos θi cos θp, (8.19)
∂Ii
∂y= − 2Vs
µoN2Ag
· sin θi cos θp. (8.20)
For the first horseshoe (θ1 = 0), (8.20) becomes zero. Therefore, the configurationshown in Figure 8.13 has no geometrical cross–coupling.
The preceding analysis involves several approximations. Since the pole face hassome curvature, the average gap would be somewhat different from the one at themiddle point. To be precise, therefore, there is a weak geometrical cross–couplingbetween axes. A more elaborate analysis showing the cross–coupling has been doneby Sortore [74].
When several coils interact in a common magnetic circuit, the current variationsin one coil can produce voltages across another. This effect is referred to as mutualinductance. The first step in finding mutual inductances is to set up a set of linearalgebraic circuit equations where the flux in any part of the stator can be determinedas a linear function of the applied coil currents [58].
Following the sign and numbering convention shown in Figure 8.14, the conserva-tion of flux and Ampere’s loop law result in 24 independent linear equations in thefluxes, consisting of :
• 7 loop equations going up a given pole, counterclockwise around the rotor, downthe next pole to the right, and back clockwise around the stator.
rpφi + rrφi+8 − rpφi+1 + rsφi+16 = nIi − nIi+1, i = 1, . . . , 7 (8.21)
81
Figure 8.14: Three–pole bearing showing sign and numbering convention used to setup flux equation
• 7 conservation equations for intersections on the rotor.
φi + φi+7 − φi+8 = 0, i = 2, . . . , 8 (8.22)
• 7 conservation equations for intersections on the stator.
φi + φi+15 − φi+16 = 0, i = 2, . . . , 8 (8.23)
• 1 conservation equation for all fluxes going into the rotor.
8∑
i=1
φi = 0 (8.24)
• 1 loop equation around the rotor.
8∑
i=1
rrφi+8 = 0 (8.25)
• 1 loop equation around the stator.
8∑
i=1
rsφi+16 = 0 (8.26)
By collecting fluxes of 24 sections from the above equations, a set of linear equationscan be written as
RΦ = NI (8.27)
82
Reluctance of each section constitutes the reluctance matrix R, and is obtained by
rp = rg + rl
rg =lg
µoAg
, rl =ll
µAl
rs =ls
µAs
, rr =lr
µAr
where l and A are the length and area of a section. Subscripts g, l, p, r and s
correspond to the air gap, leg iron, pole (sum of leg and gap effects), rotor, and stator,respectively. Note that rp is the sum of the leg reluctance and the gap reluctance.Eq. (8.27) can now be solved for the fluxes:
Φ = R−1NI (8.28)
Once the fluxes are known, inductance can then be determined. Define 24×8winding matrix T as
T =
[
nI8×8
0
]
where I8×8 is an identity matrix of order 8. The inductances of the bearing are then
L = TTR−1N. (8.29)
When the coils in eight poles are independently controlled, the inductance matrixin (8.29) is an 8×8 matrix and contains self inductances and mutual inductances. Fora horseshoe configuration as shown in Figure 8.13, the inductance matrix reduces toa 4×4 matrix by pre– and post–multiplying the interconnection matrix W.
L = WTLW , (8.30)
where W is defined as
W =
1 0 0 0−1 0 0 00 1 0 00 −1 0 00 0 1 00 0 −1 00 0 0 10 0 0 −1
83
assuming NSSNNSSN winding. The diagonal entries of L are the self inductancesof the windings, and the off-diagonal terms are the mutual inductances. For the testbearing described in Chapter 5, the inductance matrix is computed as
L =
2.31 × 10−3 2.48 × 10−6 2.39 × 10−6 2.48 × 10−6
2.48 × 10−6 2.31 × 10−3 2.48 × 10−6 2.39 × 10−6
2.39 × 10−6 2.48 × 10−6 2.31 × 10−3 2.48 × 10−6
2.48 × 10−6 2.39 × 10−6 2.48 × 10−6 2.31 × 10−3
(8.31)
Eq. (8.31) shows that the mutual inductances are extremely small compared to selfinductance, which supports the analysis with only self inductance. Leakage andfringing of flux would increase the mutual inductance, but generally, the increaseis insignificant. A more accurate assessment of leakage and fringing effects on themutual inductances can be obtained by a finite element analysis.
