Self-Bearing Motor Design & Control
by
Mohammad Imani Nejad
Submitted to the Department of Mechanical Engineering
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
ARCHrVES
MASSACHUSETTS INSTITUTE
OF T~cNLOGY
at the
/0iF
APR
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
February 2013
@ Massachusetts Institute of Technology 2013. All rights reserved.
Author ........................
.
...........
Department of Mechanical Engineering
Jan. 15, 2013
C ertified by.......................
..............
David L. Trumper
Professor of Mechanical Engineering
Thesis Supervisor
A ccepted by ...........................
.
..............
David E. Hardt
Chairman, Department Committee on Graduate Theses
f5
Self-Bearing Motor Design & Control
by
Mohammad Imani Nejad
Submitted to the Department of Mechanical Engineering
on Jan. 15, 2013, in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy in Mechanical Engineering
Abstract
This thesis presents the design, implementation and control of a new class of selfbearing motors. The primary thesis contributions include the design and experimental
demonstration of hysteresis self-bearing motors, novel segmented stator structures,
MIMO loop shaping control algorithm for levitation and commutation, hysteresis
motor analysis including frequency dependency, nonlinear hysteresis model including
loop widening, and a novel single-axis self-bearing motor, as well as a zero power
configuration for this type of motor.
In the late 1980s, the basic concepts of self-bearing motors was proposed. Since
then, different types of AC and DC electric machines have been studied as a selfbearing motor. The self-bearing system is a key technology for high efficiency and
compact systems with integrated magnetic levitation for high speed and high precision applications. One of the major disadvantages of existing self-bearing motors for
high speed application is their rotor mechanical construction. Either the permanent
magnet or induction machines rotor has mechanical features that introduces stress
concentrations. Permanent magnets have very low mechanical strength and need to
be inserted into the rotor. The assembly of magnets makes rotor vulnerable to mechanical failure at high speed. On the other hand, induction motors use soft steel
to reduce hysteresis loss. Their rotors are slotted to either carry wires or in case of
squirrel cages, having aluminum or copper bars.
As a promising alternative, this thesis demonstrates hysteresis self-bearing motors
which have a simple construction with a solid and smooth rotor. This is a very
important characteristic for some applications. This type of system can also be used
as a magnetic bearing that can apply a finite amount of torque. The rotor doesn't
have to be laminated and can be made from high strength steel. We designed, built
and tested this type of self-bearing motor successfully.
In this thesis we also introduced a new type of segmented stator for hysteresis
machines. The major advantages of this stator are: easy and low cost manufacturing,
higher filling factor and higher motor efficiency. We tested the self-bearing concept
successfully with this new configuration.
We have also introduced a novel single axis self-bearing motor that is very suitable
2
for rotors with large length to diameter aspect ratio, such as high speed flywheels for
energy storage. We implemented a zero power levitation condition along with passive
damping for this system that has several advantages for high speed systems. One
of the major advantages of zero power systems is the simple and robust touch down
bearing design which is a key element for active magnetic bearings and self-bearing
motors. This is mainly because the bearings experience minimal load in case of power
failure.
Hysteresis is a time-rate dependent phenomena which is fundamentally related to
eddy current formation in the material and the thermal agitation. Hysteresis loops of
materials with large hysteresis are highly frequency dependent. Therefore we added
hysteresis frequency dependency to hysteresis motor analysis, which is believed to
be a novel contribution . We have also developed linear and nonlinear analyses for
the stabilizing forces and moments for hysteresis self-bearing motor. The nonlinear
analysis is based on Chua's nonlinear hysteresis model that includes loop widening.
The theoretical results were verified by experimental data for three different type
motor configurations with good accuracy.
Finally, we built two identical induction machines except for the rotor material.
One rotor is a commercial squirrel cage and the other one is a simple solid rotor made
out of hysteresis material(D2 steel). We ran the IEEE standard tests for these motors
and compared their performance under different circumstances.
Thesis Supervisor: David L. Trumper
Title: Professor of Mechanical Engineering
3
0.1
Acknowledgments
First, I would like to thank Prof. Trumper for supervising this thesis. He provided
the guidance and support to make this research possible. Very few advisors would be
willing to give as much freedom and trust to explore a design as Professor Trumper
does. It has been an honor to work with him. He means more than an advisor to me
and I am so thankful for all his help.
I also want to thank the other members of my committee. Prof. Jeffrey Lang and
Prof. Sanjay Sarma were great source of enthusiasm and news ideas. I enjoyed very
much of working with Prof. Lang. He always pointed out key problems and suggested
quick solutions. He had a significant contribution on nonlinear hysteresis analysis as
well as motor modeling. I was so fortunate to have Prof. Sarma in my committee. He
is a perfect person to discuss new ideas. He was an invaluable resource in directing
this thesis.
Many people provided me with invaluable help for this thesis. I would like to
thank Darya Amin-Shahidi for his inspiration and support on almost all aspects of
this thesis. I would also like to thank Lei Zhou for helping me on editing and deriving
hysteresis motor modeling equations. I want also to thank Dean Liubicic, Ross Ian
MacKenzie, Jun Young Yoon, Zhen Sun, Minkkyun Noh for their supports and helps.
Finally, I am most indebted to my wife, daughter, son, Mom and brothers for their
patience, love and encouragement. They truly made tremendous sacrifices during last
five years for my study at MIT.
4
To
Rojina and Aryo
5
Contents
0.1
1
4
21
Introduction
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
1.2
Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
1.3
Prior Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
1.4
1.5
2
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1
Self-bearing motor
. . . . . . . . . . . . . . . . . . . . . . . .
25
1.3.2
Hysteresis Motors . . . . . . . . . . . . . . . . . . . . . . . . .
27
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Thesis Overview.
1.4.1
Hysteresis Self-bearing Motor
. . . . . . . . . . . . . . . . . .
30
1.4.2
Yokeless Segmented Stator . . . . . . . . . . . . . . . . . . . .
31
1.4.3
Single Axis Self-bearing Motor . . . . . . . . . . . . . . . . . .
35
1.4.4
Nonlinear Hysteresis Analysis
. . . . . . . . . . . . . . . . . .
38
1.4.5
Hysteresis Motor Modeling Including Frequency Dependency .
40
Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Principles of Hysteresis and Synchronous Motors
2.1
2.2
45
H ysteresis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
2.1.1
Hysteresis Loop . . . . . . . . . . . . . . . . . . . . . . . . . .
46
2.1.2
Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . .
48
2.1.3
Loop Widening . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2.1.4
Hysteresis Linearization
. . . . . . . . . . . . . . . . . . . . .
50
2.1.5
Nonlinear Hysteresis Modeling . . . . . . . . . . . . . . . . . .
50
Synchronous Motor Fundamentals . . . . . . . . . . . . . . . . . . . .
56
6
3
2.2.1
Torque Expression
. . . . . . .
. . . . . . . . . . . .
56
2.2.2
Balanced Operation . . . . . . .
. . . . . . . . . . . .
60
2.2.3
Rotating Magnetomotive Force
. . . . . . . . . . . .
61
2.2.4
Stator Leakage
. . . . . . . . .
. . . . . . . . . . . .
64
2.2.5
Stator Winding Resistance . . .
. . . . . . . . . . . .
66
Small setup . . . . . . . . . . . . . . .
. . . . . . . . . . . .
67
3.1.1
Active Magnetic Bearing . . . .
. . . . . . . . . . . .
68
3.1.2
M otor
. . . . . . . . . . . . . .
. . . . . . . . . . . .
69
3.2
Large setup . . . . . . . . . . . . . . .
. . . . . . . . . . . .
74
3.3
Radial motor
. . . . . . . . . . . . . .
. . . . . . . . . . . .
76
3.4
Power Amplifier . . . . . . . . . . . . .
. . . . . . . . . . . .
78
3.4.1
Design Specifications . . . . . .
. . . . . . . . . . . .
78
3.4.2
Architecture . . . . . . . . . . .
. . . . . . . . . . . .
79
3.4.3
Power OP Amp Considerations
. . . . . . . . . . . .
80
. . . . . . . .
. . . . . . . . . . . .
82
. . . . . . . . . . .
. . . . . . . . . . . .
87
3.1
3.4.4
3.5
4
67
Hardware
Controller design
Magnetic Material
3.5.1
Hysteresis Loop Measurement
. . . . . . . . . . . .
87
3.5.2
Somaloy . . . . . . . . . . . . .
. . . . . . . . . . . .
89
3.5.3
D2 Steel . . . . . . . . . . . . .
. . . . . . . . . . . .
91
3.5.4
Maraging Steel
. . . . . . . . .
. . . . . . . . . . . .
91
94
Active Magnetic Bearing
4.1
4.2
Basic Active Magnetic Levitation
.
. . . . . . . . . . . . . . . .
94
4.1.1
Single Axis Modeling . . . . . .
. . . . . . . . . . . . . . . .
95
4.1.2
Planar Levitation . . . . . . . .
. . . . . . . . . . . . . . . .
98
4.1.3
Force-Current Linearization
. .
. . . . . . . . . . . . . . . .
100
. . . . . . . . . .
. . . . . . . . . . . . . . . .
10 1
4.2.1
Stiffness and Damping Selection . . . . . . . . . . . . . . . . .
102
4.2.2
System Identification . . . . . . . . . . . . . . . . . . . . . . .
104
Closed Loop System
7
4.2.3
4.3
5
105
. . . . . . . . . . . .
111
Active Magnetic Bearing Design Considerations
4.3.1
Cooling Capacity . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.3.2
Bearing Geometry
. . . . . . . . . . . . . . . . . . . . . . . .
114
4.3.3
Power Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . .
115
4.3.4
Maximum Achievable Speed . . . . . . . . . . . . . . . . . . .
117
120
Hysteresis Self-bearing Motor
5.1
5.2
5.3
Single Winding Segmented Stator . . . . . . . . . . . . . . . . . . . .
120
5.1.1
Principle of Operation of Segmented Stator . . . . . . . . . . .
121
5.1.2
Control Strategy
. . . . . . . . . . . . . . . . . . . . . . . . .
122
5.1.3
Suspension Forces and Moments . . . . . . . . . . . . . . . . .
126
5.1.4
Simulation and Experimental Results . . . . . . . . . . . . . .
130
Multiple Winding Configuration . . . . . . . . . . . . . . . . . . . . . 141
5.2.1
P t 2 Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.2.2
DC Excitation (Magnetic Bearing)
5.2.3
Park and Clarke Transformation . . . . . . . . . . . . . . . . .
145
5.2.4
AC Excitation . . . . . . . . . . . . . . . . . . . . . . . . . . .
146
5.2.5
Simulation and Experimental Results . . . . . . . . . . . . . .
147
Single DOF Self-Bearing Motor . . . . . . . . . . . . . . . . . . . . .
149
. . . . . . . . . . . . . . . . . . . . . .
149
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
155
. . . . . . . . . . . . . .
157
5.3.1
5.4
Principle of Operation
Zero Power
5.4.1
6
Basic Controller Design . . . . . . . . . . . . . . . . . . . . . .
Lateral Damping Ratio and Stiffness
. . . . . . . . . . . . . . . 143
160
Hysteresis motor
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
160
6.2
M odeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
163
6.3
Motor Equivalent Circuit Parameters . . . . . . . . . . . . . . . . . .
166
6.3.1
Hysteresis Material Approximation
. . . . . . . . . . . . . . .
167
6.3.2
Flux Distribution . . . . . . . . . . . . . . . . . . . . . . . . .
169
6.3.3
Derivation of Equivalent Circuit . . . . . . . . . . . . . . . . .
173
8
Dynamic Equations of Motion . . . . . . . . . . . . . . . . . .
180
6.4
Circuit Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
180
6.5
Simulations and Experimental Verification
. . . . . . . . . . . . . . .
182
6.3.4
6.6
6.5.1
Simulations
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 2
6.5.2
Small motor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 7
6.5.3
Large Motor . . . . . . . . . . . . . . . . . . . . . . . . . . . .
200
6.5.4
Torque Constant and Back EMF
Motor Design Considerations . . . . . . . . . . . . . . . . . . . . . . .
206
. . . . . . . . . . . . . . . . . . . . . . . . . .
207
. . . . . . . . . . . . . . . . . . . . . . . . .
209
6.7.1
Rotor Materials . . . . . . . . . . . . . . . . . . . . . . . . . .
209
6.7.2
Stator materials . . . . . . . . . . . . . . . . . . . . . . . . . .
2 13
. . . . . . . . . . . . . . . . . . . . . . . . .
2 14
ANSYS Simulation . . . . . . . . . . . . . . . . . . . . . . . .
2 15
6.6.1
6.7
6.8
Optimal Motor
Material Considerations
Finite Element Analysis
6.8.1
7
. . . . . . . . . . . . . . . 206
Hysteresis and Squirrel Cage Motor Comparison
218
7.1
Induction Motor Equivalent Circuit . . . . . . . . . . . .
219
7.2
Radial Hysteresis Motor Equivalent Circuit . . . . . . . .
221
7.3
IEEE Standard Test Procedure for Polyphase Induction Motors and
. . . . . . . . . . . .
Generators.
7.4
226
7.3.1
Definitions . . . . . . . . . . . .
. . . . . . . . . . . . 226
7.3.2
Basic Requirements . . . . . . .
. . . . . . . . . . . . 228
7.3.3
No-load Test
. . . . . . . . . .
. . . . . . . . . . . . 228
7.3.4
Locked Rotor Test
. . . . . . .
. . . . . . . . . . . . 229
7.3.5
Efficiency Test Methods
. . . .
. . . . . . . . . . . . 232
7.3.6
E or El Efficiency Test Method
IEEE Standard Test Results . . . . . .
. . . . . . . . . . . . 233
. . . . . . . . . . . . 236
7.4.1
Squirrel Cage Motor Results . .
. . . . . . . . . . . .
236
7.4.2
Hysteresis Motor Results . . . .
. . . . . . . . . . . .
239
7.4.3
Comparison . . . . . . . . . . .
. . . . . . . . . . . .
244
9
8
Conclusion and Future Works
8.1
Conclusion . . . . . . . . ..
8.2
Future Works ......
245
. . . . . . . . . . . . . . . . . . . . . . .
245
. . . .
246
...........................
8.2.1
Hybrid M otors
......................
. . . .
246
8.2.2
Segmented Stator . . . . . . . . . . . . . . . . . . . . . . . . .
247
8.2.3
Single Axis Self-bearing Motor . . . . . . . . . . . . . . . . . . 247
8.2.4
Hysteresis Active Magnetic Bearing . . . . . . . . . . . . . . .
247
8.2.5
Linear Hysteresis Self-bearing Motor
. . . . . . . . . . . . . .
248
8.2.6
Sm art Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . .
248
8.2.7
Nonlinear Hysteresis Motor Analysis
. . . . . . . . . . . . . .
248
8.2.8
Finite Element Method For Hysteresis Motor Analysis
. . . .
248
8.2.9
Transient Analysis Of Hysteresis Motor . . . . . . . . . . . . .
248
250
A Electromagnetic actuator
A.0.10 Electromagnet . . . . . . . . . . . . . . . . . . . . . . . . . .
250
. . . . . . . . . . . . . . . . . . . . .
251
A.0.12 Inductance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
252
. . . . . . . . . . . . . . . . . . . . . . . . . .
255
A.0.14 Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
256
A.0.11 Electromagnetic Force
A.0.13 Eddy current
257
B MATLAB Codes
B.0.15 Axial flux hysteresis motor . . . . . . . . . . . . . . . . . . .
257
. . . . . . . . . . . . . . . . . . . . . . . .
262
B.O.16 Induction Motor
10
List of Figures
1-1
Hysteresis Self-bearing motor. The rotor ring at the center is levitated
and rotated by a hysteresis motor structure. . . . . . . . . . . . . . .
1-2
Segmented stator consisting of 12 laminated U cores.
30
This machine
uses a rotor with integral air gap, as visible in the picture. The gap is
filled with epoxy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-3
Segmented stator magnetic path. The epoxy gap separates the two
rotor flux paths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1-4
33
Proposed trapezoid pole face for segmented stator design, allowing
more efficient magnetic flux paths . . . . . . . . . . . . . . . . . . . .
1-5
32
33
Hysteresis self-bearing lower stator. The stator has 36 open slots, and
the coils are wrapped around the stator behind each slot. . . . . . . .
34
1-6
Single axis hysteresis self-bearing system. . . . . . . . . . . . . . . . .
36
1-7
Rotor axial run-out at 600 rpm. . . . . . . . . . . . . . . . . . . . . .
37
1-8
Single axis bearing rotor axial position during startup.
37
1-9
Large setup step response compared with linear and nonlinear simula-
. . . . . . . .
tion results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
1-10 Simulation results for force versus current for large setup. . . . . . . .
39
1-11 Simulation results for force versus displacement for large setup. . . .. .
39
1-12 Angular velocity versus time for small motor.
. . . . . . . . . . . . .
40
1-13 Angular velocity versus time for large motor. . . . . . . . . . . . . . .
41
1-14 Simulation results for hysteresis motor torque versus speed for small
motor neglecting eddy current . . . . . . . . . . . . . . . . . . . . . .
11
42
1-15 Simulation results for hysteresis motor torque versus speed for small
motor including eddy current. . . . . . . . . . . . . . . . . . . . . . .
42
2-1
Hysteresis definition.
46
2-2
Hysteresis loop schematic.
2-3
Hysteresis for soft and hard materials.
. . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
47
B-H is in solid line, M-H in
dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
2-4
Linear model of hysteresis. . . . . . . . . . . . . . . . . . . . . . . . .
51
2-5
Possible W functions and their effect on frequency behavior, adapted
from [11].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2-6
Chua model Simulink block diagram, which implements 2.2.
2-7
Chua model response for hysteresis of D2 steel.
52
. . . . .
53
. . . . . . . . . . . .
53
2-8
Loop widening in Chua model at 1, 5, and 20 Hz. . . . . . . . . . . .
54
2-9
Minor loops in Chua model with excitation frequency of 3 Hz. .....
55
2-10 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
57
2-11 MMF produced by a concentrated full-pitch winding. . . . . . . . . .
61
2-12 MMF produced by a concentrated full-pitch winding. . . . . . . . . .
63
2-13 Stator slot geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
3-1
Small setup has 80 mm diameter rotor levitated by upper magnetic
bearing stator and rotated by lower stator. The upper stator has U
core actuators to operate with the rotor as an active magnetic bearing
(A M B ).
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
3-2
Sensor response data for DW-AD-509-M8-390 sensor by Contrinex [13]. 70
3-3
Active magnetic bearing used as upper stator for small setup. ....
71
3-4
Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
72
3-5
Small setup in operation. Upper rotor air-gap is visible. . . . . . . . .
73
3-6
Large setup schematic view. Rotor is shown as transparent.
. . . . .
74
3-7
Segmented stator for large setup upper side.
. . . . . . . . . . . . . .
75
3-8
Radial rotor for squirrel cage and hysteresis motor. Conventional rotor
on top was replaced with solid D2 steel rotor.
12
. . . . . . . . . . . . .
76
3-9
Radial stator of commercial induction machine.
. . . . . . . . . . . .
77
3-10 Radial motor of commercial induction machine made by Oriental Mo-
tor model 51K90A-SF.
. . . . . . . . . . . . . . . . . . . . . . . . . .
77
3-11 Power amplifier designed and built for the project. . . . . . . . . . . .
79
3-12 Power amplifier schematic. . . . . . . . . . . . . . . . . . . . . . . . .
80
3-13 PA12 safe operating area from apex PA-12 data sheet [1].
. . . . . .
81
. . . . . . . . . . . . . . . . . . . . .
82
3-14 Power amplifier Block diagram.
3-15 Differential amplifier configuration.
. . . . . . . . . . . . . . . . . . .
83
3-16 Current control system bode plot. These plots are for the plant, the
controller and the loop return ratio. The loop cross over is at about
20 K H z. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-17 Closed-loop current control frequency response.
. . . . . . . . . . . .
84
85
3-18 Loop transmission for current control loop, based upon model and
experim ent.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3-19 Slew rate effect.
lim ited output.
85
Upper trace is input, lower trace is the slew-rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
86
3-20 Circuit diagram for B-H measurement adapted from Haus and Melcher
[2 1].
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
3-21 Measured B-H loop for Somaloy. . . . . . . . . . . . . . . . . . . . . .
89
3-22 Somaloy data from Hoganas [24].
90
. . . . . . . . . . . . . . . . . . . .
3-23 Measured B-H loop for Somaloy for three current excitation amplitudes. 90
3-24 Measured B-H loop for Somaloy for different excitation frequencies.
.
91
3-25 Measured B-H loop for D2 steel for three current excitation amplitudes. 92
3-26 Measured B-H loop for D2 steel for different excitation frequencies.
.
92
3-27 Measured B-H loop for Maraging steel for three current excitation am-
plitude.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
3-28 Measured B-H loop for Maraging steel for different excitation frequencies. 93
4-1
Basic one degree of freedom magnetic levitation system . . . . . . . .