8.4 Back EMF
Back electro–motive–force (back EMF) is the voltage opposing the time rate of theinductance. Since the inductance is a function of the gap between the actuator andthe suspending object, the back EMF is induced by the motion of the object. InChapter 4, the inductance for the top pair of the poles is derived as
L =µoN
2Ag
2(g − x) + lc/µr
. (8.32)
The expression relating the time rate of the inductance (8.32) to the motion of thesuspending object is obtained utilizing the chain rule :
dL
dt=
∂L
∂x
dx
dt=
2L
2(g − x) + lc/µr
dx
dt. (8.33)
In the modeling process, the voltage induced by this time rate of inductance wasneglected, assuming that the voltage drop across the inductor is dominant over otherterms in Faraday’s law (4.12). An approximate analysis is presented in this section,which confirms the assumption made in the modeling.
The ratio between the voltage drop across the inductor and the voltage inducedby the time rate of the inductance can be written as
∣
∣
∣
∣
i · dL/dt
L · di/dt
∣
∣
∣
∣
=
∣
∣
∣
∣
2i
2(g − x) + lc/µr
∣
∣
∣
∣
∣
∣
∣
∣
dx/dt
di/dt
∣
∣
∣
∣
. (8.34)
Assume that the motion of the suspending object is sinusoidal.
x = Xm sin ωt. (8.35)
Assume also that the current in the coil can be approximated as a superposition ofthe bias current, a sinusoid of the same frequency as the motion, and another sinusoidof the switching frequency.
i = Ib + I1 sin(ωt + φ1) + I2 sin(ωst + φ2). (8.36)
84
The magnitude of the current is bounded as
|i| < Ib + I1 + I2 (8.37)
Then, the speed of the motion is proportional to the frequency ω :∣
∣
∣
∣
dx
dt
∣
∣
∣
∣
= Xmω. (8.38)
Assuming that the switching frequency ωs is much higher than the driving frequencyω, one can make an approximation that
∣
∣
∣
∣
di
dt
∣
∣
∣
∣
= |ωI1 sin(ωt + φ1) + ωsI2 sin(ωst + φ2)| ≈ I2ωs. (8.39)
From (8.34), (8.37), (8.38), (8.39), the following inequality holds.∣
∣
∣
∣
i · dL/dt
L · di/dt
∣
∣
∣
∣
<
(
Xm
g
)
·(
Ib + I1 + I2
I2
)
·(
ω
ωs
)
. (8.40)
Eq. (8.40) results from an assumption that the motion of the suspending object isconfined within the small vicinity of the nominal position.
During the normal operation, the motion of the rotor supported by a magneticbearing is regulated by the bearing controller. Hence, the ratio Xm/g is usually small(less than 0.2). The amplitude of the switching waveform, I2, is a function of theapplied voltage, the inductance, and the switching frequency. For a PWM amplifier,the amplitude is approximately
I2 ≈Vs
2fsL0
. (8.41)
Generally speaking, I2 is on par with the low frequency current fluctuation I1. InChapter 6, it is assumed that
ω
ωs
< 0.1 (8.42)
Therefore, the following assumption is reasonable.(
Ib + I1 + I2
I2
)
·(
ω
ωs
)
< 1 (8.43)
Obviously, the inequality (8.43) depends on the driving frequency ω. Except at veryhigh frequency, the right hand term in (8.43) should be much less than one. Therefore,one can conclude that the effect of the back EMF is negligible for a normal operation.
In order to see the actual effect of back EMF on the performance of the estima-tor, computer simulations are carried out. An inductor model which includes theback EMF generates the current waveform. This current is supplied to the estimator.The resulting estimation is compared with the assumed actual displacement. In Fig-ure 8.15, the results of computer simulations are presented. The driving frequency
85
is 120Hz (ω = 2π · 120 rad/sec). The estimation with back EMF quickly convergesto the estimation produced from the case in which the back EMF is not considered.The result confirms that the back EMF is negligible at this frequency. As the motionof the object becomes faster, the effect of the back EMF will be more evident. Fig-ure 8.16 shows the results of the simulations when ω = 2π · 1200 (rad/sec). Althoughthe estimation error when the back EMF is considered is somewhat larger than theestimation error of the idealized model, the overall performance of the estimator isnot significantly affected by the back EMF.
8.5 Other nonidealities
There are several nonidealities, not discussed in the previous sections, that are ne-glected in the modeling process. Those include nonideal characteristics of the poweramplifier and journal growth due to temperature rise and centrifugal force.
In Chapter 4, it is assumed that the switching of polarity in the power amplifiertakes place instantaneously, and once switched, the voltage level remains constant.In actuality, the completion of switching requires a short time (60 ∼ 70 ns), and theinput voltage receiver may make the transition time longer. Furthermore, the voltagelevel in one switching interval is not constant, as shown in Figure 8.17. The waveformin this figure was obtained by measuring the output of the input receiver and dividingby the gain of the receiver. Generally, these nonidealities of the power amplifier causefew problems for the position estimator, since the simulation model of the parameterestimator is not driven by the switching logic, but by the actual voltage.
Although magnetic bearings have no friction, losses due to resistance of coil, hys-teresis, and eddy currents transform into heat that makes the stator and rotor ex-pand. Moreover, at high rotational speed, journal growth due to centrifugal forcemay become noticeable. Since the parameter estimation relies on the fact that theinductance is a linear function of the gap distance, any change in diameter would re-sult in estimation error. In other words, the estimator will detect the journal growthas displacement. The estimation error due to journal growth can be easily eliminatedby operating the estimator differentially, assuming the growth is circumferentiallyhomogenous. Differential operation means that the gap is estimated by the differ-ence between the estimators in opposing coils. Differential estimation has the addedadvantage of an improved signal to noise ratio.
86
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Time (msec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
Pos
ition
Est
imat
es(
m)
Back EMFIdealized
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Time (msec)
-50
-40
-30
-20
-10
0
10
20
30
40
50
Est
imat
ion
erro
r(
m)
Back EMFIdealized
Figure 8.15: Estimation error when back EMF is considered in the computer simula-tion. (ω = 2π · 120 [rad/sec])
87
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Time (msec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
Pos
ition
Est
imat
es(
m)
Back EMFIdealized
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Time (msec)
-50
-40
-30
-20
-10
0
10
20
30
40
50
Est
imat
ion
erro
r(
m)
Back EMFIdealized
Figure 8.16: Estimation error when back EMF is considered in the computer simula-tion (ω = 2π · 1200 [rad/sec])
88
0.0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04Time (msec)
-100
-80
-60
-40
-20
0
20
40
60
80
100
Vol
tage
(V)
Figure 8.17: Actual voltage waveform from switching amplifier
89
Chapter 9
Conclusion and Future Research
The need for self–sensing magnetic bearings is apparent in some applications suchas magnetically suspended heart pumps, where the reduction of the wire count fromcontroller to bearings is one of the most important design issues. Elimination of thediscrete position sensor also results in improved reliability, lowered system cost, andcollocation of sensor with actuator.
Several approaches to developing self–sensing magnetic bearings have been at-tempted, mainly using either parameter estimation techniques or state–space ob-servers. Previous efforts treating the gap as a parameter of the system have met withlittle success due to the problem of force feed–through. State–space observers relyon the linearized model of the bearing, which makes it difficult to design a robustcontroller and observer pair.