95
4-2
Magnetic levitation free body diagram. . . . . . . . . . . . . . . . . .
96
13
4-3
Mechanical and electromagnet stiffness. . . . . . . . . . . . . . . . . .
98
4-4
Planar levitation free body diagram.
99
4-5
Magnetic levitation in differential mode.
4-6
Block diagram for magnetic levitation.
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . .
101
. . . . . . . . . . . . . . . . .
102
4-7
Plant frequency response for axial direction(small setup). . . . . . . .
104
4-8
Plant frequency response for tip-tilt direction(small setup). . . . . . .
105
4-9
Simulink block diagram for small setup AMB using Dspacell03. Bad
link blocks simply indicate lack of a hardware connection, but are not
norm ally presents . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
108
4-10 Labview block diagram for small setup AMB using Ni PXI controller.
110
4-11 Bode plots for small setup AMB.
. . . . . . . . . . . . . . . . . . . .
111
4-12 Step response for small setup AMB. . . . . . . . . . . . . . . . . . . .
112
4-13 Magnetic bearing thermal network.
. . . . . . . . . . . . . . . . . . .
113
4-14 Coil with rectangular cross section.
. . . . . . . . . . . . . . . . . . .
115
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
4-16 Small setup power bandwidth. . . . . . . . . . . . . . . . . . . . . . .
117
4-17 Radial and tangential stress in rotating ring. . . . . . . . . . . . . . .
118
5-1
Segmented stator-rotor magnetic path in composite ring. . . . . . . .
121
5-2
FEA simulation for segmented stator in two pole configuration.
. . .
122
5-3
Segmented self-bearing motor principle in schematic representation.
5-4
Four pole segmented self-bearing motor.
4-15 Fabry chart [18].
.
123
. . . . . . . . . . . . . . . .
124
5-5
Single winding self-bearing control block diagram. . . . . . . . . . . .
125
5-6
Magnetic path for U core actuator.
126
5-7
Force generation in nonlinear bearing model including Chua hysteresis
model. ........
5-8
.............
. . . . . . . . . . . . . . . . . . .
... .
.....
..
..........
Hysteresis self-bearing suspension control block diagram.
force model is enclosed in block labeled Electromagnet.
5-9
14
Nonlinear
. . . . . . . .
Plant frequency response for axial direction at zero speed.
5-10 Plant frequency response for tip/tilt direction(6).
.128
129
. . . . . .
130
. . . . . . . . . . .
131
5-11 Loop transfer function frequency response for axial direction. . . . . .
132
5-12 Loop transfer function frequency response for tip/tilt direction (6).
.
133
5-13 Closed loop step response for axial direction. . . . . . . . . . . . . . .
133
5-14 Closed loop step response for tip/tilt direction (0) . . . . . . . . . . .
134
5-15 Hysteresis model of Chua's configuration as fitted to D2 steel experimental data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5-16 Axial position during startup with initial gap=0.5mm.
135
. . . . . . . .
136
5-17 Current during startup from zero initial current. . . . . . . . . . . . .
136
5-18 Force v.s. Current during startup. . . . . . . . . . . . . . . . . . . . .
137
5-19 Plant B-H during startup. . . . . . . . . . . . . . . . . . . . . . . . .
137
5-20 Step response. Zero reference corresponds to a gap of 0.5mm.
138
. ...
5-21 Force v.s. Current during transition of an 0.1mm step input from initial
0.5m m gap. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
138
5-22 Force v.s. Displacement during transition of an 0.1mm step input from
initial 0.5mm gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
139
5-23 Plant B-H during transition of an 0.1mm step input from initial 0.5mm
gap. .........
....................................
139
5-24 Current v.s. Displacement during transition of an 0.1mm step input
from initial 0.5mm gap . . . . . . . . . . . . . . . . . . . . . . . . . .
5-25 Force generation in self-bearing motor.
. . . . . . . . . . . . . . . . .
5-26 Moment generation in axial self-bearing motor.
140
142
. . . . . . . . . . . .
142
. . . . . . . . . . . . . . . . . .
143
. . . . . . . . . . . . . . . . . . .
144
5-29 Arbitrary space vector in complex plane for a three phase system. . .
145
5-30 Self-bearing (AC excitation). . . . . . . . . . . . . . . . . . . . . . . .
147
5-31 Step response for P+2 control scheme in x direction.
. . . . . . . . .
148
. . .
149
5-33 Single axis rotor configuration. . . . . . . . . . . . . . . . . . . . . . .
151
5-34 Single axis self-bearing setup.
153
5-27 Self-bearing motor winding diagram.
5-28 Magnetic bearing (DC excitation).
5-32 Free body diagram for single degree freedom self-bearing motor.
. . . . . . . . . . . . . . . . . . . . . .
5-35 Single axis plant frequency response.
15
. . . . . . . . . . . . . . . . . .
154
5-36 Rotor axial position during startup. . . . . . . . . . . . . . . . . . . .
154
5-37 Step response.......
155
................................
5-38 Axial run-out at 600 rpm. . . . . . . . . . . . . . . . . . . . . . . . .
156
5-39 Im pulse response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
157
6-1
Hysteresis motor operation principle, adapted from [17]. Here W is the
rotor angular velocity, 5 is the lag angle, and SS and RR are the stator
and rotor axes respectively ..
. . . . . . . . . . . . . . . . . . . . . . .
161
6-2
Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
162
6-3
Magnetic vector in the rotor with non-magnetic material core.....
163
6-4
Magnetic vector in the rotor with magnetic material core. . . . . . . .
164
6-5
Hysteresis motor equivalent circuit. . . . . . . . . . . . . . . . . . . .
165
6-6
Hysteresis approximation for Maraging steel. . . . . . . . . . . . . . .
168
6-7
D2 steel linear model approximation.
. . . . . . . . . . . . . . . . . .
169
6-8
D2 steel magnetic properties as a function of frequency. . . . . . . . .
170
6-9
Magnetic path in motor. . . . . . . . . . . . . . . . . . . . . . . . . .
171
6-10 Magnetic flux density continuity in the rotor .
. . . . . . . . . . . . .
172
. . . . . . . . . . . . . . . . . . .
178
6-12 Hysteresis motor equivalent circuit. . . . . . . . . . . . . . . . . . . .
181
6-13 MATLAB script for initial parameters ...
. . . . . . . . . . . . . . . .
183
6-14 Hysteresis motor SIMULINK block diagram. . . . . . . . . . . . . . .
184
6-15 Rotor equivalent impedance SIMULINK block diagram. . . . . . . . .
185
6-16 Air-gap equivalent impedance SIMULINK block diagram. . . . . . . .
186
6-17 Hysteresis motor stator.
187
6-11 Phasor diagram adapted from [26].
. . . . . . . . . . . . . . . . . . . . . . . . .
6-18 Drag torque measured experimentally.
. . . . . . . . . . . . . . . . .
188
6-19 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
190
6-20 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
191
6-21 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
192
6-22 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . . 193
6-23 Optional caption for list of figures .
16
194
6-24 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
195
6-25 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
196
6-26 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .
197
6-27 Small motor input power versus time. . . . . . . . . . . . . . . . . . .
198
6-28 Terminal voltage and current versus time.
. . . . . . . . . . . . . . .
198
. . . . . . . . . . . . . .
199
6-29 Hunting phenomena (Experimental results).
6-30 36 slot stator made out of soft composite materials used for large setup. 200
6-31 Large motor angular speed versus time. . . . . . . . . . . . . . . . . . 201
6-32 Large motor lag angle. . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6-33 Large motor lag angle. . . . . . . . . . . . . . . . . . . . . . . . . . . 202
6-34 Large motor torque versus angular velocity. . . . . . . . . . . . . . . . 203
6-35 Large motor torque v.s. speed zoomed in near synchronous speed. . . 203
6-36 Large motor lag angle versus speed zoomed in near synchronous speed. 204
204
6-37 Large motor output power. . . . . . . . . . . . . . . . . . . .
6-38 Large motor output power. . . . . . . . . . . . .
. . . .
6-39 Measured and simulated back EMF .
. . . . . .
. . . . 206
. . . . . . . . . . . . . . . . .
. . . . 207
6-41 Magnetic flux as a function of radius. . . . . . .
. . . . 208
6-42 Maraging and D2 steel B-H curves. . . . . . . .
. . . . 210
6-43 Maraging and D2 steel speed comparison.
. . . . 212
6-40 Torque constant.
. . .
205
6-44 Magnetic path. . . . . . . . . . . . . . . . . . .
. . . .
213
6-45 Absolute magnetic Flux density in rotor.....
. . . .
215
6-46 Absolute magnetic Flux density in stator.
. . . .
216
. . . .
217
.
6-47 Hysteresis modeling in ANSYS. Only path 1 can be modeled..
. . . . . . . . . . . . . . .
219
. . . . . . . . . . .
222
7-3
Radial hysteresis motor magnetic path. . . . . . . . . . . . . .
222
7-4
Radial hysteresis motor SIMULINK block diagram. . . . . . .
224
7-5
Rotor apparent impedance ..
. . . . . . . . . . . . . . . . . . .
225
7-1
Induction motor equivalent circuit.
7-2
Radial hysteresis motor equivalent circuit.
17
7-6
Test bed for efficiency test. . . . . . . . . . . . . . . . . . . . . . . . .
7-7
E2 form for efficiency calculation [25]. . . . . . . . . . . . . . . . . . . 235
7-8
Friction and windage loss for squirrel cage motor.
. . . . . . . . . . .
237
7-9
Core loss versus voltage for squirrel cage motor. . . . . . . . . . . . .
238
7-10 Squirrel cage motor output torque.
233
. . . . . . . . . . . . . . . . . . .
239
7-11 Squirrel cage motor output power. . . . . . . . . . . . . . . . . . . . .
239
7-12 Friction and windage loss for hysteresis motor. . . . . . . . . . . . . .
241
7-13 Core loss versus voltage for hysteresis motor. . . . . . . . . . . . . . .
241
7-14 Torque versus speed hysteresis motor. . . . . . . . . . . . . . . . . . .
242
7-15 Output power versus speed hysteresis motor. . . . . . . . . . . . . . .
243
A-1
U core electromagnetic actuator and equivalent circuit.
. . . . . . . .
251
A-2
Mutual inductance. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
253
18
List of Tables
3.1
Displacement Sensor Spec. . . . . . . . . . . . . . . . . . . . . . . . .
69
3.2
Maximum allowable inductance
. . . . . . . . . . . . . . . . . . . . .
82
3.3
Power amplifier elements . . . . . . . . . . . . . . . . . . . . . . . . .
83
3.4
Somaloy machining . . . . . . . . . . . . . . . . . . . . . . . . . . . .
91
4.1
Current factor and negative stiffness (x)
. . . . . . . . . . . . . . . .
105
4.2
Current factor and negative stiffness (0)
. . . . . . . . . . . . . . . .
106
4.3
U core thermal properties
. . . . . . . . . . . . . . . . . . . . . . . .
114
4.4
U core dimensions
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
5.1
Current factor and negative stiffness for large setup . . . . . . . . . .
131
5.2
Passive magnetic bearing parameters
. . . . . . . . . . . . . . . . . .
158
6.1
Small Motor Parameters . . . . . . . . . . . . . . . . . . . . . . . . .
188
6.2
Large Motor Parameters . . . . . . . . . . . . . . . . . . . . . . . . .
201
7.1
IEEE Std112 empirical leakage distribution for induction machines . .
231
7.2
Induction Motor Parameters . . . . . . . . . . . . . . . . . . . . . . .
236
7.3
Manufacturer data for squirrel cage motor
. . . . . . . . . . . . . . .
236
7.4
Measured no-load test variables . . . . . . . . . . . . . . . . . . . . .
237
7.5
Measured locked rotor test variables . . . . . . . . . . . . . . . . . . .
238
7.6
Calculated motor parameters
239
7.7
EE1 test results for squirrel cage motor at rated frequency
. . . . . .
240
7.8
Measured locked rotor test variables . . . . . . . . . . . . . . . . . . .
240
7.9
Calculated motor parameters
242
. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . .
19
7.10 EEl test results for hysteresis motor at 11% slip . . . . . . . . . . . .
243
7.11 Motors comparison at 11% slip
244
. . . . . . . . . . . . . . . . . . . . .
20
Chapter 1
Introduction
In this thesis we design, construct and test a novel self-bearing motors that have
superior advantages for high speed applications. This thesis presents the modeling,
design, construction and testing of different sizes and configurations of hysteresis selfbearing motors. Through the course of this thesis, three different control schemes
implemented for hysteresis self-bearing motors. A new single axis self-bearing motor
was also designed, fabricated and successfully tested in several experiments.
This
process involved several steps including:
" Structural design of the parts, manufacturing and assembling.
" Electromagnetic actuator design and its fabrication.
* Dynamic system modeling and numerical analysis.
" Power electronics design and fabrication.
" Controller design and implementation for the systems.
" Interfacing.
" Instrument and material selection.
" Nonlinear hysteresis modeling including frequency dependent loop widening for
self-bearing motor.
21
* Hysteresis motor analysis including frequency dependency.
The primary thesis contributions include the design and experimental demonstration of hysteresis self-bearing motors, novel segmented stator structures, MIMO loop
shaping control algorithms for levitation and commutation, hysteresis motor analyses
including frequency dependency, nonlinear hysteresis models including loop widening,
and a novel single-axis self-bearing motor, as well as zero power configuration for this
type of motor.
Electric motors usually have a pair of mechanical bearings to support the rotor and
maintain the air gap. However these bearings have some limitations, especially for
high speed applications. Bearing lifetime is related to operating environment, rotor
speed and load. Moreover, for high speed systems, bearings interact with rotor bending vibration. In recent years, magnetic bearings have become a realistic proposition
for high speed machines implementing contact-free support. Some applications, such
as compressors, pumps and flywheels are designed to run at high speed to minimize
weight, cost and size. In addition to high speed systems, there are other applications
such as chemical pumps, rotors running at very low or very high temperatures and
clean pump systems where conventional mechanical bearings are not suitable.
Further, electric motors and active magnetic bearings (AMB) have very similar
constructions. Thus, it seems reasonable to consolidate the motor and AMB in one
package called a "self-bearing motor". This type of motor has several advantages
compared to regular motors but with the disadvantage of a relatively complex and
expensive controller and drive system.
Typically, electric motors have permanent magnets, windings or mechanical features on the rotor which makes them vulnerable to stress concentration at high speed.
An advantage of hysteresis motors is that they have a smooth, solid rotor made out
of high hysteresis material. Fortunately, high-strength steels, such as Maraging steel
have large hysteresis and are thus suitable for rotor material.
Therefore, hystere-
sis motors are suitable for high-speed applications thanks to their simple and rigid
construction.
Thus in this thesis we designed and built different type of hystere-
sis self-bearing motors to combine the advantages of both self-bearing systems and
22
hysteresis motors for high speed machines.
Typically, magnetic bearing rotors are made out of laminated soft steel to reduce
eddy current and hysteresis loss. However soft steels have low mechanical strength
which is not suitable for high speed applications. On the other hand, a hysteresis
self-bearing rotor can be made out of high strength steel, which also doesn't have to
be laminated. Therefore hysteresis self-bearing motors can also be used as magnetic
bearings that can apply both torque, and levitation force which may be specially
useful for high speed applications.
Aside from actuation, sensing is another major issue in high speed systems. Torque
measurement is not an easy task for high speed machines. Commercially available
hysteresis brake dynamometers (HD series) are used for testing in the low to middle
power range (max.14 KW intermittent duty). Hysteresis dynamometers can provide
precise torque loading independent of shaft speed. Typically these dynamometers
run on mechanical bearings which are limited to relatively low speed. This thesis also
provides the foundation for designing and building high speed hysteresis dynamometers.
1.1
Motivation
As we progress into
2 1"
century global warming has become a major issue.
The
rising C02 concentration is driven by carbon-based fuel that we use on daily basis.
To maintain and develop energy-based technologies, renewable energy resources are
necessary. Growing use of intermittent energy sources, such as wind and solar, increases the need for energy storage devices. In addition, one of the major problems
with electricity supply is the demand fluctuation throughout the day. However the
energy that can be produced is limited. Thus reliable, fast-response energy storage
devices are needed for supporting the electric power system. Many types of energy
storage have been proposed and implemented over the years.
High speed flywheels store energy in kinetic form. They have many advantages
compared to other energy storage systems. They are benign to the environment, have
23
no memory effect, have fast response, long life, high efficiency and are not significantly
influenced by temperature variation. They can be charged and discharged at high
rates for many cycles. The main drawbacks of high speed flywheel system are high
cost, and safety issues. Flywheel storage is a relatively old idea and has been used
in many different systems such as in internal combustion engines. As the technology
has advanced in different fields high speed flywheels have again become a major
interest of many researchers and companies for energy storage. High speed flywheels,
using composite materials for higher efficiency and lower cost are extensively under
development.
On the other hand self-bearing motors are a key technology in high
efficiency and ultra high speed motors. Thus, the focus of this thesis is to explore
a novel actuator (hysteresis self-bearing motor) that can be used for suspension and
driving of high speed flywheels. This design is also very suitable for high speed active
magnetic bearings as it has a simple rigid rotor.
We concentrate herein on axial flux type motors which we believe have advantages
over radial flux designs for ultra high speed machines. High speed rotors undergo a
large expansion due to centrifugal force. This significant expansion leads to air gap
variation in radial flux motors that reduces the efficiency and stability. For this reason
we propose axial flux type motors in which the expansion happens in a non-sensitive
direction.
1.2
Application
Besides high speed flywheels, we believe that hysteresis self-bearing motors have broad
applications.
High speed and maintenance free electric machines are necessary for
high precision machine tools, turbo molecular pumps, compressors, etc. Introducing
cogging-free, permanent-magnet-free self-bearing motor can open a new window for
many applications such as heart pumps, clean pumps for semiconductor industries,
hysteresis brake, hysteresis torque transducers, high speed magnetic bearings, etc.
A simple solid rotor that can be made out of hard and strong steel is a distinct
feature of this design which makes it very attractive for many high-speed applications.
24
As we know, induction machines become very competitive in terms of efficiency to
permanent magnet machines as the size increases. Therefore we envision a hysteresis
motor which also can operate as an induction machine can be a viable alternative for
large size machines.
1.3
Prior Work
A significant amount of effort has been devoted to research and development of selfbearing motors during the past two decades.
There are many self-bearing types
reported in the literature using a variety of actuating mechanisms and configurations. On the other hand, there are significantly fewer published papers and notes on
hysteresis motors. To the best of our knowledge, no one has yet studied hysteresis
self-bearing motors, providing the basis for this thesis.
Thus in the next two sections we present the prior art work that has been done
on self-bearing and hysteresis motor separately. The extensively studied induction
self-bearing motor is related to hysteresis machines, thus we include a review of prior
art for that field.
1.3.1
Self-bearing motor
The AC self-bearing motor concept was apparently first introduced in the mid 1980's [3].
This type of research was also pursued in Japan, especially by a group in University
of Tokyo [9] and [37]. Self-bearing motors are also known by other names such as
bearing-less motor, bearing motor, levitated motor, floating actuator and integrated
motor bearing. There are as many types of self-bearing motor as there are electric motors. Based on their working principles, prior art self-bearing motors can be
categorized mainly as two types: reluctance and Lorentz. There are also different categories in the literature summarized as permanent magnet, reluctance and induction,
as detailed below:
Permanent magnet self-bearing motors are based on permanent magnet motors,
and have several advantages: a) high efficiency b)high power factor, and c) small
25
size. Bichsel and Hugel introduced the first permanent magnet self-bearing motor
in 1990 [3]. They showed that with additional suspension windings the radial force
can be controlled so the rotor can be levitated and rotated with a single stator.
Since then many researchers in Switzerland, the United States, Japan and other
countries have developed different types of PM self-bearing motors. Okada's group
in Ibaraki university have done extensive work on permanent magnet self-bearing
motors [35] and [36].
magnet machines.
They presented the multiple winding method for permanent
They also developed a simpler method which uses DC flux to
control radial forces in self-bearing motors [36].
Reluctance motors have generated significant interest among researchers and industrial engineers recently. In particular, high temperature applications have been
under investigation for this type of motor. Reluctance motors are potentially low
cost because the rotor has neither permanent magnets nor windings. Some authors
proposed this type of motor for high speed application as it has a rigid rotor [5]. We
believe that the hysteresis type can have superior advantage for high speed application compared to reluctance motor. This is mainly because the hysteresis motor has
even more rigid and high strength rotor, since it can be constructed with solid steel
and no mechanical feature on the rotor.
Induction motors are the most widely used type of electric machine. These types of
electric machines which are low cost and robust, become competitive with permanent
magnet motors for larger sizes. This is mainly because of the scaling of volume and
surface area as function of rotor diameter. Since the ratio of rotor volume to surface
grows linearly with radius, both rotor and stator equivalent surface current densities
increase as the machine radius grows.
Induction self-bearing motors have also been under investigation for the last two
decades.
There are wide range of induction machine such as squirrel cage, rotor
wound and hybrid motors. Schoeb did his thesis at ETH on asynchronous machines
in 1993 [44].