When a switching amplifier is driving a magnetic bearing, the current switchingwaveform carries gap information through its amplitude. An amplitude demodulationresults in a signal which is a function of the displacement as well as the duty cycle ofthe amplifier. In this dissertation research, a new approach for position estimation isproposed with a goal of overcoming the duty cycle dependency. The method uses aparameter estimation technique. By introducing a real–time simulation of the bearingand a feedback loop forcing the simulated current to track actual current waveform,the force feed–through is significantly reduced.
The simulated current is generated from an idealized model of the inductor andis driven by the same voltage waveform as the actual bearing. Actual current andsimulated current pass through separate but identical nonlinear filters which performamplitude demodulation. The error between the two outputs of the filters providesan input to a proportional and integral compensator, and the output of the controllercorrects the simulation model.
Using analyses, computer simulations, and experiments, the performance of theestimator is assessed in terms of linearity, bandwidth, and signal to noise ratio. Statictests show that the position estimation is remarkably linear to the actual displace-ment (the maximum error from the linearity is less than 2 %). Through idealizedanalysis and computer simulation, it is found that the bandwidth of the estimator iscomparable to that of eddy current probes. The limitation of the test rig prohibitsdynamic tests above 500 Hz, but, within the capacity of the rig, the experimental
90
results corroborate the analysis and simulation. Bandwidth of the estimator is lim-ited not only by the phase lag of the estimation to the actual motion, but also by thesignal to noise ratio. Analysis indicates that the signal to noise ratio is larger than 60dB until 2 kHz and reduces to 20 dB beyond the bandwidth of the estimator. Thisis a significant improvement over the amplitude demodulation.
After approximating the internally complicated nonlinear estimator as a linearsystem with a time–varying gain, a stability analysis is performed, which determinesthe stability of the feedback loop with respect to the proportional and integral gains.
Nonidealities that were neglected in the modeling were investigated in order toidentify their effects on the performance of the estimator. Magnetic saturation de-creases the permeability of the core material. This decrease in the permeability resultsin a constant estimation error. If the material is highly saturated (the situation whichcan occur with high bias current and small air gap), the sensitivity of the estimatorto the amplitude of the current waveform may change its sign. As a result, the es-timation is unreliable without additional information. Hysteresis and eddy currentsinduce an anomaly in the switching waveform which can be modeled as jump dis-continuity. This discontinuity produces another constant offset error. However, theeffect of the jump can be easily fixed by injecting a scaled switching voltage signal atthe output of the simulation model. It is also verified that cross–coupling and backEMF have minimal effect on the performance of the estimator.
As stated in Chapter 1, the fundamental goal of this dissertation research is todevelop a position estimator which can replace discrete position sensors without no-ticeable degradation of the system performance. To achieve this goal, several issuesdemand further investigation and research efforts.
1. As discussed previously, the current test rig is only capable of dynamic tests inthe low frequency range (less than 500 Hz). Therefore, an apparent next step ofthe research would involve a redesign of the test rig so that the dynamic testsup to several kilo hertz are possible.
2. In Chapter 8, it has been shown that hysteresis and eddy currents induce thejump discontinuity in the current switching waveform. The embedded simula-tion model can be modified to accommodate this discontinuity. An experimentalconfirmation of such modification should be worthwhile.
3. By the introduction of the simulation model, force feed–through was substan-tially reduced. However, the estimation is still affected by the force change.Further reduction can be achieved by employing high order (possibly higherthan third order) high–pass filter. The author was recently made aware of aroot–mean–square chip at a relatively low cost. This chip can replace the recti-fier and the low–pass filter simultaneously. Thus, higher order of the high–passfilter can be achieved without increasing the part count in the circuitry.
4. The controller used in the estimator is a simple proportional and integral typecompensator. The bandwidth of the controller relies on the characteristics ofthe circuit components. The controller can be refined with increased complexity.The refinement can improve the bandwidth of the estimator.
91
5. In Chapter 7, the stability of the estimator is investigated from the input andoutput standpoint. Nonidealities have no influence on the stability of the esti-mator. However, these nonidealities may affect the stability of the whole system(estimator+bearing controller+magnetic bearing). Given a specific controllerfor the bearing, a stability analysis of the system must precede the implemen-tation of the estimator.