Chiba's group in University of Tokyo have conducted extensive work
on induction machines that still is under investigation [8] and [23]. Peng and Tseng
developed a hybrid induction self-bearing motor for a blood pump [39]. Katou also
26
developed a squirrel cage self-bearing motor [32].
There have also been many efforts to develop control schemes for self-bearing
motors. Herman proposed the P + 2 scheme in which the stator has two different sets
of windings to generate P and Pi2
poles [22]. In 1988 Williamson proposed a different
pole combination for stator winding for static force [52]. One year later Chiba and
his group in university of Tokyo developed the self-bearing motor general concept [9].
They showed that most electric motors can operate as a self-bearing motor with
additional windings. We implemented their methodology for our hysteresis motors
and compared it with our design scheme. Okada's group also investigated the basic
pole combination self-bearing motor [31]. In this method, they developed a simpler
self-bearing motor that controls the radial force by means of a DC flux.
In this
work, they combined a hybrid type magnetic bearing with an AC motor. The hybrid
AMB has a bias permanent magnet between the two radial magnetic bearings. The
Lorentz force type of self-bearing motors have been also proposed [46].
This type
of self-bearing motor can use thick permanent magnets which leads to high torque
density motor and large levitation force.
1.3.2
Hysteresis Motors
Steinmetz apparently introduced the first hysteresis motor in 1917 as stated in [30].
This type of motor has a simple solid rotor of material with large magnetic hysteresis. They also typically have a polyphase stator winding similar to an induction
machine.
Like induction machines, a slotted stator having distributed windings is
used to produce as nearly as possible a sinusoidal magnetic flux distributed spatially.
The rotating magnetic field creates magnetic poles in the rotor by means of induction.
These poles follow the stator magnetic field. Because of hysteresis, the rotor poles lag
behind the inducing magneto motive force wave by the hysteretic lag angle. Before
the rotor reaches synchronous speed, a given region of the rotor experiences a cyclic
hysteresis loop at the slip frequency. Under this circumstance, the lag angle is only a
function of the rotor material, and it remains at its maximum value. When the rotor
reaches synchronous speed with an initial load, due to the rotor inertia interacting
27
with magnetic stiffness, the lag angle starts to oscillate about its equilibrium point
until the hysteresis torque is sufficient to carry the load. Typically the period of these
oscillations is about 1-5 seconds. This phenomena is called hunting.
There are a few papers published about hysteresis motors in the literature. Only
a few books on electric machines dedicate a couple pages to describing its basic
operating principles [17].
The first hysteresis motor modeling was apparently done by Teare in 1940 in his
PhD dissertation [47]. He derived analytical expressions for hysteresis starting torque.
He showed that the maximum net work available is proportional to the area enclosed
by the hysteresis loop of the rotor material.
Copeland and Slemon developed an
analytical model for a motor equivalent circuit based on harmonic response methodology [14], [15]. They proposed an equivalent circuit model with nonlinear hysteresis
elements.
Miyairi and Kataoka presented the equivalent circuit by linearizing the hysteresis loop into an ellipse [33], [34].
They used field theory to explain the torque-slip
behavior. Miyairi and Kataoka's papers inspired Nitao et al to derive an equivalent
circuit in stationary and rotating dq0 frames [30]. Nitao et al derived the governing
equations from first principles and presented a hypothetical simulation. They also
modeled the hunting phenomena in their analysis. They proposed a sets of equations
that model the hunting phenomenon.
Rahman studied the dynamic response of hysteresis motors to various small-scale
disturbances such as supply voltage and load variation [41].
hysteresis reluctance motor [42].
He also proposed the
Qin and Rahman proposed the hybrid permanent
magnet and hysteresis motor for electric vehicles [43]. In their hybrid design, they
added permanent magnets to the rotor of a hysteresis motor.
Okelly proposed an equivalent circuit model for single phase hysteresis induction
machines [38].
He derived a general equivalent circuit for a family of single phase
motors with magnetic and electrical symmetry in one or two windings. Badeeb studied dynamic performance and stability of hysteresis motors [2]. Although he refers to
Miayri and Kataoka's papers as his source; his equations are different. We were not
28
able to derive his equations nor confirm his results. Gavril and Mor investigated the
asynchronous behavior of hysteresis motors [19].
They considered the eddy current
effect by solving Maxwell's equations with elliptic approximation of hysteresis. However their analysis is not valid for the zero slip condition, i.e when the motor becomes
synchronous.
There are also a few papers and a Master's thesis on the hunting phenomenon.
Clurman used a second order model to simulate hunting [12]. Truong did his Master
thesis on modeling hunting by assuming the lag angle as a dynamic degree of freedom [49]. In his analysis the rotor apparent current considered to be constant which
is not quite true. We believe that the mechanical and electrical equations should be
coupled and thus must be solved simultaneously.
Our hysteresis motor analysis is mainly based on a report from Lawrence Livermore National Laboratory published in 2009 [30] and also two papers from Miyari [33]
and Ishikawa [26]. Our contribution to motor analysis is adding frequency dependency
of the hysteresis loop to the analysis and also conducting experimental verification.
Also to the best of our knowledge no one has looked into hysteresis self-bearing motor
before this dissertation.
1.4
Thesis Overview
In this dissertation we designed, built and tested three different types of self-bearing
motors. Chapter 5 presents these self-bearing motors.
We integrated Chua's nonlinear hysteresis model for force and moment analysis
of hysteresis self-bearing motors.
We also studied the hysteresis motor including
hysteresis frequency dependency, as presented in chapter 6. The simulation results
were verified experimentally.
Finally we studied a hysteresis and an induction squirrel cage motor of identical
sizes. We did this by replacing the rotor of a commercial induction machine with
a hysteresis material.
We ran IEEE standard tests to compare these two motor
performances.
29
1.4.1
Hysteresis Self-bearing Motor
In the first step, we implemented the multiple winding method successfully for a
hysteresis self-bearing machine. This method is well known for permanent magnet
machines; and our work is described in chapter 5. Figure 1-1 shows the first setup
that was built for this purpose. This axial flux self-bearing motor consists of a rotor
sandwiched between two stators on the top and bottom. There are four inductive
sensors to measure axial position as well rotor tip-tilt angles. These degrees of freedom
are actively controlled as the open loop modes are not stable.
Figure 1-1: Hysteresis Self-bearing motor. The rotor ring at the center is levitated
and rotated by a hysteresis motor structure.
We believe that axial flux type motors are more suitable for many high speed
30
applications, as the rotor expansion due to high centrifugal force takes place in a
non-sensitive direction and thus doesn't affect the motor gap significantly. For this
reason we first focused on axial flux type hysteresis self-bearing motors. It should be
noted that in this configuration the rotor is passively stable in planar plane. However
for the sake of motor comparison we also built two identical radial flux motors with
hysteresis material and squirrel cage rotor.
1.4.2
Yokeless Segmented Stator
Figure 1-2 shows the novel segmented stator that we designed, built and tested for
driving one of our self-bearing motors. Figure 1-3 shows the magnetic path for such
a motor. In order to improve the motor efficiency we segmented the rotor as well.
In practice this is an advantage for high speed machines where we can over wrap
the hysteresis material with high strength composites such as carbon fibre. Carbon
fibre has larger mechanical strength relative to steels; thus higher speed is attainable.
There are several advantages for such a stator design including: high filling factor,
easier coil winding, manufacturing and assembling. Obviously this part needs to be
investigated in more details and is left for future works as explained in chapter 8.
It should be noted that since we had limited manufacturing resources we used U
core actuators for our segmented stator for the proof of concept. However the stator
pole face can have a trapezoid shape as shown in figure 1-4.
In this design we have wired each core separately to a connector for more flexibility.
Since we have access to each core, the segmented stator can be easily configured for
different number of pole motor. A novel control scheme is implemented for this type
of motor successfully in this thesis.
Figure 1-5 shows the more conventional axial flux stator that we built for the lower
side of the self-bearing motor. The stator is built out of soft magnetic composite
material (SMC) which used sintered powder metal to allow for 3D flux path. Coil
winding for such a stator is significantly more difficult than for the segmented stator
especially for mass production, because the coil has to be wrapped around the yoke.
31
Figure 1-2: Segmented stator consisting of 12 laminated U cores. This machine uses
a rotor with integral air gap, as visible in the picture. The gap is filled with epoxy.
32
D2 steel
Epoxy
Figure 1-3: Segmented stator magnetic path. The epoxy gap separates the two rotor
flux paths.
Figure 1-4: Proposed trapezoid pole face for segmented stator design, allowing more
efficient magnetic flux paths.
33
Figure 1-5: Hysteresis self-bearing lower stator. The stator has 36 open slots, and
the coils are wrapped around the stator behind each slot.
34
1.4.3
Single Axis Self-bearing Motor
One the main drawbacks of self-bearing motors and active magnetic bearings is their
complex controller and relative large number of sensing and actuating elements. There
are valuable benefits for practical application if one can reduce the number of axes
that have to be actively controlled. For this purpose we designed, built, and tested a
novel single axis controlled self-bearing machine as shown in Figure 1-6.
In this design we control the axial axis actively and all other axes are passively
stable by means of attractive forces applied with a pair of permanent magnets on the
stator and rotor at each end. Figure 1-7 shows the rotor axial run-out (only actively
controlled direction) at 600 rpm. The peak to peak run-out is about 50pm. Figure
1-8 shows the rotor axial position during startup as the rotor is levitated.
35
Figure 1-6: Single axis hysteresis self-bearing system.
36
30
20
10
0
C
0
-10-20
-30
0
0.1
0.2
0.3
Time[sec]
0.4
0.5
Figure 1-7: Rotor axial run-out at 600 rpm.
E
0
0
CL
0
0.05
0.1
0.15
Time[sec]
0.2
0.25
Figure 1-8: Single axis bearing rotor axial position during startup.
37
1.4.4
Nonlinear Hysteresis Analysis
After reviewing the literature we chose Chua's nonlinear hysteresis model [11] to
simulate the self-bearing motor stabilizing forces and moments. We used the Chua
hysteresis model because it is a mathematical model that can simulate the frequency
dependency of hysteresis as well as minor loops. The challenge with this model is
correctly selecting the various functions to match experimental responses.
We developed a SIMULINK model for hysteresis self-bearing motor force and
moment analysis. Figure 1-9 shows the step response of the large motor setup along
with linear and nonlinear simulation results.
1.8
1.6-
-
1.4 -
-
-
Experiment
Linear
Nonlinear
.20.8-D
i5
0.4
0.2
0
-0.2
0
'
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Time (sec)
Figure 1-9: Large setup step response compared with linear and nonlinear simulation
results.
Figure 1-10 and 1-11 shows further simulation results for the large motor setup
shown in figure 1-1.
38
2.5
2
1.5
LL
1
0.5
0 0.1
0.2
0.3
0.5
0.4
0.6
0.7
0.8
Current[A]
Figure 1-10: Simulation results for force versus current for large setup.
2.5
2
1.5
a)
C)
0
LL
1
0.5
0
5.5
6
Position[m]
6.5
x 10~4
Figure 1-11: Simulation results for force versus displacement for large setup.
39
1.4.5
Hysteresis Motor Modeling Including Frequency Dependency
Our hysteresis motor analysis is mainly based on a report from Lawrence Livermore
National Laboratory published in 2009 [30] and also two papers from Miyari [33]
and Ishikawa [26]. However these references don't support their analysis with experimental results. Our contribution to this part is adding frequency dependency of the
hysteresis loop to the analysis and also verifying the modeling results experimentally.
We developed a SIMULINK model as well MATALB code for both axial and radial
flux motors. We tested two different axial flux motor sizes and pole configurations
to verify model accuracy. We also compared the simulation and experimental results
for a radial flux motor. As we can see in figure 1-12 and 1-13 the model provides
acceptable results for both tested motors. As we can see the model can't predict the
motor behavior at the beginning where the rotor experiences large acceleration. We
expect that this is mainly due to the underlying model assumption of steady-state
or quasi-steady-state operation. Thus as the rotor approaches steady-state speed the
model has better accuracy.
2500
Experimental
Model
-------
2000
15000
a)
0
5
10
15
2'0
Time(sec)
25
30
35
40
Figure 1-12: Angular velocity versus time for small motor.
40
1400
1200
E
IVIUUe
1000
S800--2
600 -
2200$
00
0
U
-
5
10
15
20
25
Time(sec)
Figure 1-13: Angular velocity versus time for large motor.
Another distinct feature of our simulation is modeling the hunting phenomena
as steady-state speed is reached. In other references, hunting is modeled and solved
separately, but our simulation model includes it automatically.
The frequency dependency of hysteresis has been included in the simulation based
upon experimental measurements of the rotor material. For this purpose we measured
B-H curves of toroidal samples of rotor material at different frequencies. We built
lookup tables based on these experimental data and incorporated them into our analysis. As can be seen in figure 1-14 the hysteresis torque is not constant at all speed.
The hysteresis torque variation is due to the fact that lag angle and permeability
varies with slip frequency. Figure 1-15 shows the hysteresis torque including eddy
current which is what really takes place.
41
x 10,
3
2.5
2:
2
0)
=3
0r
1.5 -
1
0
500
1000
1500
Angular velocity(rpm)
2000
2500
Figure 1-14: Simulation results for hysteresis motor torque versus speed for small
motor neglecting eddy current.
5
x 10-3
4
E
3-
2
1L
0
500
1500
1000
Angular velocity(rpm)
2000
2500
Figure 1-15: Simulation results for hysteresis motor torque versus speed for small
motor including eddy current.
42
1.5
Thesis Outline
This thesis is organized as follows:
Chapter 1: Presents a brief overview of the self-bearing motor types and active
magnetic bearings. The thesis objectives, motivations, thesis outline and prior
work are also presented.
Chapter 2: Contains the fundamentals of hysteresis and electric machine modeling. In this chapter we first present a brief introduction to the hysteresis phenomenon. Then we develop the linear and nonlinear hysteresis modeling as a
foundation for hysteresis self-bearing motor analysis. In the second part of this
chapter we discuss induction machine governing equations and modeling,which
are relevant to the inclusion of induction effects in the hysteresis motor model-
ing.
Chapter 3: Presents the details of the different hardware and materials that we used
in this thesis. It describes the two different setups for the axial flux motors and
radial motors as well as our power amplifier design and configuration. It also
explains the measurement procedure of magnetic properties for key materials
that were used.
Chapter 4: Presents active magnetic bearing design and analysis. In this chapter
we discuss the detailed design of an AMB that was built for two different setups.
Chapter 5: Addresses three different types of self-bearing motors that were built
and tested in this dissertation. In this chapter we focus on the control scheme
and stabilization as well as suspension forces and moments. The motoring and
torque production are discussed in chapter 6.
Three different types of self-
bearing motors that were designed and built along with linear and nonlinear
analyses are explained.
Chapter 6: Discusses the hysteresis motor analysis and experimental verification. In
this chapter we present the motor theoretical modeling and governing equations.
43
The simulation results are then compared with experimental data that were
collected from two setups.
Chapter 7: Hysteresis and induction machine comparison is presented in this chapter. We studied two machines with an identical commercial motor stator. One
machine has a standard squirrel cage rotor. In the second machine the rotor is
replaced with solid D2 steel. The IEEE standard tests are performed for both
the hysteresis and the induction squirrel cage motors. Thus all motor parameters except rotor material are identical. The radial hysteresis motor analysis is
also discussed as both motors are radial flux type.
Chapter 8: Presents the conclusions and recommendations for future work and related ideas.
44
Chapter 2
Principles of Hysteresis and
Synchronous Motors
In this chapter we present the principles of hysteresis phenomena as well as electric machine modeling as a foundation for this thesis. The first section contains an
introduction to general aspects of hysteresis, hysteresis loop properties, magnetization curves, frequency dependency and modeling. In the second section, we present
synchronous electric machine fundamentals and the general governing equations.
2.1
Hysteresis
Hysteresis phenomena exist in many different areas such as mechanics, optics and
magnetism.
hysteresis.
In general, all electric machines are affected by particular aspects of
Typically, hysteresis causes power loss in electric machines and other
devices that use a varying magnetic field. However, loss is only one aspect of this
complex phenomenon. Webster defines hysteresis as " a retardation of an effect when
the forces acting upon a body are changed".
Physically speaking, we can say that
when hysteresis is present an output is not a single-valued function of an input. In
other words, a function H(u) can be called hysteretic if it has a multi-valued output
for a single input.
It should also be noted that hysteresis phenomenon is not simply a loop. When the
45
Y(t)
X(t)
H(u)
-
Y(t)
')X(t)
Figure 2-1: Hysteresis definition.
input varies cyclically such as in a sinusoidal wave form, then the looping takes place.
The width of the loop also depends on the input frequency which is known as loop
widening. It is very important to understand the physics of hysteresis loops before
moving to any further analysis. In the next section we discuss hysteresis behavior in
more detail.
2.1.1
Hysteresis Loop
As described in [20] at the quantum mechanic level, any magnetic material can be
considered as an assembly of magnetic moments mi. For example a pure iron atom
can carry a magnetic moment of
4
2 2
. pb where the Bohr constant in SI units is yUb
=
1
9.27 x 10- J- T. A paramagnetic material forms the most simple situation where
each magnetic moment mi are independently shaken by thermal agitation and do
not interact with each other [20]. Therefore, these magnetic moments take random
orientations spatially and consequently the total net magnetization is zero. However
if a magnetic material is excited by an external magnetic field He, then a non-zero
magnetization is produced. In this situation there are two type of energy that interact
with each other in opposite directions. The magnetic potential energy (-pomiHe)
tries to align the magnetic moment with the external field, whereas the thermal
agitation has an opposite tendency. If the potential energy is greater than the thermal
energy then we have a net magnetization in the material [20].
46
In ferromagnetic
materials the magnetic moments are strongly coupled with an internal field called
the molecular field. Indeed the molecular field acts as a positive feedback system and
tries to align all magnetic moments.
In the early 1900s, Weiss presented the spontaneous magnetization term for ferromagnetic material.
He suggested that this type of magnetic materials can have
a very large spontaneous magnetization because their magnetic moments are completely independent [20]. For pure iron at room temperature this magnetization is
about 2 tesla. Having said that, when we apply an external magnetic field to a magnetic material, at high field the entire magnetized material is aligned to the external
magnetic field. At this point the average of magnetization is close to the spontaneous
magnetization. When the external field is reversed progressively, the magnetization
path is somewhat different from the original one and looks as shown in figure 2-2. In
this figure the horizontal and vertical axes represents the magnetic field intensity and
magnetic flux density respectively.
B*
B,
Hc
H
Figure 2-2: Hysteresis loop schematic.
There are three important quantities associated with hysteresis loops; remanence
B,
coercivity He and hysteresis loss. The remanence is the magnetization remaining
in the material after removing the material from a saturating applied magnetic field.
It is a natural property of ferromagnetic material that shows it can be spontaneously
magnetized even in the absence of external field.
47
The coercivity is the amount of magnetic field needed to eliminate the remanence.
Unlike the remanence, the coercivity can take a wide range from 1Am'
to 106 Am-'.
There are a significant variety of chemical, structural and metallurgical factors that
affect the coercivity. For instance, for a specific material such as D2 steel, the coercivity is somewhat proportional to the hardness. Hysteresis loss is the amount of
energy dissipated as heat in each loop cycle. This loss is proportional to the area
enclosed by the hysteresis loop.
2.1.2
Magnetic Materials
Magnetic materials can be subdivided into two different categories based on their
remanence and coercivity values.
Soft: This type of magnetic material has very low coercivity and is easy to magnetize. The coercive field is about 1-100 Am- 1 . For example, the grain oriented Si-Fe
alloys used in transformers have a coercivity of 10 Am-
1
while the the non-oriented
one used in motors take this value up to 100 Am'. Nickel-based alloy can provide
the most soft magnetic material with 7 Am-
1
coercive field. In general this type
material has low power loss due to the fact that their hysteresis loops are small.
Hard: The coercivity of this type of material is in the order of 50-1000 KAm
1
and it is being used in those application where a permanent source of magnetic field is
required. The rare-earth magnets such as Nd-Fe-B have the largest coercivity among
permanent magnets with a value of about 1100 KAm
.
One of the most important properties of hard magnetic materials is the maximum
energy product (B x
H)ax
which is the indication of available energy for external
work. In general, the relation between magnetization M and induction B is given
by B = po(H + M). Here M is the average magnetic moment per unit volume and
has a unit of Am-'. We also define the magnetic polarization I = poM which has
the same unit of magnetic induction measured in tesla. In soft magnetic materials
the field associated with hysteresis is much smaller than magnetization therefore we
can ignore the H and with a very good approximation the magnetic induction is
just simply B '
poM(H). However in hard magnetic materials both fields are on
48
the same order of magnitude and the B-H field is different from the M-H loop. As
it is shown in figure 2-3 there are two different He depending if we are focused on
magnetization or induction. The magnetization loop explains the intrinsic behavior
of the magnet while the induction loop provides the system properties under the
working condition [20].
M,B
B(
BR
M(H)
IfI
H
Figure 2-3: Hysteresis for soft and hard materials. B-H is in solid line, M-H in dashed
line.
The maximum energy product or (B x H)ma, is related to the B-H loop. The
theoretical maximum value for this energy is given by "'MfA where M, is the spontaneous magnetization. The poM, is about 2T for pure iron which gives the 800
3
maximum energy [4].
2.1.3
Loop Widening
Hysteresis is a time rate dependent phenomena which is fundamentally related to
eddy current formation in material and thermal agitation as described in [20]. As
we will see in chapter 3, the loop area increases with increasing the magnetization
frequency. The loop area is proportional to the amount of energy that transforms into
heat in each cycle. Thus as we increase the frequency, the eddy currents induced in
the specimen increase accordingly which in turn causes more dissipation and therefore
a larger loop area.
49
To identify a dynamic hysteresis loop, it is not sufficient to specify the fixed
magnetic field H corresponding to constant current. The magnetization frequency
must also be known in order to have clear picture of hysteresis. We have measured
the hysteresis loop for different frequencies for the two desired material in chapter 3 for
a constant current in figures 3-26 and 3-28. We use these results for hysteresis motor
modeling in chapter 6. We have also used Chua's hysteresis modeling to incorporate
the loop widening in the self-bearing motor analysis in chapter 5.
2.1.4
Hysteresis Linearization
In some literature, hysteresis is defined as when the output is lagging the input. If
the input is in sinusoidal form as it is in typical motors in sinusoidal steady state,
then we can write:
H (t) = Ho cos(wt)
(2.1)
B(t) = Bo cos(wt - 5).
Equation 2.1 provides elliptical loops in B as a function of H under time varying
excitation as in figure 2-4. We use this linearization in chapter 6 to solve the motor
governing equations.
2.1.5
Nonlinear Hysteresis Modeling
We searched the literature for the existing magnetic hysteresis modeling. Much work
has been done in this field including by Preisach [401, Jiles [29], Biorci and Pescetti [4].
To the best our knowledge, most of these approach doesn't include loop widening.
Among the reviewed models, Chua's modeling [10) appears to be the only dynamic
model that allows extensive control over the frequency dependency of the hysteresis
loop. In addition, it can model the DC input i.e. w = 0 as would occur with a step
input. The Chua hysteresis model is given by [11]
d
di
-=tW
dt
dt
)
x h (y(t)) x g [x(t) - f(y(t))].
50
(2.2)
H
B
Figure 2-4: Linear model of hysteresis.
In this equation x(t) is the input and y(t) is the output. The model is specified
by two monotonically increasing functions:
f
the restoringfunction which describes
the energy storage, and g, the dissipation function which describes the energy dissipation. The W and h functions provide the loop widening and weighting functions
respectively. These four functions should be tuned such that the desired hysteresis
loop is attained. Chua and Bass have illustrated the hysteresis loop for two different
types of material in [10]. In this paper they also present several possible functions for
loop widening shown in Figure 2-5.
Figure 2-6 shows a SIMULINK block diagram for Chua hysteresis modeling. This
block diagram is created based upon equation 2.2. It should be noted that all functions are similar to those Chua and Bass presented in their paper with some tuning
and adjustment to achieve a fit for D2 steel. In this block diagram we used a MATLAB/SIMULINK lookup table to generate those functions, however as described
below and in chapter 5, we defined logarithmic and trigonometric functions to fit the
hysteresis model to our experimental data.
As an example, functions
f and g were
replaced by the logarithmic functions given
below to simulate the D2 hysteresis loop shown in figure 2-7. We use this model in
chapter 5 to simulate the hysteresis self-bearing motor.
51
W (Function)
Behavior Description
dx
W(W )
Loop widens as frequency increases. This is a
purely nonlinear lag or delay.
dx
dt
dx
W(W )A
dx
This model eliminates the effect of frequency
variation on loop width. The loop remains
constant under any frequency.
dt
dx j
dt
In this model loop widening happens at certain
frequency. At low frequency the loop is not
function of frequency.
dx
dt
Up to certain frequency the loop remains constant
but at the threshold the loop widening occurs with
specific slope.
dx
dt
dxA
Three different region can be defined for loop
widening. At low frequency the loop is not
sensitive to frequency Loop widening becomes
more sensitive progressively.
Figure 2-5: Possible W functions and their effect on frequency behavior, adapted
from [11].
52
Figure 2-6: Chua model Simulink block diagram, which implements 2.2.
0.8
0.6 0.4 0.2 -
rn
0
-0.2 -0.4 -0.6 -
-0.8
L
-2000
-1500
-1000
-500
0
H [A/m]
500
1000
1500
2000
Figure 2-7: Chua model response for hysteresis of D2 steel.
53
~
1
=sign(y) n
f()
(2.3)
g(u)
=
-50ln
I -
1 1500.
ul
According to figure 2-5 if we replace the W function with an absolute function
then the model is not frequency dependent. However if we use a linear function then
the model has frequency dependency. We also chose unity gain for function h. Figure
2-8 shows the frequency dependency of Chua's model with our chosen frequencies.
The inner loop is at 1Hz, the middle loop is at 5Hz and the outer loop is at 20Hz.
E
03
0
1500
H[ANm]
Figure 2-8: Loop widening in Chua model at 1, 5, and 20 Hz.
Figure 2-9 shows the minor loops that Chua model can produce. These loops
are created by varying the sinusoidal, excitation amplitude in H, at on excitation
frequency of 3Hz.
54
1
0-5
-0-5-
-1
4
-2
0
H[A/m]
2
4
x 10 3
Figure 2-9: Minor loops in Chua model with excitation frequency of 3 Hz.
55
2.2
Synchronous Motor Fundamentals
Considering the fact that hysteresis motors operate sometimes as a synchronous machine, it is appropriate to examine this type of motor first. In this section, we first
describe the torque expression for a synchronous machine. Following this we introduce
some important characteristics of general induction machines that we will use later in
this thesis. This section is mainly based on the book "Electric Machinery" [17] and
Prof. James Kirtley's notes [28] used in course 6.685 at MIT.
2.2.1
Torque Expression
This section is adapted from [28] chapter 4.
Figure 2-10(a) shows the hysteresis
motor as a synchronous machine model. Here 6 is the angular difference between the
rotating magnetic field and the rotor magnetized pole.
In this configuration, the stator has three separate identical windings with spatial
distribution by electrical angle of 1. These three windings have the same self inductance (La = Lb = Lc).
The mechanical angle between any pair of windings is the
same and characterized by cos
(2)
=
thus
-j,
1
La.
2
Lab = Lac = Lbc =
There are also mutual inductances between the rotor and stator windings.
Mar = M cos (PO)
Mb, = M cos (PO
27r
+ -
Mc, = M cos (P0 -
27r
-
)
where M is the magnitude of mutual inductance and p is number of pairs of poles.
Figure 2-10(b) shows a simple model of a synchronous motor that consists of a
polyphase stator with three identical coils spaced uniformly 1200 apart from each
other and an equivalent single rotor winding called a field winding.
56
ia(
a
*C-
*Ar
b'C
ib/
AC
'a'
(a) Flux linkages
A
Air gap
A'
(b) Polarized rotor
Figure 2-10: Synchronous motor schematic.
If we assume that the flux linkages are sinusoidal functions of rotor position then
the flux linkages associated with these coils are as follows:
57
Aa = Laia + Labib + Lacic + M cos(pO)i,
Ab = Lbaia + Lbib + Lbcic
+ M COS
27r
i
(0--
(2.4)
277r7
Ar = M COS(PO)ia + M C
0-
ib ±
M CO
p0 +
ic + Lrir
Here, we use an energy approach to find torque. The electric motor stores energy
in magnetic fields, the energy stored depends on the state of the system defined by
three variables: flux linkage (A), current i, and rotor angle. The magnetic field energy
in this system is given by:
dWm
=
iadAa + ibdAb + icdAe - TedO
(2.5)
or
d( aAa + ibAb + icAe - Wm)
Aadia + Abdib + Acdic + TedO
=
(2.6)
Typically, we prefer to describe systems in terms of inductance as opposed to its
reciprocal. We also prefer to work with current instead of flux. Therefore, it is more
convenient to define co-energy for motor as follows:
W' =
Ai - Wm
(2.7)
Thus we can write:
dWm' = Aadia + Abdib + Aedic + TedO
Then for specified currents, torque is given by:
58
(2.8)
Te
(2.9)
"'
dO
=
Now we need to find co-energy in this system.
Since polyphase motors have
multiple excitation, it is important to choose the right integration path.
system, there are five dimensions:
ia, ib, i,
In our
4, and 0; however since we can position
the rotor with all currents set to zero, there is no contribution to co-energy from the
rotor position. Suppose the rotor is at angle 0 as it is shown in figure 2-10(b) with
four currents ia0, b0, ico and iro. Now we can find co-energy as follows:
W'
I
=
'aO
Laiadia
0
+
+
( Labia
J
SJ
/ir0
|-Laib)dib
(Labiao + Labibo + Laic)dic
COS
(PO)iao + M cos
P0-
27r
2-7r
+ -
ib0 + M Cos (
ico + Lrir ddIr.
(2.10)
Taking the integrations results in:
W
=±
aa
+io +
0)
+
Lab(iaibo - iaOicO + iboicO)
Z Z
Miro iao cos(pO) + ibo cos
+
0-27r
--
-p
+ c O
+ ico cos (P
+27r
+
-
+1
+
Li
L.
(2.11)
Considering the fact that inductances don't vary with rotor angle then the torque
is given by:
Te =
m = -pMiro iaO sin (pO) + ib sin
59
p0-
2
+ ico sin p0+
7
(2.12)
2.2.2
Balanced Operation
Suppose the motor is running with a three phase balanced current source of amplitude
I as:
iao = I cos(wt)
27r
i40 = I coS
Wt
Wt
ico = I coS
3
27r
+
3
ZrO = I,.
For a synchronous machine we have the rotor synchronized to the traveling stator
flux:
p0 = wt
+ J.
(2.13)
Further, we have the identity:
cos x sin y
=
1sin(x - y) +
1
2
2
sin(x
+ y)
(2.14)
By using the above relation we get:
Te
=
-PMII [ sin( ) +
+
sin(6) +
+
sin(6) +
1
2
1
-sin
sin
(2wt
2
sin(2wt +J)
+
2wt +
47r
3
1~w±
1
47
+ 3r
(2.15)
Simplifying the above equations yields:
Te =
3
PMII, sin(j)
2
(2.16)
Equation 2.16 shows that synchronous machine can produce steady torque under
applied sinusoidal excitation and constant rotor current and lag angle.
60
2.2.3
Rotating Magnetomotive Force
To understand how the polyphase machines including hysteresis motor work, it is
important to learn how the magnetomotive force is being produced.
We use the
results of this section in chapter 6 to derive the hysteresis motor governing equations.
Finding the air gap magnetomotive force produced by a full-pitched concentrated
single coil is discussed in many electric machines books. This section is adapted from
[17] section 4.3. In this section we avoid presenting the details and simply use the
necessary equations.
N tun coil
0
Fundamental FM
NI
7
0
NI
-2
1 Rotor
I"T T ""
Stator
Figure 2-11: MMF produced by a concentrated full-pitch winding.
Figure 2-11 shows the magnetic flux produced with a single concentrated N turn
coil that carries current I in a rectangular slot. This is a full-pitch coil as there is a
180 electrical angle between the two coils legs. We can find the fundamental MMF
61
by applying Fourier series analysis to the rectangular magnetomotive force shown in
figure 2-11. The fundamental component is then given by:
FM-
4 /Ni\
-i cos(O).
2
7r
(2.17)
Figure 2-12 shows the individual components and the resultant magnetomotive
force generated by a three phase motor. Since magnetic axes of individual coils are
not aligned with the resultant therefore the sum of individual coils is less than the
resultant. Therefore, the magnetomotive force for a distributed coil is modified:
4 (Kw.
FM = - (
where K,
(2.18)
i cos(O),
is the winding factor and for most AC machines falls between 0.8 to
0.9 due to finite coil density.
For a single phase system the current is:
i = Icos(Wt)
Therefore we can write:
KwN) I Cos(0) Cos(Wt)
FM =
(2.19)
Using a trigonometric identity:
4 (KN)
[1
P
p
2
FM = -I
7r
Cos(O - Wt) +
1
-
w)
cos(O + Wt)
.
(2.20)
2
We can decompose the MMF for a single phase system described above into two
components.
(2.21)
FM = FM + Fwhere
=
4
F= -r
KwN
I
62
1 Cos(9+Wt) '
-2co(
+
B
A'
A
a7
a,
al
b4
b3
C
c12
Figure 2-12: MMF produced by a concentrated full-pitch winding.
and
=
KwN
- cos(0 - wt)
The two components given above represent the positive and negative traveling
waves respectively. In the single phase case these two components are equal thus
there is no net torque. However for a balanced three phase system where the currents
are 120 degree apart we can write:
63
FM
=
(FM)A + (FM)B + (FM)c
(2.22)
It is easy to show that the summation of negative components is zero and the
summation of positive traveling waves gives:
FM = 3 ( KwN)I cos(9 - wt)
2 (7r
p)
(2.23)
We can define W as conductor distribution as
W= 2KwN
p1r
Thus the MMF is simply:
3
FM = -IW cos(9 - Wt)
2
2.2.4
[A.turn1
etura
[elect.rad_
(2.24)
Stator Leakage
The stator leakage represents those fluxes that are not linked by the rotor. There
are a number of of components of stator leakage. The most prominent component of
stator leakage is slot leakage which we discuss below. Other stator leakages such as
belt, zigzag, end winding and skew have minor contributions and are also complex
to analyze. Typically, numerical methods are required for calculating these leakages.
Stator leakage can be also found experimentally by using no-load and locked-rotor
tests which are described in Chapter 7.
Slot leakage is related to those flux in the stator slots that links with the conductors
as opposed to the rotor. Figure 2-13 shows the stator slot geometry. We first calculate
the magnetic energy stored in the slot by assuming the total current in each slot is
represented by I.
1
1
2
Let the number of conductor in each slot be N,, then the current density in each slot
64
ht
y
x
Figure 2-13: Stator slot geometry.
is given by:
NJ
hw
By applying Ampere's law about the dotted loop shown in the figure we can get:
H-NjI
y
w h
then the total magnetic energy stored in the slot can be calculated:
Wm
122
2=
1
2
1h
_pfo2
-oH,2dy = 1po
3wNI
k3 wJ
2
The permeance is:
0 = lpo1h
(3w
and the total inductance is:
Li
=
i
If there are n slots per pole per phase and N,
represents the number of slots per
pole then for a three phase machine the self inductance is:
Ls, = o [4Ns2p(n - Nsp) + Ns2pNs,]
65
The total number of slots per phase is 2pNp,, thus the mutual inductance is:
Lm, = -gpNspNs
The total slot leakage is then:
Li = e4pn 2 Ni
(2
n
5 N"s
4 n 2)
(2.25)
The stator leakage inductance can be approximated by the above equation.
2.2.5
Stator Winding Resistance
It is straightforward to calculate the stator winding resistance for a known geometry.
The total length of winding per phase is given by
1W= 2N 8 ,
where i. is the average length of wire around the slot. The cross section area of
N turn wire with diameter d, is:
A=
7r2
-d N
4w
Winding resistance per phase is:
R= 1*
o-A,
66
Chapter 3
Hardware
This chapter gives an overview of the hardware that was used in this project. As we
mentioned earlier, one of the major goals of this project is to derive scaling laws for
hysteresis self-bearing motors. For this purpose, we built two different sizes of these
motors for scaling laws verification. However, we were also looking for testing different
types of stator and motor configurations. The first setup that hereafter we call the
small setup has an 80 mm rotor with an active magnetic bearing on the top and a
regular axial stator on the lower side. The second setup (large setup) that we built
has an 120mm rotor with a novel segmented stator on the top and regular open slot
stator on the lower side. We also built a novel single axis self-bearing motor that will
be discussed in Chapter 5. Finally, in order to compare the hysteresis motors with a
regular induction machine, we replaced the rotor of a commercial squirrel cage motor
with a D2 steel rotor and examined these using the IEEE standard test for induction
machines. These three setups as well as linear power amplifiers and a hysteresis loop
measurement system are described in the following sections.
3.1
Small setup
The small setup consists of a magnetic bearing and a hysteresis motor. Figure 3-1
shows a cross section exploded view of this setup. For clarity, we have not shown the
center post that holds the AMB on the top.
67
Displacement
Sensor
Rotor
U core
Stator
Figure 3-1: Small setup has 80 mm diameter rotor levitated by upper magnetic
bearing stator and rotated by lower stator. The upper stator has U core actuators to
operate with the rotor as an active magnetic bearing (AMB).
3.1.1
Active Magnetic Bearing
The AMB has four eddy current displacement sensor that measures the gap and the
inclination (two tip-tilt angles).
Displacement eddy current sensors are also called
"inductive" sensors in industrial practice. Generally, eddy current sensors refers to
either precision displacement instruments or nondestructive probes and inductive sensors refers to inexpensive proximity sensors. In both cases a high frequency magnetic
field is generated through the sensor head by running an alternating current in the
sensor coil.
Inductive sensors are typically operated at 5 to 100 KHz while eddy
current sensors operating frequency lies in a range of 1 to 2 MHz.
Inductive displacement sensors measure the distance between the sensor head and
a metal target, based on high frequency current status. When a conductive target is
placed in front of sensor head, electromagnetic induction generates an eddy current on
68
the surface of the target. This eddy current changes the inductance of the sensor head
coil. As the gap between the sensor head and the target decreases, the amplitude
of the alternating current becomes smaller, whereas the phase difference from the
reference current becomes larger. An electronic circuit called a "linearizer circuit"
converts these changes into an analog voltage typically in range of 0-10 V or current
in range of 4 to 20mA. The linear output is proportional to the distance between the
sensor head and target.
We used a Contrinex DW-AD-509-M8-390 with specifications given in table 3.1.
Table 3.1: Displacement Sensor Spec.
Sensing Range
0 ~ 4mm
Repeat Accuracy
0.01mm
Bandwidth
1600Hz
Temperature Drift
Sensor Diam.
< %5at700 F
8mm
Figure 3.1 shows the sensor responses for aluminum and steel given by manufacturer
[13]. We could verify the same data for Maraging steel. For this purpose we
used a micrometer to change the gap between the rotor and the sensor head while
measuring the output voltage. As we can see, the sensor output is close to linear for
2mm gap. For this reason we adjusted the sensor such that the gap sensing was about
2mm so it can operate within its near linear range.
This setup also has four U core actuators that act like a conventional magnetic
bearing as shown in figure 3-3. As we will discuss later, the large setup has a novel
design for the magnetic bearing. The U cores are made out of grain oriented silicon
iron laminated steel. Each core is wound with 350 turn of AWG 24 copper wire. We
will discuss the electromagnetic specification of these actuators in the next chapter.
3.1.2
Motor
The motor consists of a stator and casing shown in figure 3-4(a) interacting with the
rotor that is shared with the AMB. The stator material is soft composite material from
Hoganas called Somalloy [24]. The material specification is given in section 3.5.2. It
69
Alurninium/aluminium/aluminum
UA M/l
5
Stahl
acier
steel
4
3
2
0
0
0,5
1
1,5
2
2,5
3
3,5
4
S [mm]
Figure 3-2: Sensor response data for DW-AD-509-M8-390 sensor by Contrinex [13].
is worth mentioning that the Somaloy material is very brittle and thus machining requires extra care. In section 3.5.2 we present the tooling and machining requirements
for Somaloy. Such composite magnetic materials are made of electrically insulated
fine magnetic particles and thus are suitable for relatively high frequency applications
or where 3D flux paths are needed. In an axial-flux air gap motor this is a significant
advantage; however composite materials suffer from relatively low permeability and
saturation flux density as compared to laminated silicon steels.
We built two different stators: semi-open and open-slots shown in figures 3-4(c)
and 3-4(d). The open-slot stator has 24 slots filled with 40 turns AWG 26 enameled
copper wire for the two pole motor and 10 turns for the four pole motor. The semiopen stator has 12 slots with 50 turns AWG 24 enameled copper wire for the two
pole motor and 15 turns for the four pole motor. The rotor is just a simple ring made
out of either Maraging steel grade 350 or D2 steel. We tested both rotor materials.
Figure 3-5 shows the small setup.
70
Figure 3-3: Active magnetic bearing used as upper stator for small setup.
We also modified the small setup to show the proof of concept for a novel single
axis self-bearing motor discussed in Chapter 5.
71
(b) Casing.
(a) Stator in its casing.
(d) Semi-open slot.
(c) Open slot.
Figure 3-4: Lower stators for small setup. Both open-slot and semi-open slot configurations were built.
72
Figure 3-5: Small setup in operation. Upper rotor air-gap is visible.