6. Chapter 8 showed that the saturation may force the reversal of the sensitivityof the estimator with respect to the actual displacement. Since the problemis identified and modeled, it may be possible to compensate the effect of thesaturation, by utilizing the time history of the estimation and the measurementsof the average current. The solution of this problem is critical for applicationswhich demand extreme performance of magnetic bearings (such as jet engines).
7. Finally, the estimator needs to be tested under actual operating conditions. Theestimator can be installed on an existing magnetic bearing system equipped withdiscrete position sensors. Employing a mechanism which makes it possible toswitch between the position sensor and the estimator, one can determine thefeasibility of self–sensing magnetic bearings.
92
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98
Appendix A
Harmonic Analysis of Switching
Waveform
Define Ib, Isn, and Icn
as follows.
Ib =
∫ T
0
i(t)dt =nc−1∑
k=0
∫ tk+τ
tk
ik(t)dt (A.1)
Isn=
∫ T
0
i(t) sin nωtdt =nc−1∑
k=0
∫ tk+τ
tk
ik(t) sin nωtdt (A.2)
Icn=
∫ T
0
i(t) cos nωtdt =nc−1∑
k=0
∫ tk+τ
tk
ik(t) cos nωtdt (A.3)
The current ik(t) during the kth switching cycle is given as (4.18). Fourier coefficientsare given in terms of these integrals.
a0 =1
TIb, an =
2
TIcn
, bn =2
TIsn
In the above equations, T is the period of one duty cycle variation, given as T =1/f , and ω = 2πf . As the equation for each switching period is available, analyticexpressions for Fourier coefficients can be obtained. Define [f(x)]ba = f(b) − f(a).Then,
∫ tk+τ
tk
ik(t)dt = Ikτ +τ 2
2β(2αk − 1) (A.4)
99
∫ tk+τ
tk
ik(t) sin nωtdt = (Ik + βtk)
[
− cos nωt
]tk+t+
tk
− β
[
1
n2ω2sin nωt − 1
nωcos nωt
]tk+t+
tk
+ Ik − β(1 − αk) − βtk[
− cos nωt
]tk+t−
tk+t+
+ β
[
1
n2ω2sin nωt − 1
nωcos nωt
]tk+t−
tk+t+
+ (Ik + 2βαk + βtk)
[
− cos nωt
]tk+1
tk+t−
− β
[
1
n2ω2sin nωt − 1
nωcos nωt
]tk+1
tk+t−
(A.5)
The integration involving the cosine can be obtained similarly.
100
Appendix B
Harmonic Analysis of
Demodulation
The inductance model derived in Chapter 4 is rewritten below.
di
dt=
2(g ± x) + lc/µr
µoN2Ag
(V − Ri) (B.1)
Assume that the reluctance of the core is negligible and only consider the minus sign.The minus sign represents the top pair of poles in Figure 4.1. Then, the switchingdynamics are represented by
di
dt=
g − x
gL0
(V − Ri) (B.2)
where L0 is the nominal inductance of the coil and is defined as
L0 =µoAtN
2
2g
Neglecting the voltage drop by the resistance of the coil, the time history of thecurrent is given as
i(t) =
∫ t
0
V (ξ)
Log[g − x(ξ)]dξ (B.3)
The voltage applied to the coil V (t) is fixed at either −Vs or +Vs, determined bya particular switching logic. Fourier series approximation of the voltage switchingwaveform is expressed by
V (t) =∞
∑
n=1
(an cos nωst + bn sin nωst) (B.4)
where ωs is the switching frequency and is equal to 2π/τ . Assume that the duty cycleof the switching is fixed at 50%. The actual waveform would be similar to the oneshown in Figure B.1. Since the switching frequency is already high enough (20 ∼ 50
101
-
6
τ t
Vs
V
−Vs
Figure B.1: Voltage waveform at 50% duty cycle
kHz), the higher harmonics are assumed to have negligible effects. Since the voltagewaveform is defined as an even function, (B.4) can be approximated as
V (t) ≈ −4Vs
πcos ωst (B.5)
Substituting (B.5) into (B.3), the current can be obtained by integrating
i(t) = − 4Vs
πLog
∫ t
0
[g − xo sin ωξ] cos ωsξdξ (B.6)
where the motion of the mass is assumed to be sinusoidal with a frequency of ω andamplitude of xo. Successive integration by parts results in
i(t) = − 4Vs
πLogωs
[
g sin ωst −xo
1 − (ω/ωs)2
− ω
ωs
+ω
ωs
cos ωt cos ωst +
ω
ωs
sin ωt sin ωst
] (B.7)
If the ratio (ω/ωs) is less than 0.1, it is reasonable to make the following approxima-tion.