73
3.2
Large setup
The large setup has two different stators for the top and bottom. Figure 3-6 shows
the large setup that we built. As shown in this figure, in this design we could also
add a radial self-bearing motor with an outer runner stator to control lateral motion
as well. However we found out that the reluctance force produced by the two axial
flux stators is enough to stabilize the rotor passively in the lateral axes. However
the passive radial stiffness is not high, thus some applications may require the active
radial bearing. For clarity we haven't shown the sensors nor the center post that
holds the upper motor on the top.
Rotor
Outer runner
stator
Stator
Figure 3-6: Large setup schematic view. Rotor is shown as transparent.
Unlike the small setup, instead of an upper conventional AMB, the large setup
has a novel segmented stator for the top side as shown in figure 3-7. The segmented
stator has 12 U core actuators wound with 250 turns of AWG22 enameled copper
74
wire.
Figure 3-7: Segmented stator for large setup upper side.
The lower stator for the large setup has the same configuration as the small one
except it has 36 slots filled by 40 turns AWG24 enameled copper wire. It is also
constructed using Somaloy magnetic material.
75
3.3
Radial motor
Squirrel cage induction machines are widely used for driving loads such as fans,
compressors and pumps, in many devices such as manufacturing machines, appliances, HVAC systems and so forth.
These types of motors have a simple struc-
ture and are very cheap compared to permanent magnet motors.
They become
very efficient and competitive with permanent magnet motors for large size and
power. However, there are many applications that even use small size induction machines. We decided to compare the performance of hysteresis and induction machines
at relatively small scale. However it should be noted that the hysteresis machine
is synchronous/asynchronous, while the induction machine can only operate asynchronously.
Figure 3-8: Radial rotor for squirrel cage and hysteresis motor. Conventional rotor
on top was replaced with solid D2 steel rotor.
For this purpose, we purchased two identical induction motors and replaced the
rotor of one with a simple D2 steel rotor as shown in figure 3-8. All other parameters
including the stator, gap, and drive are the same. Figure 3-9 shows the radial stator
in its original casing.
76
Figure 3-9: Radial stator of commercial induction machine.
Figure 3-10: Radial motor of commercial induction machine made by Oriental Motor
model 51K90A-SF.
77
3.4
Power Amplifier
This section presents the design, analysis and control of a linear high power, currentcontrolled amplifier. We chose to design and build our own amplifier so as to drive
the relatively large number of coils with linear mode drive at reasonable cost. The
power amplifier task is to drive both the AMB and the motor. This power amplifier
is a modular design and each channel can be controlled independently. The power
amplifier consists of two stages: power(voltage loop) and control (current loop) stage.
In the voltage loop we used an APEX power amplifier PA12 with a unity gain [1]. The
current loop controls the current that passes through the amplifier. The overall gain
of the amplifier is O.1A/V which means for a 1 volt input it provides 0.1A. This gain
can be easily changed by means of changing a resistor on the board. An air-cooled
heat sink is used to extract the heat generated by the power amplifier. Figure 3-11
shows the power amplifier that we designed and built for this project. We constructed
18 power amplifiers, as required to drive the coils of the AMB and self-bearing motors.
In the next sections we first present the required specifications, power amplifier
architecture and the compensation design respectively. At the end, thermal consideration will be discussed.
3.4.1
Design Specifications
For this lab-based project we decided to design a relatively large bandwidth current
controlled amplifier to ensure that the amplifier doesn't add significant phase lag
to the system.
Also a linear amplifier doesn't generate high frequency switching
waveforms, which can be a source of noise. For the small set up, we chose the smallsignal closed-loop current control bandwidth to be 20 KHz. Each channel can provide
up to 5A continuous current with voltages up to ±45V. For the small setup we used
4 channels for the AMB and 6 channels for the self-bearing motor (3 for the two pole
and 3 for the four pole motor).
For the large setup we used 12 amplifiers for the
upper stator and 6 for the lower stator.
78
Figure 3-11: Power amplifier designed and built for the project.
3.4.2
Architecture
Broadly speaking, there are two types of power amplifiers: linear and switching. The
switching type is more power efficient, but in this research we used the linear amplifier
for several reasons. First of all we were mainly interested in the self-bearing motor
performance, not the drive efficiency. By using a linear amplifier we could also get
rid of noise generated by power switching which could deteriorate our results. The
overall power amplifier architecture is shown in figure 3-12.
As we can see in figure 3-12, the PA12 power amplifier is configured as a unity gain
voltage amplifier. The current that passes through the load is converted to voltage
by a sense resistor value of 0.10. This voltage is fed back to a differential amplifier
with gain of 100. The error signal goes to two control stage to achieve the appropriate
specification. The PCB is designed based on schematic given in figure 3-12 such that
79
Vil-
R4
R,,_,
10K
P12
R
VV
10K
10K
Figure 3-12: Power amplifier schematic.
a typical PID and lead lag controller can be easily implemented. In this figure, the
first op-amp in the left hand side is configured as differential amplifier for the input
signal. As shown in the figure the other two op-amps are used to implement a lead
lag controller.
3.4.3
Power OP Amp Considerations
Power op amps are electronic components that can source or sink current in excess
of +100mA . These components are typically rated for supply voltages greater than
± 20V . Typically, general purpose op amps have a wide safe operating area (SOA).
Their SOA can be satisfied if they meet the maximum absolute rating supply voltage.
However power op amps have additional constraints due to their high current output
transistor stages.
There is a phenomenon called secondary break down that can
damage the transistor even without passing the emitter-collector break down voltage.
The absolute maximum junction temperature is also limited to about 200*C. These
constraints are represented in the SOA curves provided by the manufacturer shown
in figure 3-13.
In this graph, the x-axis represents the voltage drop across the PA-12 output drive
transistor, i.e. supply to output. The maximum safe current can be read from y axis.
The relevant thermal line is chosen based on the case temperature, which depends
80
15
30
1
5
-0
+
S3-0
U.
zW 1.5
D 1.0
0K&
SECOND B
O
DOWN
50 60 70 80 100
10
15
20 25 30 35 40
SUPPLY TO OUTPUT DIFFERENTIAL VOLTAGE VS -Vo (V)
Figure 3-13: PA12 safe operating area from apex PA-12 data sheet [1].
on the thermal path to the heat sink. We will discuss this issue in the next chapter.
The characteristic of the inductive load is very important for power amplifier design.
The inductive load create problems that need to be taken into account:
" The power op amp can be easily destroyed by the flyback effect.
When the
current changes in an inductor, a reverse voltage is being created which can
damage the op amp. We use a PA-12 in our design which has built-in flyback
diodes that protect the output from over voltage due to flyback.
" Another problem with inductive loads is that rate of change of the current in
an inductor is limited by:
di
dt
AV
L
where i is the inductor current and AV us the inductor voltage.
As we can see from equation 3.1, the higher the inductance the slower the current change for the given voltage. This imposes a voltage stress on an amplifier
during reversing of the output voltage, because the op amp must sink the current that is generated by inductor while the transistor is still on. The PA12
data sheet recommends the maximum inductance based on the rail voltage and
81
carrying current given in table 3.2 . The maximum allowable inductance for our
design, with 35 V rails is 50 mH, which is more than our actuator inductance.
Table 3.2: Maximum allowable inductance
Voltage Inductive load
[I=10A]
[I=5A]
2mH
5mH
±50
3mH
15mH
+40
5mH
50mH
+35
10mH
150mH
±30
20mH
500mH
+25
30mH
1000mH
+20
50mH
2500mH
+15
* The third problem with an inductive load is continuous current. The DC resistance of our actuator is 1.5Q therefore the voltage drop for Ic = 2A is 3V. As
we can see from SOA chart we are in the safe area for case temperature about
85 .
3.4.4
Controller design
Figure 3-14 shows the control loop block diagram for the power amplifier. A lead lag
controller was designed to get the desired bandwidth and phase margin.
in
+
9G
Figure 3-14: Power amplifier Block diagram.
Figure 3-15 shows how the differential amplifier forms the summation block. According to this configuration we can write:
82
VoR
R1
V1+
1+
R
R2
V2
4
R1) (Ra+R4)
(3.2)
Since we need gain of 100 in the feedback we choose the following values for the
resistors:
R 1 = R 4 =1KQ
R2=
R3=
100KQ
R2
R1
Vi
Vout
V2
R4
Figure 3-15: Differential amplifier configuration.
For the controller we simply use a lead-lag controller. Figure 3-16 shows the bode
plot for the plant, controller and overall system. The resistors and capacitor values
are given in table 3.3
Table 3.3: Power amplifier elements
10KQ
Rin-lead
100KQ
Rf-lead
10KQ
Rin-PI
1OnF
20pF
Cf -roll-off
As it is shown in figure 3-17 the desired cross over frequency with sufficient phase
margin is attained.
Figure 3-18 shows the loop transmission without Hc = 100.
The power amplifier bandwidth that was mentioned above is for small signal. For
large signal the the slew rate limit has to be considered. According to equation 3.1
83
Gm = Inf dB (at Inf Hz),
200
-
100
0
Pm = 75.2 deg (at 2.05e+004 Hz)
-
- -
-- -- -- --- -- - ...
- - - - .- .- -. -. -..- . -. --.
..
-
-
Plant
Controller
-Loop
.--.-.-.-- -- ---.-
--- -
CD
CO
-100
-200
0
0)
a)
3:k
(D
Mn
-45
................. ----------------- - - - - --- - -
WO
(L
-
-
- - -
-135
-180
100
10 2
104
10
Frequency (Hz)
Figure 3-16: Current control system bode plot. These plots are for the plant, the
controller and the loop return ratio. The loop cross over is at about 20 KHz.
the rate of change of current is inversely proportional to inductance. For a reference
current I = Io sin(wt) the voltage across the inductor is AV = LIow cos(wt). Thus
the voltage drops by either increasing the input amplitude or frequency. This can be
seen in figure 3-19 for our current controlled amplifier. For a reference current of 1A
the slew rate starts to limit performance at 7Khz.
84
. .. Experiment
.
.2
10
..
---
.
.
=- -=---
.. . .. .
.
|
103
L*
102
.
.el...
M..od..
4-
-
.
|
.
|
+H4---
. . .
|
±
1
|
1041
0
-50
0
. 100
++-
Cd
0150
-------------+-- + + +++-|+
...---..
... ,. . I.
------------ ----
.-----. .---..
--.- -------.
-.-. .-.-.-. ---.-.-.
|-
. . .
++
104
Frequency [Hz]
103
105
Figure 3-17: Closed-loop current control frequency response.
102,
.)s
Experment
M odel
..
.
100
.
.,
.
..
....
.
~
.Ii.~~
----..
.~~~~~~~. .---....
. .
10-2
-ru
-80
M -100
- -120'
103
102
101
-
-
-
--
T-
'
"
T-9------T
----
---
T
r
104
-"----
1 05
--
98
-- "
..-.-.
++--++-+ - - -+--- -++------
-140
10
103
102
104
+
1
Frequency [Hz]
Figure 3-18: Loop transmission for current control loop, based upon model and experiment.
85
Figure 3-19: Slew rate effect. Upper trace is input, lower trace is the slew-rate-limited
output.
86
3.5
Magnetic Material
This section discusses the key magnetic materials that were used in this thesis research. In addition to a conventional material such as grain oriented laminated silicon steel, we used three specific materials for the stator and rotor that need to be
discussed. As we mentioned earlier the soft composite material, Somaloy was used
for the stator, and two types of large hysteresis material, Maraging steel grade 350
and D2 steel, were used for the rotor. Before discussing these material properties, we
present the measuring B-H curve methodology which we used to acquire the material
magnetic properties.
3.5.1
Hysteresis Loop Measurement
This section represents the system that we built to measure the magnetic properties
of our desired materials. Figure 3-20 shows the circuit of the measuring system. We
adopted this method from Haus and Melcher [211.
A small ring of the desired material is wound with two sets of wire: a primary coil
for applying magnetic field and a secondary coil for measuring the induced voltage
and thereby ±B. A current controlled power amplifier provides current to the primary
coil. The induced voltage is integrated with an op amp to yield a signal proportional
to B. A common problem with this circuit is drift due to voltage and current offsets.
A large resistor (10Mg) is paralleled with the capacitor to prevent low-frequency
voltage drift of op amp. However we still need to adjust the 50K potentiometer so
that, with no signal in to the integrator, the integrator op amp output remains steady.
The magnetic field intensity can be found via Ampere's law:
H = Npp.
(3.3)
1,
where N, is the number of turns on the primary and I =
(Do+Di) r
magnetic path length in the core. The magnetic flux density is B
=
is the average
- assuming no
eddy currents, and thus uniform flux density. According to Faraday's law, the voltage
induced in the secondary coil is:
87
Current controlled Amplifier
C2
Figure 3-20: Circuit diagram for B-H measurement adapted from Haus and Melcher
[21].
d$b
V = N.9-d
dt
Since we approximately integrate this voltage by the op amp, then we have:
dVo
V
dt
C5R4
Therefore we have
VO =-
N,#b.
Cf 4'
Hence we can write:
VC58 R
Ns
88
(3.4)
Once we have
#
we can calculate the magnetic flux density:
(3.5)
A
For a linear material, permeability is then given by y = f and the relative permeability is p,
=
-.
In the case of hysteretic material, the relationship is multi-valued
[21].
3.5.2
Somaloy
Figure 3-21 shows the experimental acquired B-H curve for Somaloy. The measured
coercivity is about 200A/m and the relative permeability is 440. As we can see, the
B-H curve is very close to data that were provided by the manufacturer in Figure
3-22. The coercivity and relative permeability is highlighted in blue. The maximum
was also measured to be 1.2T which is very close to the
flux density at 4000 A
m
manufacturer's data.
Figure 3-24 shows the Somaloy behavior as a function of frequency.
ca
0
H(A/m)
4000
Figure 3-21: Measured B-H loop for Somaloy.
Figure 3-23 shows the measured B-H loop for three different excitation currents
for Somaloy. We also measured B-H loop for different excitation frequencies shown
89
s
[P]80
I R Cabin
7,45
7,30
r75
280
1,19
1,46
280
0 A (T1,26
B @10A1,53
73
75
260
1,23
1,49
_________________
....
....
.. H Alm______________________________
Figure 3-22: Somaloy data from Hoganas [24].
in figure 3-24.
1.5
I=0.4
1=1
1=1.5
1
0.5
-
- --..---
.
.. ..--..
-...
. . .. . . .. . .. .. .. . .. . ..
. . .. . . .. . .. . . .. . .. . . .. . ..
..........
.....
-1.
-
-3000
..........................................
-2000
0
-1000
1000
2000
3000
H(A/m)
Figure 3-23: Measured B-H loop for Somaloy for three current excitation amplitudes.
Somaloy Machining
As we mentioned before, soft composite material (SMC) materials are very brittle
and difficult for machining. Table 3.4 gives the recommended tooling and feed rates
for machining.
90
F=2Hz
F=10OHz
0.8
.6
0.4
- 0.E8--- - -
---
--..
.-----
F=400Hz
..
- - .-. - -.-0 -.-..
--
-.-
-.--.
--..
f=25Hz
F=50Hz
F=100OHz -----
02
CO
----..-.
-.
-2000
-1000
- -.-.
.
-.--
-.--
-- ..-.-.-
- - - - ..-
-- --.--
0
H(A/m)
1000
2000
Figure 3-24: Measured B-H loop for Somaloy for different excitation frequencies.
3.5.3
Process
Turning
Table 3.4: Somaloy machining
Cutting Speed
Feed Rate
Tooling
Cermet insert 100 - 200m/min 0.1mm/rev
Milling
Carbide
100 ~ 125m/min
0.05mm/tooth
D2 Steel
Figure 3-25 shows the B-H curve for D2 steel for three different current excitation
levels. We will use these graphs when we present motor modeling.
3.5.4
Maraging Steel
Figure 3-28 shows the B-H curve for Maraging 350 steel for various frequencies of
excitation. We will use these graphs when we analyze the hysteresis motor.
91
1
-- - -1=0.5
.- - -- - ..-- - - - - - - . .- - - - - - - . . - - -
0.8
1= 1
0.6
1=1.5
- ----.---.0 -
--
- - ---.
0.4 -- -.-- -.-. .-. .-.--
-
--
.-
..
- -.. -.15 -.
0.2 -..
-- - - -.
.
-.
.
.
--.
.
. -.. -..
-0.2
0
-0.4
.
---- ----- --- .- --.---.-...-. ..-. ..- .
.
.. - -- -- --.. .-
-.----- -.-..
. . -.-.-.-.
. . . . . .
. . -.-.-.-.. .
-0.6 -- . -.
-.-.-
.-
-0.8
- i'
-3000
-2000
-1000
0
1000
2000
3000
H(A/m)
Figure 3-25: Measured B-H loop for D2 steel for three current excitation amplitudes.
......................
M
0
H(A/m)
2000
Figure 3-26: Measured B-H loop for D2 steel for different excitation frequencies.
92
---- - 1=0.5
1=
0 .6
1
-.-.-.-.---
--- . .. ..- . . . .. . .-.
.
- -1=1.5
0 .4 -...
0.2
-
-
--
--.. . -
-1000
- 0.2 -.. . . . . .-.
0
.-.
-
-100
20
-
.- -.-.-.-.-.-. .-.-.-.-.- .
- 0.4 - . . . . . . - . . - . ..
-.... . .
-.. . ..-. .
-0.E
-0.0
-5000
5000
0
H(A/m)
Figure 3-27: Measured B-H loop for Maraging steel for three current excitation amplitude.
F=25Hz
=25Hz
-..
-- -F=50Hz
F=100OHz
-
0.6
0.4
.2
---.
F=400Hz
-0.4
.. - - - 0.2 - - - - - --..
-0.8
- - - - - -- - -
- --.-
-------
-2000
-1000
0
H(A/m)
1000
2000
Figure 3-28: Measured B-H loop for Maraging steel for different excitation frequencies.
93
Chapter 4
Active Magnetic Bearing
In this chapter we present the modeling, design and control of active magnetic bearings. In general, there are two types of magnetic bearings: active and passive. They
can also be categorized in terms of their force generation: Lorentz and reluctance.
There are several cons and pros for each type, however, reluctance active magnetic
bearings have been widely used in industries. Earnshaw's theorem states that it is
impossible to levitate a rigid body in all six degree of freedom by means of permanent magnets. It is worth mentioning that this theorem applies only to static, i.e.,
non-rotating systems. Besides this, passive magnetic bearings suffer from very low
damping force. Therefore, those systems that use passive magnetic bearings need to
add damping by either floating the magnet in a fluid or using a conductor to provide damping by means of eddy currents. In this research we use active reluctance
actuators to levitate and rotate the rotor.
4.1
Basic Active Magnetic Levitation
Figure 4-1 shows the basic elements of a simple magnetic levitation system.
The
desired gap is measured by a displacement sensor and is fed back into a controller.
The controller signal goes to a current amplifier and energizes the electromagnet. The
actuator consists of an electromagnet and power amplifier that are tightly interdependent. The electromagnet geometry, material and number of turns as well as power
94
amplifier voltage-current characteristic determine the actuator dynamics.
Curent
E
Controller
sensor
Figure 4-1: Basic one degree of freedom magnetic levitation system.
In the next sections we first present the single axis magnetic levitation and then
we extend it toward a real magnetic bearing design.
4.1.1
Single Axis Modeling
Figure 4-1 depicts the single degree of freedom magnetic levitation which is the most
simple case. However this simple model suffices for preliminary analysis that leads
to an understanding of basic issues. We first model the simple one axis magnetic
suspension. Our goal is to derive a differential equation with x as the output variable
and i as the input variable.
The actuating force of a single reluctance actuator can be modeled most simply
as:
)2
(4.1)
where fm is the actuator force and r. is the constant that reflects the geometry
and number of turns in the coil, i is the current in the coil, and x is the actuator
95
air gap. A real actuator has more complex dynamics, for example due to saturation, magnetic hysteresis, and eddy currents, but the above model suffices to design
a controller. It is common to use a nonlinear control law to attempt to linearize the
above relationship, but this requires a very good model for the actuator. In practice,
it is difficult to develop a sufficiently perfect linearizing control model, given that
small-scale displacements need to be controlled at high frequencies.
Figure 4-2 shows the free body diagram of a single axis magnetic levitation.
x
mg
Figure 4-2: Magnetic levitation free body diagram.
Using Newton's law:
= mz
mg-f
by substituting in equation (4.1) we get:
mg -
(i)
2
=mz
(4.2)
Since the force is dependent on both current and air gap, we must linearize about
two variables.
96
i=io+i
x =
xo
(4.3)
+ z
Our desired operating point is the equilibrium position
(ao,
io). The tilde-notation
represents small deviations from equilibrium.
Thus we have;
io
2 +++
iZ
ffm
Ox
xh
Ofm
io
=2
We can define two important factors that will be used widely in magnetic bearing
design as follows:
"
Negative stiffness:
-2
K, = 2,
[N]
[ ]
* Force current factor Ki = 2,
By substituting in equation (4.2) we get:
mg -
-
\;o
,/
+ Kzil
K,z
=m.,
(4.4)
At the equilibrium point (xo, io) the rotor weight is compensated by bias current (io)
which means:
.