i(t) ≈ − 4Vs
πLogωs
(g − xo sin ωt) sin ωst (B.8)
Eq. (B.8) clearly shows that displacement modulates the amplitude of the switchingnoise. Thus, a proper demodulation will result in a signal directly related to thedisplacement. A simple demodulation can be done through a rectification and a low–pass filtering. Since the gap is always positive, the rectification of (B.8) would resultin
|i(t)| =4Vs
πLogωs
(g − xo sin ωt)| sin ωst| (B.9)
102
Based on the previous assumption that the dynamics of the mass are much slowerthan that of the switching, the effect of low–pass filtering is essentially an averagingaction over a certain time period. The cut–off frequency of the filter needs to behigher than the bandwidth of the mass but lower than the switching frequency. Theoutput of the filter can be approximated by considering the harmonic contents of therectified signal. The Fourier series expansion of | sin ωst| would be
| sin ωst| =1
2a0 +
∞∑
n=1
(an cos nωst + bn sin nωst) (B.10)
Only the DC component of (B.10) will be significant after the low–pass filtering, TheDC component is obtained by evaluating a0 in (B.10).
a0 =1
π
∫ π
0
| sin θ|dθ =4
π
With the above approximation, the idealized output of the filter is
u =8Vs
π2Logωs
(g − x) (B.11)
In terms of the switching interval, the output is
u =4Vsτ
π3gLo
(g − x) (B.12)
103
Appendix C
Hysteresis Model
A magnetization model is a model that relates the magnetic field intensity H to theflux density B. This relationship can be approximated as a linear one in which aproportionality constant µ called permeability is defined as
µ.=
B
H
Fundamentally, however, the magnetization process is nonlinear in nature due tosaturation and hysteresis. Hysteresis nonlinearity usually means the input–outputrelationship is a multi–branch nonlinearity for which a branch–to–branch transitionoccurs after each input extremum. For a nonlinear hysteresis model, the definition ofthe permeability should be modified as the following equation.
µ.=
B
H(C.1)
As presented in Chapter 2, there have been numerous efforts to model this hys-teresis phenomenon, notably by Coleman and Hodgdon, and Jiles and Atherton. Al-though these models are capable of describing complicated magnetization processes,they are usually given as differential equations or integral equations. Therefore, thenonlinearity only becomes obvious after solving the equations either analytically ornumerically. This complexity makes it difficult to employ these models for the purposeof assessing the effect of magnetic nonlinearity on the performance of the estimator.In this dissertation research, a different nonlinear magnetization model is proposed.The model is presented by a set of analytic functions (not differential or integral equa-tions). Once the parameters describing the model are identified, the model properlypredicts the magnetization process without needing to solve additional equations.