2
-
mg = r,
Thus the plant transfer function with input of incremental current () and incremental output of position (z) is:
X(s)
I(s)
Ki
ms
97
2
-
K
(4.5)
Although these linearized equations work only at operating point but we can use
it for a wide range of applications. It is very important to remember that the K,
term represents a negative stiffness. This tells us that electromagnets behave in an
opposite way of mechanical springs. Figure 4-3 shows the electromagnet behavior
versus a mechanical spring.
F
saturation
fmechanical
NK,
Xelectromagnet.
x'
Figure 4-3: Mechanical and electromagnet stiffness.
It should be emphasized that the value of K, is proportional to - and i; therefore
small changes in gap or current can cause significant change in negative stiffness. For
instance, manufacturing imperfection or thermal growth can change the gap. Varying
the static load will change the i0 which in turn leads to variation in K,.
Taken
together, designing a magnetic bearing needs special attention to consider variation
of K,.
4.1.2
Planar Levitation
Modeling of a single axis magnetic levitation system was discussed in the previous
section. Now we try to expand our model into three degree of freedom. In order to
levitate one ring type rotor we need to actively control at least three axes:(x, Oy,, 62).
These three axes are shown in figure 4-4. By symmetry the tip(pitch) and tilt (yaw)
axes modeling are the same.
Using Newton's law:
T= JN
98
fi
x
Do
jL
7-
I
f3
3
*
Figure 4-4: Planar levitation free body diagram.
#
(4.6)
Ra
(4.7)
R
- f3) =
(Ui
Ravg
where:
Ravg =
2
and
(.
2
1)
(.
)2
X3
Ravgn
We linearize about the equilibrium point as we did for the single axis:
i1 = io1
+
Z1
i3 = i03
+
21
X1 = ioi
+ -3
X3 = Z03
+ z3
Thus:
fl
-
f3
=
I
\Xol
2
+Kiiis~i]+ Kiti- Ks.zi -
)2
0 20
\X03/
At equilibrium point we have:
99
+ KJi3 - Ks
3
(4.8)
2
-0
2
-o
X01 }
Xo 3 J
Thus we have:
Kii - Ki
- K03 + KX
3
=-
Ravgro
(4.9)
Since planar levitation is done differentially we have:
Z1
-Z3
also since the rotor is rigid we have:
521 =
-23
By substituting in 4.9
e(s)
I(s)
2 Ki_(4.10)
2RvgK2
- RavgK(
As expected, the lateral and axial plant has the same transfer function with different coefficient values.
4.1.3
Force-Current Linearization
In general, two counteracting magnet pole forms the magnetic bearing. As we discussed in previous chapter we have both suspension and commutation on both sides
of our axial self-bearing motor. Figure 4-5 shows the concept of differential driving
magnetic levitation. This configuration enables us to generate force in both directions. This configuration is also essential in self-bearing motors in order to control
the shear force(torque).
In such a setup the electromagnet in one side is driven with 10+ i, while the other
side is magnetized by 1o - ix. The Io is a bias current that pre-loads the bearing or
can control the torque in the self-bearing motor. The i. component however is the
100
ix
Figure 4-5: Magnetic levitation in differential mode.
control current that stabilizes the system. Neglecting the iron magnetization leads
to a linear force-current relation. Force fm is the resultant force exerted by both
actuators. Thus we have:
f=
[(I
. ±
[(X0 _ X
(4.11)
I
-
(X0
X)2
Having x0 >> x we can linearize the above equation:
K10.
fm=-
4.2
0I
i
xo
-
3 x
(4.12)
Closed Loop System
As we can see from equation (4.5) the open loop magnetic levitation system has two
poles located at: si,2
-
±
which means the open loop system is unstable. From
a frequency response point of view the phase margin is zero i.e. (zero damping). Thus
we need to add damping by the controller to the system. On the other hand we also
need to adjust the stiffness of the system which we can use a proportional controller.
101
Figure 4-6 shows the block diagram for a single axis magnetic levitation.
In this
figure, we have just shown a generic PID controller, however any other appropriate
controller can be used. In this design the current and implicitly the force is related
to position according to equation 4.13.
f oci=PAX+I Adt+Ddx
(4.13)
dt
J
By analogy to a simple mass-spring-damper system we can see that P corresponds
to stiffness and D corresponds to the damping coefficient. The integral part tries to
eliminate the steady state error, or in other words it brings the rotor position to the
desired point. Thus, it seems that the mechanical equivalent for the integrator is the
bearing housing. However the integrator controls the desired position actively and is
maintenance free. This is a unique behavior of active magnetic bearings that can't
be attained by regular mechanical bearings.
Magnetic
Levitator
10
P
Current
^^^
-- - -- - -- -
Amplifier
Ref.
Rotor
(mass)
i=aV
Sensor
Figure 4-6: Block diagram for magnetic levitation.
4.2.1
Stiffness and Damping Selection
Before designing the controller we need to decide about stiffness and damping. As
we discussed earlier the parameters P and D determine the stiffness and damping
of the system. Therefore before computing these parameters we need to define our
requirements which depends on application. Typically a high precision and high force
102
system requires high stiffness, whereas devices with small external loads such as high
speed flywheels, compressors, etc not only don't require high stiffness but also gain
some benefits from low stiffness. In such applications the magnetic bearing task is
just to provide a non-contact rotation. Having said this, we can categorize magnetic
bearings into three regime: low, moderate and high stiffness.
" Low stiffness:
In this case we choose P such that we only, compensate for the negative stiffness
K, and stabilize the system. Thus the value of P ~K-9. As we discussed earlier,
the value of K, is subject to uncertainty, thus we need to be cautious in choosing
K,. For such a system we need a very exact knowledge of the system, because
the unstable pole can easily make the system unstable. If negative stiffness of
the system is not known accurately or it changes during operation (heat, large
forces, etc.) then the system is vulnerable to instability. This is another reason
that we have chosen axial self-bearing motor for high speed application. From
rotor dynamics we know that in order to decrease the load on the bearing we
need to minimize the stiffness, however, because of large centrifugal force the
gap changes which in turn the negative stiffness varies significantly and the
system becomes unstable.
" Moderate stiffness:
For this system the the overall stiffness is in the order of magnitude of negative
stiffness. In special case when K = K, the value of proportional gain is P
2-,*
=
Under this condition the open loop and closed loop natural frequencies
are the same and are equal to
=
-
-.
From robust control design point
of view this P provides the most robust system for magnetic bearings[1].
" high stiffness:
This regime is not easy to attain, mainly because of different limitations: power
amplifier saturation, controller bandwidth, magnetic flux saturation, sensor
bandwidth and etc.
Typically, high stiffness magnetic bearings produce an
audible chattering of the rotor.
103
The choice of damping D or velocity feedback depends on stiffness. The higher
stiffness needs the higher damping. For high damping systems the noise level should
be considered. In practice a value of D
=
which corresponds to ( =0.5 is a
good choice.
4.2.2
System Identification
The plant transfer function is a key element for designing a closed loop control system.
Beside the theoretical approach we need to find the plant transfer function experimentally for a good controller design. We used an HP dynamic analyzer to find the
frequency response of our desired systems. We will use the experimental frequency
response function to design controllers with loop shaping method in the next section.
Figure 4-7 shows the plant frequency response in axial (x) direction for small
setup.
Table 4.1 compares the calculated and measured current factor (Ki) and
negative stiffness (K,). The phase at high frequency doesn't behave as we expected
which could be related to eddy current effect.
100
-E 10
12
OE
Experiment
* Model10
100
101
10
103
102
103
-160 -180(0D
CO,
CO
-200
0r-
-220
-240
-
100
101
frequency [Hz]
Figure 4-7: Plant frequency response for axial direction(small setup).
104
Table 4.1: Current factor and negative stiffness (x)
Calculated
Measured
Current factor Kj[i]
6.2
6.8
Negative stiffness K,[2]
4800
4000
Figure 4-8 shows the plant frequency response in the tip-tilt direction for the small
setup. Table 4.2 shows the calculated and measured factors comparison. The calculations are based on the modeling presented in this chapter as well as electromagnetic
equations developed in Appendix A.
100
=3
1
E
Experiment
Model
0)
-
10
101
100
102
103
-140
-160
CO
-C
iL
- -
- -
-
-
--
. ... ........ ...
%2.
-200
--
- - -
-
--
--
-
--
--
---
-220 .......................-- -240
100
102
10
103
frequency [Hz]
Figure 4-8: Plant frequency response for tip-tilt direction(small setup).
4.2.3
Basic Controller Design
Since the open loop magnetic levitation is inherently unstable, therefore the first
and most important task of controller is stabilization.
From mechanical point of
view, the system needs a spring to provide restoring force and a damper to attenuate
105
Table 4.2: Current factor and negative stiffness (0)
Calculated
Measured
Current factor Kj[[]
3.3
3.8
Negative stiffness K,[[]
1900
1500
oscillations. The required resulting force can be written as:
f = -(kx +b)
The above expression can be rewritten:
(K - K,)x+bd
(4.14)
Ki
Equation (4.14) tells us that the most simple controller that can stabilize the
system is a PD or a lead compensator. We could also get to the same conclusion by
simply looking at the plant frequency response. In the case of the PD controller, the
gains for desired stiffness K and damping b are given by:
K
P _ K Ki
b
Ki
We also know from control theory that with only a PD controller, the gap would
be increased by applying external force. In order to eliminate the steady-state error
we need to add an integral controller.
As a practical approach, we use the loop
shaping method to design a lead-lag controller for each axis separately. The derivative
controller magnitude is proportional to the frequency; however in practice we bound
the output by adding a pole. This is because the actual signal has noise and high
frequency noise at an input terminal will be amplified significantly.
Figure 4-9 shows the Simulink block diagram of the three degree of freedom levi-
106
tation controller. As we explained in the previous chapter, there are four inputs from
displacement sensors. The average of all four sensors is used to measure the gap in
x direction. By a simple math manipulation the inclination in tip-tilt directions are
calculated. The error signal goes to a lag compensator with a pole at origin first. The
lag compensator is implemented such that it applies anti-windup to the controller [48].
The output is fed into a lead compensator and then properly distributed to the output channels. Since the electromagnetic force doesn't see the current sign due to the
square dependency on current, we have to limit the outputs to one direction.
107
Bad Unk
sna
-0
--- --
+lh
-++
KPir
ta1
tu +
+a
Bad Uk
pugain
AV1
Pithrtf
KJ.
Saturation2
Intagtor-
but
alphttuP+
CP
00
M-
-
fPitvh_Centroller
S
P1
L
DS103DAC.C2
Integrato P
t:
--
Bad Uni*
-
Sturations3
L
OS1103DAC.C3
P2
Y
Y
+
Saturaton
OW 03AC
C
~CJ
+
y1
alphAtauYjfi
ap
1
Yavw_C*Otoller
$a-Y integratorY
Figure 4-9: Simulink block diagram for small setup AMB using Dspace1103. Bad link blocks simply indicate lack of a hardware
connection, but are not normally presents.
Figure 4-10 shows the three degree of freedom levitation controller in Labview.
We used NI real time controller (PX16281) for our project. As is shown in this figure
we have added an FIR filter to the input signal. It should be noted that the time unit
in the Labview PID block is minutes; this block needs to be programmed accordingly.
109
SensorInput
Figure 4-10: Labview block diagram for small setup AMB using Ni PXI controller.
For this system we designed a lead-lag controller by loop shaping method for a
cross over frequency of
f
= 100Hz and phase margin of <5m = 35'. The bode plots
for plant, controllers and overall system is given in figure 4-11.
Gm =-9.21 dB (at 38.2 Hz) ,Pm
= 35 deg (at 100 Hz)
50
0
M
0
-1 0C
Gp
-- G-lead
---- Gjlag
CD
15C
SysLoop
-20C
901
'.a
0
0D
-0
-90|r- CD
U)
C13
.C
CL
"
ow
~'.
-o
_180
-270
10
10
10
10,
10
Frequency (Hz)
Figure 4-11: Bode plots for small setup AMB.
Figure 4-12 shows the step response of the levitated rotor.
4.3
Active Magnetic Bearing Design Considera-
tions
In this section we briefly discuss the physical limitations on AMB. Indeed, these limitations confine the design boundary and AMB performance.
Among the classical
constraints, load capacity is the most important one. The load capacity is the maximum force that AMB can tolerate. Obviously this is a dynamic force that depends
on many parameters such as: magnetic properties of material, actuator geometry,
111
0.18
Experiment
Designed
0.160.14
0.12
E
E E 0 .1 -- .
- - -.
. . -.-.
.. . -.- - - ..--
ai)
-o
. 0.08 a.
E
< 0.06
0.04
0.02
0
0
0.005
0.015
0.01
Time (sec)
0.02
0.025
Figure 4-12: Step response for small setup AMB.
power amplifier, controller and maximum admissible magnetomotive force NI. The
magnetomotive force itself is limited by heat dissipation rate. The maximum heat
dissipation depends on cooling type and actuator configuration. Maximum stiffness,
bandwidth and mechanical constraints such as maximum surface speed are other
limitations that will be discussed in the next sections.
4.3.1
Cooling Capacity
The total loss in an AMB is contributed by rotor, stator and gap. The stator loss is
also composed of copper (winding) and iron (core) losses. At low excitation frequencies the core loss is negligible compared with copper loss. The thermal network of
our AMB is shown in figure 4-13. By analogy to an electrical circuit, for a coil with
a total thermal resistance of Req we find the power dissipation:
PCU = T~
Tair
Req
(4.15)
The equivalent thermal resistance of the coil can be found from figure 4-13; how-
112
ever since the thermal resistance of iron and aluminum is much less than that insulations we can ignore them. The thermal conduction of a coil with a cross section of
AC, length of le and thermal conductivity A is given by:
UIns
(4.16)
A A
=
'C
HFE
HH
Figure 4-13: Magnetic bearing thermal network.
The thermal conduction from a surface A8 to air is given by:
(4.17)
U8 = AsaLr
Where aai, is the air thermal conductivity.
Having the thermal network we can find the total power that can be dissipated
from the coil. The small setup magnetic bearing is constructed with U core actuators.
Table 4.3 gives the thermal properties of a U core actuator.
The maximum admissible temperature for coil is 200C' and we assume the room
temperature to be 25C'. We can calculate the maximum power dissipation for a U
core actuator by using equation 4.15 and data given in table 4.3. The maximum
power dissipation calculated to be Pe, = 1.3W.
Now we can find the admissible current density as follows:
J =
_
""C
V V
Thus the maximum current density can be J = 3.6
113
(4.18)
For copper wire AWG24
Table 4.3: U core thermal properties
Coil cross section Ac
760 [mm 2 ]
Coil length lc
18 [mm]
Insulation thermal conductivity A
0.2 [
Coil surface area As
1620 [mm]
Air thermal conductivity aai.
0.15 [
Air thermal conduction Hs
3.2 x 10-
Insulation thermal conduction Ii,,
7.2 x 10-3
]
-g]
[4 )
with diameter of 0.4mm the maximum current is 0.7A. Therefore we limit the output
current to 0.6A for the small setup magnetic bearing.
4.3.2
Bearing Geometry
The bearing space is shared with copper and core; thus the maximum load capacity
depends on the optimal space distribution between coil and core. The design of an
electromagnet coil is based on fundamental work by Fabry. He showed that for a coil
with rectangular cross section shown in figure 4-14 , the magnetic field intensity at
the coil center (Ho) is given by:
Ho = G
pai
Where:
P: is ohmic loss power [watt]
A: is space factor =
Copper cross section
Total cross section
p: Copper resistivity [Q.cm]
G is a dimensionless factor called "Fabry factor" given by:
114
(4.19)
a2
Figure 4-14: Coil with rectangular cross section.
G
/Z
G
5
#~
a2-
+ Va2 +#)2
/1+1#2
in a
1+
(4.20)
Where
a-
a
2
a1
b
a1
He showed that for a fixed intensity of magnetic field, the optimal coil geometry
that will consume the least electrical energy corresponds to a maximum value of
G.
Figure 4-15 shows a family of curves of G.
The maximum value of G which
corresponds to minimum power loss in the coil is G = 0.179 that occurs at a
=
3 and
3= 2.
Our U core actuator dimension is given in table 4.4 which shows a good agreement
with Fabry criteria.
4.3.3
Power Bandwidth
Unlike mechanical bearings, both stiffness and damping of magnetic bearings depend
on current, frequency and position. Therefore it is appropriate to add dynamic terms
115
2.0
_____
10
10
2.0
3,0
0
4,0
6.0
7.0
8.0
a
Figure 4-15: Fabry chart [18].
Table 4.4: U
ai
a2
b
a
#
core dimensions
0.32cm
0.70cm
0.96cm
2.2
3
for stiffness and damping of AMB. The power amplifier has also influence on the
dynamic properties of the magnetic bearing.
The output current and voltage of
the power amplifier are limited. The maximum output voltage (Vmax) depends on the
power amplifier design. The maximum current at dc level is Imax =
which will
drop at higher frequencies caused by inductance L at cut-off frequency(wt_of
=
D
In general a fraction of total current is used for static load, namely rotor weight, and
the rest for dynamic load.
The fraction of current that is used for dynamic load
determines the power bandwidth. The power bandwidth, i.e., ( the highest frequency
that the actuator can still operate) plays a great role in designing the closed loop
control.
116
It can be shown [451 that for typical magnetic bearings the power bandwidth can
be estimated as:
wpor
= 0.92
xofmax
where
fmax
is the maximum force acting on the bearing. In our small setup for an air
gap of 0.4mm and amplifier of 28 watt a force of 20N can be generated up to 500Hz.
In order to verify the above equation we measured power bandwidth experimentally
shown in figure 4-16.
9 9
-15
a)
-o
-20 I-
Y: -15.97
-25
101
102
13
frequency [Hz]
-20
cc)
0- -40
-60
101
102
103
frequency [Hz]
Figure 4-16: Small setup power bandwidth.
4.3.4
Maximum Achievable Speed
As soon as the rotor spins, it starts experiencing two types of stress namely radial
and tangential given by equations 4.21 and 4.22. In these equations the ri,r,, and
ra are inner, outer and average radius respectively.
qualitatively in figure 4-17.
117
These two stresses are shown
r
Figure 4-17: Radial and tangential stress in rotating ring.
(3 + v) pQ2
O-,r
r 2 + r2 i
o-t =pQ2 (3 + v)(r2 + r2) + (3 + v)
8
a - r2
(4.21)
2
r~
i
0r2
- (1 + 3v)r2
1
(4.22)
In practice, it is essential that a reference stress of rotor is less than its material
yield strength. As an example this reference stress can be defined based on Tresca's
shear stress criteria given below:
max [Iot(r) - o-,(r), Io-t(r)|, Io.(r)|]
o=
(4.23)
It can be shown that the maximum reference stress happens at the inner radius
obtained by:
0-rmax
=
lot(ri)| =
pQ2 [(1 - v)r2 + (3 + v)r ]
(4.24)
The maximum theoretical achievable speed can then be derived from equation
4.24.
118
Vmax = (r
80-a
m
=
(3"mv
V(3 + v)p
(4.25)
Obviously the practical achievable speed is less than theoretical one for different
reasons. Typically, rotors are made of laminated steel that are either press or shrink
fitted in a casing. The stress concentration at assembled parts reduces the maximum
surface speed. We believe that hysteresis self-bearing motors which can be used as
an AMB have the advantage that rotor can be made out of simple rigid ring. We
use Maraging steel grade 350 which is the strongest steel built so a maximum speed
of 570;- can be obtained. On the other hand the manufacturing cost and assembly
decreases drastically for a single ring.
119
Chapter 5
Hysteresis Self-bearing Motor
This chapter presents three different methods that we implemented for hysteresis
self-bearing motors. In the first section, we propose a novel control scheme with a
single winding segmented stator. In the second section, a multiple winding scheme is
discussed and implemented for hysteresis motors. Finally in the last section, another
novel self-bearing motor with a single axis control is investigated. This third variation
is a very practical design for a high speed flywheel. In this section we also present zero
power implementation for flywheel applications. In this chapter we focus on control
strategy and suspension forces and moments. In the next chapter we investigate the
details of motoring and torque production by hysteresis motors.
5.1
Single Winding Segmented Stator
Different types of multiple winding self-bearing motors have been presented in the
literature. One of the main disadvantages of such a system, especially for axial flux
type motors is the low filling factor for the stator. This is a practical issue because
winding of such a stator is not an easy task.
On the contrary, U core actuators
have a very good filling factor. Moreover, these type of actuators are much easier
to manufacture and less expensive compared to regular axial flux stators. Another
advantage of a segmented stator is the short magnetic path in the stator, which will
be discussed in the next sections. The segmented stator can be easily wired for a
120
desired number of poles for the motor.
5.1.1
Principle of Operation of Segmented Stator
Before discussing the principle of proposed idea for self-bearing motors, it is very
important to explain how the magnetic path is formed in the rotor. Even though a
simple solid ring would work in this design, instead we propose a composite ring such
as shown in figure 5-1.