The model assumes that the B−H curve is confined by two envelopes representedby
H1(B) =B
µoµ0r
+σ
µo
(
1 − 1
µ0r
)
log(1 + η · e(B−Bs)/σ) + Hr (C.2)
H2(B) =B
µoµ0r
− σ
µo
(
1 − 1
µ0r
)
log(1 + η · e(−B−Bs)/σ) − Hr (C.3)
104
Parameters ValuesNominal relative permeability µ0
r 50942Saturation flux density Bs 1.25Envelope parameter σ 0.17579Envelope parameter η 0.00474Approach factor β 8.2
Table C.1: Parameter values used in simulation
Close inspection of (C.2) and (C.3) reveals that they are composed of two asymptotes.One of these asymptotes represents the linear portion where flux density is less thansaturation. The other asymptote represents the saturation region in which the slopeof the B−H curve converges to the permeability of free space. The parameter σcontrols the sharpness of the transition from one region to the other. The value ofη can be determined from the intercept of the asymptote for large flux density on Baxis. The actual magnetization H as a function of flux density B is given by
H(B) =
H1(B) − [H1(B0) − H(B0)] · e−β|B−Bo| if B ≥ 0
H2(B) − [H2(B0) − H(B0)] · e−β|B−Bo| if B < 0(C.4)
where Bo is the flux density when B changes its sign. The parameter β controlsthe shape of the actual B−Hcurve. A “soft” magnetic material can be representedby (C.4) with smaller β, whereas a “hard” magnetic material will be described withlarger β. Given the actual B−H for the core material, one can easily identify theseparameters. Figure C.1 shows the nonlinear magnetization curve produced by (C.4)using the parameters in Table C.1. The saturation flux density Bs can be eitherchosen as a free variable in the fitting process or fixed to a manufacturer’s data.With the selected parameters, the model satisfactorily represents the experimentaltest data, which is obtained by running 60 Hz sinusoidal current wound on a stack of0.3556 mm (14 mil) laminations made with silicon iron.
When considering only the major hysteresis loop, the reciprocal of the perme-ability, 1/µ, can be approximated as the partial derivative of the average of twoenvelopes (C.2) and (C.3) with respect to B.
1
µr
=1
µ0r
+ sgn(B)
(
1 − 1
µ0r
)
η · e(|B|−Bs)/σ
1 + η · e(|B|−Bs)/σ(C.5)
Since a switching amplifier is assumed to drive a magnetic bearing, the time historyof the flux density B and the magnetization H forms minor hysteresis loops, as shownin Figure C.2. It can be seen from this figure that the permeability of the minor loopsis substantially lower than that of the major hysteresis loops. Therefore, one wouldunderestimate the core reluctance if the permeability obtained from (C.5) is used.Figure C.3 illustrates that the relative permeability based on (C.5) is one order ofmagnitude larger than the relative permeability of the minor loops at low flux density.At large flux density, the difference becomes smaller, and in the limit, the relative
105
-1000 -800 -600 -400 -200 0 200 400 600 800 1000Magnetization (A/m)
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Flu
xde
nsity
(T)
ExperimentModel
Figure C.1: Nonlinear magnetization curve obtained by (C.4).
permeability of the minor loops would be the same as that of the major hysteresisloops.
106
−500 0 500 10000
0.5
1
1.5
Magnetization (A/m)
Flu
x de
nsity
(T
esla
)
Figure C.2: Minor hysteresis loops.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Flux density (T)
102
2
5
103
2
5
104
2
5
105
Rel
ativ
epe
rmea
bilit
y
Major loopMinor loop
Figure C.3: Relative permeability based on major and minor hysteresis loops.
107
Appendix D
Circuit Diagram of the Estimator
Shown in Figure D.1 is the cicuit diagram of the estimator used in this dissertationresearch. Connections to supply voltages are ommited due to space restriction. Thetable below lists the pin numbers of each component for supply voltage. All connec-tions to DC supply are bypassed with 1.0 µF capacitors. Two single op amps usedto realize the inductor model are offset–adjusted as shown below.
Component Function Manufacturer +V −VUAF42 Universal filter Burr Brown 10 9MPY634 Analog multiplier Burr Brown 14 8TL084 Quad op–amp Texas Instrument 4 11TL071 Single op–amp Texas Instrument 7 4
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Figure D.1: Circuit diagram of the position estimator
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