D2 steel
Epoxy
Figure 5-1: Segmented stator-rotor magnetic path in composite ring.
As we can see in this figure, the gap between the two rings causes a circumferential
magnetic field in both rings. We used epoxy to fill this gap, however for real application this gap can be filled with composite material such as carbon fibre. This is
an advantage for high speed flywheels, where we can use high hysteresis material for
the rings and over wrap it with a strong material such as carbon fiber. In this case,
steels with large density provide the mass whereas the carbon fibre gives the strength
to the rotor structure. The other advantage of the segmented stator configuration
which can be seen in figure 5-1, is that the magnetic path in stator is relatively short.
121
In our experimental setup we used 12 U cores of laminated silicon iron to build the
segmented stator discussed in chapter 3. This configuration enables us to implement
either a two or a four pole motor. For the larger rotor, one can increase the number
of cores and wire them like a standard motor. Figure 5-2 shows the FEA simulation
for the two pole motor configuration. The FEA analysis shows that magnetic flux in
the rotor is similar to that generated in the regular stator.
Figure 5-2: FEA simulation for segmented stator in two pole configuration.
5.1.2
Control Strategy
Figure 5-3 depicts the principle of operation of the proposed self-bearing motor. This
is the schematic top view of the segmented stator with 12 U cores. We divide these
cores into 4 equivalent sections shown as A, A, E and E, i.e., each section has three
cores. The three degree of freedom levitation is implemented in differential form, such
122
that the stabilizing moments about the rotating z and y axes are added to section A
and E and subtracted from
A and
. This is exactly like what we explained in chapter
4 for magnetic bearings. Once the rotor is levitated, the output signal is commutated
with a three phase sinusoidal waveform for desired number of pole motor. As we will
discuss in the chapter 6, in AC motors both the amplitude and the phase angle of
currents need to be regulated to control the torque. Therefore we need to have a
stator on the both sides of the rotor in order to apply a bias current for the torque
control, and thus decouple net levitation force from torque production.
3
4
E
CA
2
5
+Iz
+Iy
61
7
JJ12
-IY
-Iz
8
11
Figure 5-3: Segmented self-bearing motor principle in schematic representation.
In order to clarify the concept, we have shown a four pole motor configuration
in figure 5-4. The rotor is levitated by four magnetic fields which are arranged in
differential form. The dotted circles represent the differential part of opposite magnetic fields. Once the rotor is levitated their corresponding signal is commutated by
a three phase sinusoidal waveform. Thus we have a regulated rotating magnetic field
that maintain the suspension force as well as torque and speed.
Figure 5-5 shows the control block diagram for such a system. As shown in this
figure there are three control loops for levitation control i.e. the x and the two tip
tilt 0, and O6 axes. Then the commutated signals are added and subtracted to the
123
Rotating
frame
0
Sv
Stationary
frame
s
,.>I
/
/
Figure 5-4: Four pole segmented self-bearing motor.
upper and lower stator respectively. The torque is controlled by a bias current that
is shown by T. This bias current is added differentially to top and lower stator; thus
it enables us to vary the shear magnetic field. Since hysteresis motor is synchronous,
we can simply use
control strategy for speed regulation. However we can run the
system with a closed loop velocity control as needed.
124
CCD
C)q
0
C-
C+
t
-w
CD
CD
or
t
(D
5.1.3
Suspension Forces and Moments
In this section we present the analysis of suspension forces and moments for hysteresis
self-bearing motors. In this analysis we use a Chua nonlinear hysteresis model as
discussed in chapter 2. We derive the governing equations and present the SIMULINK
block diagram for axial force; however it can be easily used for inclination moments
as we discussed it in chapter 4. For this purpose we only need to replace the force,
displacement and mass by moment, inclination and moment of inertia respectively.
A
I
A
V
Magnetic
Pah
xgap
Rotor
Figure 5-6: Magnetic path for U core actuator.
Figure 5-6 shows the magnetic path for a U core actuator and a segment of the
rotor. In case of homogenous magnetic field the magnetic normal stress o- is given:
o- =
B2
.
2po
(5.1)
Therefore the force is given by:
f =A
B2
2po
(5.2)
where A is the total surface area of the electromagnet. Meanwhile we can write
Ampere's law along the magnetic path
126
Ni = HRR + Hglg + Hsls,
where
1R,
1g
(5.3)
and 1s are the length of magnetic paths in rotor, gap and stator
respectively. In this analysis we assume that the stator permeability is much larger
than the air and rotor. Thus we can write:
HR- =(Ni
- Hgig)
(54)
1R
Figure 5-7 shows the block diagram of force generation. The block is constructed
based on the equations explained above. The block has two inputs: current and
displacement (inclination). The magnetic field intensity HR for the rotor is calculated
by using equation 5.4. The magnetic flux density is then computed through Chua's
model. For homogenous magnetic flux density and ignoring fringes the magnetic flux
density is constant i.e. B9 = BR. Thus we can calculate the magnetic field intensity
in gap by:
= BH
B
A0
(5.5)
Once we have the magnetic field density, force is obtained by using equation 5.2.
Now we are able to construct the system block diagram shown in figure 5-8. In
this block diagram, the rotor is in its initial condition and then it is perturbed to
some displacement.
127
aI)
0
0
Uj-
z
CM
E
U.
C0
E3
7D
0
E
-C
X
'3-
Figure 5-7: Force generation in nonlinear bearing model including Chua hysteresis
model.
128
0
Figure 5-8: Hysteresis self-bearing suspension control block diagram. Nonlinear force
model is enclosed in block labeled Electromagnet.
129
5.1.4
Simulation and Experimental Results
In this section we present the simulation and experimental results for the large setup
explained in chapter 3. In the first part, we show the linear simulation that is based on
the linear analysis discussed in chapter 4. Then we present the nonlinear simulations
based on Chua's model explained in the previous section.
Linear
The linear analysis is quite similar to the one-axis magnetic bearing except for the
commutation. We use a balanced three phase current for the hysteresis motor. Thus
we can just simply multiply the output current to a unity gain balanced three phase
vector. For example, in DC condition one phase current could be zero while the other
two are multiplied by sin(2)
and sin(-).
Therefore the commutation effect on
suspension forces is just a gain.
Figure 5-9 and 5-10 show the plant frequency response for the x and tilting axes,
respectively.
10
CL 10
E 102
-0
10 0
Experiment
Model
-
101
-
- -
102
103
102
103
0)
-C
a_
-220'
100
101
frequency [Hz]
Figure 5-9: Plant frequency response for axial direction at zero speed.
130
10
10-
c- 010
Experiment .
10210
101
100
-100 -- -- 00
-180
- -
---
-
---
Model-
10 3--
-
- -:
-
--
--
-
-3-
--
-
-200
S-220
. .
-240..
-2601
100
. .. .
102
101
103
frequency [Hz]
Figure 5-10: Plant frequency response for tip/tilt direction(9).
The current factor Ki and negative stiffness K, of the linearized models are given
in table 5.1 .
Table 5.1: Current factor and negative stiffness for large setup
tilt
x
Axis
Current factor Kj[ ]
6.5
4.2
Negative stiffness K,[2]
4300
3400
A lead-lag controller is designed for a cross over frequency of
phase margin of <,m
=
fc
=
60Hz and
40' which provides approximately ( = 0.4 damping. The loop
transfer function frequency responses for both vertical and tilting axes are given in
figure 5-11 and 5-12. As we can see in figure 5-12, the tip tilt phase differs significantly
from theory at low frequencies. We believe that this is due to lateral mode coupling.
Since the rotor is passively stable in lateral axes thus it can couple with a lateral
mode. We can also see a resonance at 200Hz which is a structural resonance.
131
102
:F
70
-D
100
E
102 100
101
100
101
102
103
102
103
a)
(a
CU
-C
frequency [Hz]
Figure 5-11: Loop transfer function frequency response for axial direction.
The step responses for x and the tip tilt are shown in figure 5-13 and 5-14.
132
102
32 10
--- -- -.---
0)
-.-.
Experiment..- -M o--e - l -
E 10
...
- -- -.-- - -.-.
- - - .--
10-
10 0
10 1
102
103
102
103
-10 0
-15 0
U)
c -20
0
cI
101
100
frequency [Hz]
Figure 5-12: Loop transfer function frequency response for tip/tilt direction (0).
1.8
1.6
1.4
E
(D
c
1.2
1
a)
0.8
C.
CL
0.6
E
0.4
0.2
0
-0.2'
0
a
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
Time (sec)
Figure 5-13: Closed loop step response for axial direction.
133
co 0.8
-S0.60.4
0.2
0
0
0.02
0.04
0.06
0.08
0.1
Time (sec)
Figure 5-14: Closed loop step response for tip/tilt direction (0).
134
Nonlinear Modeling
The nonlinear analysis is based on Chua's model presented in section 5.1.3. The B-H
curve used in this analysis is the Chua's model fitted for D2 steel shown in figure
5-15.
1
+
0.8
Experiment
---- --- --- --- ------- ------------ -- -
- -- - - -
0.6
------------------+ --------- --- -- ----------- - - - - - 0.4 ----------- - - - - - - - - - -- - - - -0.2
0
0
-0.2
--------
----------+---------
- ------
----- - ---------
- -
---------- - ----------
-----------------
-0.4
-0.6
-0.8
-1
0o
-2000
-1000
0
H(Nm)
1000
2000
3000
Figure 5-15: Hysteresis model of Chua's configuration as fitted to D2 steel experimental data.
In the first step we set the initial current to zero and we run the simulation to
find the current when the rotor becomes stable at gap = 0.5mm. Figure 5-16 shows
the rotor axial position during startup. Figure 5-17 presents the current during this
transition. The final DC value of current is the initial current for final simulation.
Figure 5-18 and figure 5-19 show the current versus force and the plant B-H curve
respectively during start up.
Figure 5-20 compares the step response of linear and nonlinear model with experimental results for the large setup.
As we can see from the step response, the nonlinear model provides an acceptable solution. Now we can generate interesting simulations for various parameters
including hysteresis shown in figures 5-22 to 5-24.
135
X 10,4
5.2
5
4.8
a_
S4.44.243.80
0
0.02
0.06
0.04,
Time[sec]
0.08
0.1
Figure 5-16: Axial position during startup with initial gap=0.5mm.
0.5
0.4-
T 0.3
0 0.20.1
0'
0
0.02
0.06
0.04
Time[sec]
0.08
0.1
Figure 5-17: Current during startup from zero initial current.
136
2
1.5
1
LL
1
0.5-
0
0
0.1
0.4
0.3
0.2
0.5
Current[A]
Figure 5-18: Force v.s. Current during startup.
0.14
0.12
0.1
_,
0.08
0.06
0.04
0.02
0
L
0
60
100
150
H[A/m]
20D
250
Figure 5-19: Plant B-H during startup.
137
300
1.8
1.6
1.4
1.2
1
C:
c)
0.8
E
co 0.6
-E 0.4
0.2
-
0-0.2
0
I
'
'
0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04
I
Time (sec)
Figure 5-20: Step response. Zero reference corresponds to a gap of 0.5mm.
2.5
2
1.5
IL
1
0.5
0 L_
0.1
0.2
0.3
0.5
0.4
Current[A]
0.6
0.7
0.8
Figure 5-21: Force v.s. Current during transition of an 0.1mm step input from initial
0.5mm gap.
138
2.5
2
1.5
03
V
0-
1
0.5 0'
5.5
6
6.5
X 10
Position[m]
4
Figure 5-22: Force v.s. Displacement during transition of an 0.1mm step input from
initial 0.5mm gap.
0.16
0.14
0.12
m 0.1
0.08
-
0.06 -
0.041
0
100
200
300
H[Am]
400
500
600
Figure 5-23: Plant B-H during transition of an 0.1mm step input from initial 0.5mm
gap.
139
1
0.8 0.6
o 0.4 0.2 0
6
5.5
Position[m]
6.5
x 10~4
Figure 5-24: Current v.s. Displacement during transition of an 0.1mm step input
from initial 0.5mm gap.
140
5.2
Multiple Winding Configuration
In this section we first present P ± 2 principles and then we introduce the suspension
force/moment mechanism for hysteresis self-bearing motors. The multiple winding
electromagnets was first apparently proposed by Hermann [22].
Later, Chiba and
Fukao showed based on field oriented theory that most electrical machines can run
on a multiple winding design [6]. The general concept is well explained in [7].
We implemented P ± 2 for hysteresis motors by means of multiple windings on a
single stator, as well as for the U-core stator design explained in the previous sections.
For the sake of simplicity and flexibility we wired all 12 cores separately, with 12 power
amplifiers and implemented the control scheme in the software. In the first step we
run the DC excitation that provides the levitation force component. Then we add
rotation (AC excitation) to add the motor torque self-bearing component.
5.2.1
P + 2 Principle
In this section we focus on general principles of multi winding self-bearing motors.
These machines can generate force/moment for levitation as well as torque with only
one stator by means of multiple windings. Figure 5-25 shows the principle of force
generation in such a system. This specific configuration consists of a two pole and a
four pole winding on a single stator. For this specific design we assume a two pole
motor for commutation torque and a four pole motor for suspension control. In this
figure the big circles depict the two pole motor whereas the small circles represents
the four pole motor. The mechanical phase shift between the two windings is 450
which means they are electrically orthogonal.
In this particular position, the two magnetic fluxes on the right hand side of the
motor amplify each other, while there is attenuation on the opposite side. Therefore
in case of a radial flux motor a force is generated toward the right direction. Figure
5-26 shows the moment generation for axial flux gap motor. In this configuration also
the two pole motor provides the commutation whereas the four pole motor controls
the suspension. As it is shown in figure 5-26 the two magnetic flux amplifies each
141
y
Figure 5-25: Force generation in self-bearing motor.
other on the right hand side and attenuates on the opposite side. In this case the
moment is generated about Z axis. Having said that, we can control the force or
moment vector by controlling the angle between the two fluxes and their amplitudes.
4z~7
z
B
Figure 5-26: Moment generation in axial self-bearing motor.
142
Figure 5-27 shows the winding pattern for such a system. This pattern can be
readily implemented for either radial or axial flux motors. The dot shows the current
direction out of the page and the cross sign means the current going into the page.
The stator has 24 slots, distributed for both four and two pole windings. Symbols
u4, v4, and w4 represent the four pole windings for phase u, v and w respectively.
The same notation is used for the two pole windings. The subscript s represents the
star or Y connection point.
U2 8
Wt
VT
W4
N2s<:
TT
*
1
=
V2hn
Figure 5-27: Self-bearing motor winding diagram.
5.2.2
DC Excitation (Magnetic Bearing)
Before we discuss the self-bearing motor, it would be helpful to investigate the static
operation of such a system. Fortunately, even without a rotating magnetic field, selfbearing motors are still useful and can be used as an AMB. A self-bearing motor can
be easily converted into a magnetic bearing by simply running a DC current into the
system. Figure 5-28 shows the control scheme for the DC excitation of a self-bearing
143
motor.
Xr
Figure 5-28: Magnetic bearing (DC excitation).
The actual gap is measured by averaging the signal of four displacement sensors
(please refer to chapter 2).
By subtracting this signal from desired gap, the error
signal is fed into a proper filter and controller.
The control effort signal is then
amplified by a simple three phase power inverter and energizes the electromagnetic
actuators. The two inclinations (tip-tilt) are measured by the same four displacement
sensors and are fed back to the summation point where we can get the error signal.
By proper compensation, we get two control effort signals associated with the two
inclinations in cartesian coordinates. These signals then can be transformed to the
three phase domain by means of a Park transpose matrix. We can again use a regular
three phase power inverter to amplify these signals and drive the actuators.
The
major advantages of such a magnetic bearing are listed below:
" Low-cost, off-the-shelf power inverter can be used as power amplifier.
" The magnetic field can be synchronized to shaft speed to decrease eddy current
and hysteresis losses in the rotor.
" We have less wiring for three phase systems compared to orthogonal systems.
144
* Three phase systems are well known, so we can implement the associated algorithm for magnetic bearings.
5.2.3
Park and Clarke Transformation
The Park transformation matrix is a well known tool in three phase electric machine
analysis [51]. The Park and Clarke transformation is typically used to transform the
three phase system to dq axes to simplify the analysis [17]. In this project we use the
transpose of the Park transformation to go in the opposite direction. In the previous
section we used the Park transpose matrix in order to transfer signals from cartesian
coordinates to the three phase domain. In this section we present this matrix and its
derivation.
Im(P)
A
b
X
a
....,Re(a)
C
Figure 5-29: Arbitrary space vector in complex plane for a three phase system.
An arbitrary space vector as shown in figure 5-29 can be described in terms of its
magnitude JX| and angle ( [50]:
I
=|iXlek = Xa + jxp.
For convenience, we label the real and imaginary axes with o and
(5.6)
#
respectively.
Thus x, is the component of the space vector along the real axis and xg is in the
imaginary axis.
145
The general Clarke transformation for a three phase system with 1200 phase shift
is given by:
1
]
xO
[1
k 0
XO
][XA
x
--
-1
1
The set of variables defined by
-
-j
1
1
XA, XB, xC
(5.7)
X:B
zc
are the phase variables. The transformed
variables represent a vector defined in the stationary reference frame. Indeed, a three
E(variable)=
phase system has two degrees of freedom because
0. Therefore we can
simplify the general Clarke transformation as given below:
[0
1
k
a
[303
1i-
XA
2
XB
-1
2
-0
T2
The inverse of the transform is given by
2
J
(5.8)
[XC]
= T 1 Xem; however a 2 x 3 matrix
Xold
is singular. There are several ways to find the inverse or in some cases equivalently
the transpose. One can find such a transpose for the Clarke transformation as given
below:
1
XB
XC
01
3
(5.9)
X-
11
-
There is a choice of k to make the transformation given in equation 5.9 be magnitude or power invariant. For magnitude and power invariant k is k
=
and k
=
respectively as given in [51] and [50]
5.2.4
AC Excitation
In order to produce torque, we need to run AC current in the coils of the system
described in the previous section. The three phase balanced AC currents produce
a revolving magnetic field that creates shear forces on the rotor. Figure 5-30 shows
146
the control algorithm of such a system. In order avoiding confusion in this figure,
we assume that the air gap and the two inclinations are measured and fed back as
we explained in the DC excitation section. The variable I'
is the bias current that
can be used for torque control, noticing from the previous chapter that torque is
proportional to current. The IBy, is also the bias current for the two inclinations and
can be typically set to zero. It should be emphasized that in this figure we have not
shown the DC current for rotor weight compensation. In real controller, this current
is added to the upper actuator.
Ax
Motor
Controller
r
motor
E
Controller
- +Z
Figure 5-30: Self-bearing (AC excitation).
5.2.5
Simulation and Experimental Results
In this section, we present the simulation and experimental results for the 12 U core
design that was setup for P t 2 control scheme. The simulation is based on Chua's
hysteresis model and quite similar to what was explained in the previous section. The
147
analysis of a nonlinear model for multiple winding for regular stator is left for future
work. Figure 5-31 shows the step response for this system.
2
1.5
C)
1
E
CO 0.5
CL)
01
-0.5
0
'
0.01
0.02
0.03
0.04
0.05
0.06
0.07
time (sec)
Figure 5-31: Step response for P+2 control scheme in x direction.
148
5.3
Single DOF Self-Bearing Motor
In this section, we present the idea and a simple proof of concept for a novel selfbearing motor with only a single axis actively controlled. This design is very suitable
for high speed flywheels, as it is very simple and reliable. In this thesis we do not
provide any detailed analysis for such a system and it is left for the future work. But
the design principles are described below.
5.3.1
Principle of Operation
In this section, we present the first order analysis of single DOF self-bearing motor.
This simple analysis does not cover the whole physics; however it is sufficient to
describe the principle of operation.
Figure 5-32 shows the free body diagram of
levitated cylinder with only single axis control.
m1
fm2I
2
Figure 5-32: Free body diagram for single degree freedom self-bearing motor.
For small 0 we can write:
149
X 1 = l9 X2=19
R9
+
(5.10)
RO
The moment M is then:
M = (fmi- fm 2 ) R
(5.11)
The magnetic force fm can be approximated by:
fm
(j)2
Thus we have:
M
=
KR
2 -
(5.12)
)2
By substituting equation 5.10 into above equation we get:
M = KRi 2
[1
(l+
I(1g - RO) 2
Since 6 is small therefore we can neglect the
02,
RO)2
(5.13)
thus:
KR 2 i 2
(5.14)
9
The negative stiffness produced by motor Ke is then approximated by:
Ke=
KR 2 i 2
1
9
On the other hand the mechanical positive stiffness Km produced by rotor weight and
length is given by:
L
Km = Mg-
2
If the Ke
+ Km is positive, then the rotor can be stable with only a single axis
control. For those systems where the rotor has a large aspect of ratio the cylinder is
150
large enough that the mechanical stiffness is dominant. The ideal rotor for proposed
self-bearing motor is thus a long and light weight rotor with a large mass at the lower
end. The performance of such a system can be improved by using permanent magnets
in the center of rotor as shown in figure 5-33. The upper magnet reduces the current
in the electromagnets, which in turn minimizes the negative stiffness by canceling the
rotor weight out. The reluctance force applied by the two pair of magnets also helps
the tip-tilt stabilization.
Copper
Magnet
Ah
Magnet
Copper
Figure 5-33: Single axis rotor configuration.
For the proof of concept we modified the small setup rotor shown in figure 5-33.
For clarity we have just shown the rotor; however the rest of the system is the same
151
as small setup discussed in chapter 3. We used permanent magnets at each end to
increase the stability in radial direction. The gap between upper magnets is smaller
than the lower ones to compensate for the rotor weight. All four magnets are adjusted
such that rotor is almost floated, thus we use a very small current just to stabilize it.
The system tested successfully for the purpose of proof of concept. As we mentioned
earlier, in this thesis we just proposed the idea and detailed analysis is left for the
future work. The real hardware is shown in figure 5-34. This is not presented as the
optimal design, and is just built for the proof of concept. For the real high speed
application the supporting column that connects the upper and lower stator can be
replaced by an aluminum casing. The casing can have multiple tasks including safety
in case of rotor failure. It also creates a vacuum chamber for rotor as well as providing
alignment between the two stators.
Figure 5-35 shows the plant frequency response. Figure 5-36 shows the rotor axial
position during startup.
152
Figure 5-34: Single axis self-bearing setup.
153
10
0
-ee.
I
Experiment
Model
*
2 S10-2
E
-4
10,4
0
101
10
-100
15
-0 --~r
r - : -r -r i
a-10 -------4---+ -+-|-
102
n-- -- r-- r -r - 4
|-'r------'-
-
-250 ----------01
--
--
r
--
-
4
N
- -----2
3
10
frequency [Hz]
Figure 5-35: Single axis plant frequency response.
0.6
0.5 .........
-- -- ---.
0.4
0.3 ..........
E
...........
0.2
0
0.1 ...........
0. ...........
-0.1 ....
-0.2-
0
. ......
0.05
0.1
0.15
Time[sec]
0.2
Figure 5-36: Rotor axial position during startup.
154
3
I T-.1
. --- I
i
10
10
10
i
1
0.25
5.4
Zero Power
Having permanent magnets on the system described in the previous section, encouraged us to test the zero power condition for this setup. We tuned the magnet gap to
cancel out the rotor weight. Under this circumstance, we need a very small current
just to stabilize the rotor. The permanent magnet in such a system introduces significant passive radial stiffness. We also covered the permanent magnets by copper
to add radial damping by means of eddy currents.
As we can see from step response shown in figure 5-37 the rotor has 10 micron
axial vibration when it is at standstill. However when it rotates the axial vibration
increases. Figure 5-38 shows the rotor axial run-out at 600rpm which is about 50
micron peak to peak. The overall system performance can be improved by proper
design which is left for future work.
0.54
0.53
0.52
0.51
_f 0.5
E
0.49
0
CL
0.48
0.47
0.46
0.45
0.44
0
0.1
0.2
0.4
0.3
Time[sec]
0.5
Figure 5-37: Step response.
155
0.6
0.7
30
20
E
0
10
0-
0
-10-
-20
-30
0
0.1
0.2
0.3
Time[sec]
0.4
Figure 5-38: Axial run-out at 600 rpm.
156
0.5
5.4.1
Lateral Damping Ratio and Stiffness
The rotor lateral movement is constrained passively by means of two pairs of permanent magnets and copper cover.
The reluctance force produced by permanent
magnets provides the lateral stiffness whereas the copper cover generates damping
by means eddy current. We use impulse response in order to determine the lateral
properties of passive magnetic bearing. Figure 5-39 shows the impulse response of
the rotor upper end.
0.5
-
-
0 .4 -- --
- -
-
-
0.3 ...... ...... ............
--.........
0.2 .......--
E
C
0
0
.........
.......
0.1 .......
- .-- ...
-.
CA,
0
CL
-0.1
............
-
-0.2 ......
--............
-0.3 .......-0.4 ......-0.5
0
-
-
----
0.2
-............
0.4
0.6
0.8
Time[se
1
1.2
1.4
Figure 5-39: Impulse response.
By definition logarithmic decay (A) is the natural log of two successive peak. The
damping ratio is then given by equation 5.17:
A
v4ir 2 ± A2
(5.15)
The lateral frequency can be obtained from equation given below:
27r
(5.16)
Td
Where Td is the time delay between the two successive peak. Then the lateral
157
stiffness is given by:
2
Katerai = mw
(5.17)
Table 5.2 gives the measured passive bearing parameters:
Table 5.2: Passive magnetic bearing parameters
Stiffness (K)
1203 [[]
Damping ratio (()
0.08
Natural frequency (Wn)
6.8 Hz
Damping factor (C)
4.6 [N
]
There are a few analytical solution for eddy current damping in literature which
are very similar. We chose Cunningham's equation given below [16]:
C-
B 2 X 2 wcLi
9
p
PI'o
SN
(5.18)
crm
where:
B. Flux density in the gap
Li Inductivity per unit length
W, Conductor thickness
X, Pole face width
po Permeability
p Conductor resistivity
The Cunningham's equation gives the damping ratio of (=
be reasonable.
158
0.065 which seems to
One of the major advantage of zero power system is a simple touch down bearing
design. Since most of rotor weight is compensated by permanent magnet, the touch
down bearing doesn't have to tolerate too much force. Moreover the passive damping
produced by eddy currents makes the system more robust and stable. Therefore it
makes it much easier to design a touch down bearing.
159
Chapter 6
Hysteresis motor
In the previous chapter, we developed hysteresis motor control schemes and stabilizing
control via force-moment modeling. In this chapter we introduce hysteresis motor
torque production mechanisms and motor modeling. The motor analysis presented in
this chapter is deduced from fundamental laws of electromagnetism. This analysis is
largely based on a report from Lawrence Livermore National Laboratory published in
2009 [30] and also on two papers from Miyari [33] and Ishikawa [26]. However these
references don't support their analysis with experimental results, whereas we have
experimentally examined the analyses with multiple prototypes.
Another contribution in this thesis is adding frequency dependency of hysteresis
loops to the analysis. We have also verified the simulation results experimentally.
6.1
Introduction
In general, electric motors can produce torque using two independent principles. The
first is using two interacting magnetic fields and the second is having one magnetic
field and a magnetic circuit whose reluctance is a function of rotor position. Motors
can use only one or can use both of these principles to produce torque. These two
source of torque production, namely mutual and reluctance, can be well defined by
three fundamental laws: Faraday, Lenz and Lorentz.
Hysteresis motors use the combination of mutual (eddy current) and reluctance
160
(hysteresis of material) mechanism for torque production. All ferromagnetic materials
have some degree of hysteresis which means they retain a portion of magnetization
even after removing the external magnetic field. This residual magnetization depends
on the material and its structural condition. Typically, high cobalt or high chrome
steels have a large hysteresis loop and by proper heat treatment one can attain the
desired magnetic properties. Indeed, hysteresis motors are a combination of a weak
permanent magnet motor and an induction motor. However unlike induction motors,
the accelerating torque is due to both eddy currents and reluctance force. Once the
rotor becomes synchronous with the stator field, the eddy current portion vanishes
and the motor operates as a permanent magnet machine.
S
o>
R
Stator
Rotor
R
Figure 6-1: Hysteresis motor operation principle, adapted from [17]. Here W is the
rotor angular velocity, 5 is the lag angle, and SS and RR are the stator and rotor axes
respectively.
The fundamental magneto motive force (MMF) in the air-gap and rotor is shown
in figure 6-1. The revolving MMF produced by stator is indicated by axis SS that
rotates with angular velocity w. The hysteresis of the rotor material causes a lag angle
(J) between the stator and rotor magnetic fields. This lag angle is the heart of the
161
hysteresis torque production mechanism. Indeed the hysteresis torque is proportional
to the product of stator and rotor magnetic field and of the lag angle. In practice, if
the maximum available the hysteresis torque is more than the load torque then the
rotor will synchronize with the stator frequency. However if the hysteresis torque is
not sufficient to overcome the load torque then the rotor will slip until the total eddy
current and hysteresis torque can drive the load.
Before the rotor reaches synchronous speed, each region of the rotor experiences
a hysteresis cycle at the slip frequency. As we discussed in chapter 2, the hysteresis
phenomenon is dynamic and depends on the excitation frequency. Therefore the lag
angle and in turn the hysteresis torque varies with the slip frequency. The frequency
dependency of hysteresis torque is included in the machine analysis for the first time
in this thesis.
The stator is just like a regular polyphase machine that can create a rotating
magnetic field. The rotor consists of a hysteresis material and its support. Hysteresis
motors can be segregated into different types based upon magnetic flux generation in
the air gap or in the rotor itself. Also like other electric machines there are two classes
of hysteresis motors: radial and axial flux motors, depending on the orientation of
the air gap.
Retainer
Rotor
(Hysteresis Material
Air gap
Stator
Air gap
Rotor
Stator
RJ
r
R.
(b) Axial.
(a) Radial.
Figure 6-2: Radial and axial flux Hysteresis Motors.
Radial flux: Figure 6-2(a) shows a cross section of this type of motor. The rotor
162
is made out of hysteresis material that can have either an inner or outer core.
The magnetic flux is in the radial direction in the air gap.
Axial flux Figure 6-2(b) shows a schematic of an axial flux air gap motor. These
motors, also known as disk-type motors, have a ring type rotor made out of
hysteresis material and supported with either magnetic or non-magnetic material. The magnetic flux in the rotor is mainly axial in the former case shown
in figure 6-3 and is circumferential in the latter case (figure 6-4). Hence we can
categorize the disk type machine into two different types: axial rotor flux and
circumferential rotor flux motor.
I7
Figure 6-3: Magnetic vector in the rotor with non-magnetic material core.
6.2
Modeling
In this section we try to find the governing equations and the equivalent circuit of
the hysteresis motors. Before deriving these expressions, the following assumptions
have been made to simplify the calculations:
* We assume sinusoidal steady-state operation.
163
Figure 6-4: Magnetic vector in the rotor with magnetic material core.
" The permeability of the stator is infinite so the magnetic flux intensity (H) is
zero in the stator.
" The air gap between the stator and rotor is small compared to the rotor diameter.
9 We ignore space harmonics of the stator as well as fringing in the rotor.
" The hysteresis characteristic of the rotor is represented by a tilted ellipse in the
B-H plane. Thus we ignore the saturation region.
In this section we will also use the following symbols.
R, Stator resistance
L, Stator leakage inductance
M Mutual inductance
Rg Gap resistance
Re Eddy current resistance
164
I, Rotor apparent current
Ig Gap apparent current
6 Lag angle
Wr Rotor electrical angular velocity
p Number of pairs of poles
m Number of phases
J Rotor moment of inertia
Te Electrical torque
Td Drag torque
In this study we use a motor equivalent circuit to simulate motor behavior. For
this purpose we take the following steps:
1. Determine motor equivalent parameters: The motor equivalent circuit is
shown in figure 6-5.
R,
L,
I
Rg
V
L
M
Re A
Figure 6-5: Hysteresis motor equivalent circuit.
The first step in our approach is to find these parameters from the fundamental
electromagnetic equations.
165
2. Solve circuits for apparent currents: Once we have the equivalent parameters, we can use Kirchhoff's circuit laws to find the rotor and gap apparent
currents for given terminal voltage and frequency.
3. Assign the initial condition: We define the proper initial conditions for angular velocity and lag angle in order to solve the governing differential equations
given in steps 5 and 6. It should be emphasized that lag angle will vary with
frequency.
4. Calculate torque: we use equation (6.1) to compute the torque at any speed.
mP
Te
=
(6.1)
sin(S)
MII
-2 2
5. Calculate speed: The motor dynamic equation given below is used to calculate
the speed.
Te - Td =
(6.2)
p dt
6. Calculate lag angle: The lag angle is given by:
Wb -Wr
p
63
_dJ
dt
=
-
(6.3)
It should be mentioned that phasor analysis is only valid for circuits with balanced
sinusoidal excitation in steady state. If we are after the vector control of induction
machine then a dynamic analysis is recommended. For this purpose, one can use the
space vector analysis which is presented in different books and literature. However the
traditional equivalent circuit analysis is sufficient for steady-state operation. Modeling
of machine transients is not addressed in this thesis.
6.3
Motor Equivalent Circuit Parameters
As we mentioned in the previous section, the first step in our approach is to find
the motor equivalent parameters.
From Ohm's law we know that the equivalent
166
impedance can be found simply by dividing the induced emf (e) by current. Thus we
need to find the induced voltage and associated currents. To do so, we can take the
following steps:
1. Calculate induced emf: This step can be divided to three steps given below:
" a)Finding magnetomotive force produced by stator.
* b)Approximating the hysteresis curve by a tilting ellipse to find B and H
in the rotor.
* c)Use Ampere's law to find the flux linked by coils.
* d)Use Faraday's law to calculate the induced emf.
2. Calculate currents: This step can also be followed as:
" a)Applying continuity law.
* b) substitute in magnetomotive equation to find the rotor and gap equivalent currents.
3. Ohm's law: We use Ohm's law to find the equivalent impedances. The real
part is resistance while the imaginary part is the inductance.
In the next sections the above steps are described in detail.
6.3.1
Hysteresis Material Approximation
The hysteresis loop can be approximated by a tilted ellipse [30]:
H
=
(
cos60
(6.4)
p
B = B, cos(0 - 6max ),
where p is permeability, 0 is a parametric variable and
field lags the H field.
167
(6.5)
5 is the angle that the B
expanding equation 6.5 we can get:
B
=
(6.6)
a 1 cos 0 + b1 sin 0
where:
a1 = Bm COs( 6 max)
b1 = Bm sin(5 max).
The maximum lag angle can be found as
6max=
tan- 1 k
(6.7)
.
al
For Maraging steel the data were found as follows shown in figure 6-6:
6max = 0.66 rad(380 )
Bm = 0.9T
.
1
-
0.5 F---
...
7
-4,
-0.5
-1
L---
-6000
-
---------
-4000
-2000
0
H(Am)
2000
4000
6000
Figure 6-6: Hysteresis approximation for Maraging steel.
As we discussed in chapter 2, the hysteresis loop is a function of excitation fre-
168
quency. We also presented the hysteresis loops for D2 and maraging steel as a function
of frequency. Figure 6-7 shows the hysteresis loop and linearized model for D2 steel
at two different frequencies.
0.8-
--
0.6
+
+
-- ExperimelF=3Hz Experimental
F=50Hz Experimental
F=3Hz Linear Model
F=50Hz Linear Model
-----
0.4 -------- --------- -------- - -- ----------0.2 -------------------
---- ---- ----------------- - -------- 0 -----
- - - -
-0.6 -- -
-2000
-1500
--
--
----
- ---
- ------ -------- ------L--------
-------
-0.4 -------
-----
- - -- ------- ------------ ---------
-1000
- -------- ---------- --------- -------------- -
-500
0
H(A/m)
500
1500
1000
200)
Figure 6-7: D2 steel linear model approximation.
Figure 6-8 gives the lag angle and relative permeability for D2 steel as a function
of frequency.
We will use these data in the next sections to include the effect of
frequency on hysteresis torque.
6.3.2
Flux Distribution
Figure 6-9 shows a cross section of stator-gap-rotor of hysteresis motor. As we mentioned earlier the magnetic field intensity in the stator is assumed zero. By applying
Ampere's law to the closed magnetic path shown in figure 6-9 with dotted line we
can write:
Fm.dp = -(Hg + dHg)g + Hgg + Hr dO
P
169
(6.8)
350
300
£2-
a)X
1-1
CD
E)
250
L-
(D
CD
-J
200
150 L
0
I
10
I
I
I
140
50
40
30
20
Frequency(Hz)
Figure 6-8: D2 steel magnetic properties as a function of frequency.
Which we obtain the magnetomotive force (FM):
FM = g
r
0H
+ H-.
0p
(6.9)
p
By applying continuity law in figure 6-10 we get:
Ro
t
(B + dB)dr - t,
Bdr=
R
0
Bgr
(d
)dr
(6.10)
By neglecting fringes in the rotor we have:
tr(B + dB)dr - trBdr = Br
( d p ) dr
(6.11)
Thus we can find the magnetic field density in the gap:
B -
(6.12)
170
r
dI#
-= rd6
P
Figure 6-9: Magnetic path in motor.
Noticing that within the air gap:
Bg = j oHg
(6.13)
o9BB
H9 Hg=pt,
= Pt
por O@
(6.14)
We have:
Noticing that the linearized hysteresis loop can be written as follows:
(6.15)
B = Bm cos(O)
H= ( m)cos(
pA
+ 5)
(6.16)
In order to solve equations (6.9) and (6.12) we approximate the B field by a
sinusoid functions:
B = Bm cos(Wt - 0 -
171
o)
(6.17)
Where 6 = Wbt
-
4,
and Oo is the phase shift.
Therefore the corresponding magnetic field intensity is:
H =
B
(6.18)
*cos(wbt -
B+dB
tr
I
Figure 6-10: Magnetic flux density continuity in the rotor.
By substituting eq.(6.17) into eq.(6.12):
B9 =
(6.19)
PtrB, sin(Wt - @-0)
9
From eq.(6.17) and eq.(6.14) we get:
__H
o
-
Bmptr
o
(6.20)
s(wbt-@-o)
By substituting eq.(6.18) and eq.(6.20) into eq.(6.9):
FM-Bmgptr
'r cos(wt Fm =
por
Bmr
-
0)
+
pp
cos(wt -
4'-
0
+
6
(6.21)
)
We obtained the magnetomotive force in chapter 2:
m
Fm = -IW
2
cos(wbt
(6.22)
-4)
Using 6.22 we get:
m
-IW
2
cos(wbt
-
> Bmgptr cos(wbt
por
--
-4o)
+
cos(wbt
pP
172
-
'
-
0
+ 6) (6.23)
In order to solve equation 6.23 we multiply both sides by
.Bmgpt
m
j-IW sin(wt-/) =rj
2
por
sin(Wt - ---
o)+j
.Bmr
pP
j
and add I to
sin(wt -4/-4'o+
6
4.
) (6.24)
Now we add both sides of equation 6.23 and 6.24:
2
IWestb-*) =
J
t +
por
Bme(Wb--0)
(6.25)
pp_
We find Bi:
Bm =
(mIWejPO)
gp,+
poT
(6.26)
irej6)
PA
Since Bm is real we can write:
gptr + rej 6 1 ejO=]
por
pP)
Im
0
(6.27)
Thus we can find Bm and 0o as follows:
Bm(r)
L~,
jp cos(6) +
+ 2
r
o(r) =
6.3.3
(6.28)
mIW
=
tan- 1
g2p2t2]
2
j
por
sin(6)
(6.29)
ptrp
+
+
-11cos(6)
Derivation of Equivalent Circuit
As we mentioned earlier, to derive the equivalent circuit, we need to follow the next
three steps
" Finding the induced EMF (E) from the Lorentz law.
" Calculating apparent currents in the rotor (Ir) and air gap (Ig).
" Impedance is given by Ohm's law: Z = -
173
I.
Induced EMF
To find the induced voltage, we first need to calculate the total flux linked by the
stator coils. Then we can use the Lorentz law to find the EMF. Flux linked to a one
coil between electrical angle V) and 0 + 7r is:
#=
(-B,)rdodr
jR+
R,
(6.30)
p
p
We can write equation 6.17 as follows:
B = Bm cos(wt -4,--0)
=
Bm [cos(@0o) cos(wbt - 4/) + sin(@o) sin(wbt - 4,)] (6.31)
By using equation 6.12 we have:
Bg - pt, &B
Spt
-r
r 8@
[cos(@o) sin(wbt - 4,) - sin(@o) cos(Wbt
-
4)] Bm
(6.32)
By taking the second integral of equation 6.30:
(B,)
+s
d@o
= t,
tr [cos(@o) cos((wbt -
sin(wbt - 4,)d4 + sin(@o)
I- cos(o)
4,) + sin(@o) sin(wbt
cos(wbt
-
@)d
4)] Bm
-
(6.33)
Now we can substitute equation 6.33 into equation 6.30 we have:
p
JR trBm
[cos(to) cos(wbt
-
4,) + sin(to) sin(wbt - 4)] dr
Ri
(6.34)
Ro
=trL
[oRi
cos(to)Bmdrcos(wbt - 4,) +
174
sin(4o)Bmdr sin(wbt
R.
-
4,)]
Bm
Now we define:
Co
cos(@o)Bmdr
= f
Ri
(6.35)
So
sin(@o)Bmdr
=
Ri
Then we have:
Co COS(Wbt - @) + So sin(Wbt -
4)
(6.36)
C + S2 and also defining:
Dividing both sides of equation 6.36 by
sin(#1) =
(6.37)
cos(*1) =
0S+ CO
We can find:
4=
sin(wbt
-
4+
(6.38)
1)
Where:
> =2tC2 + S02
ID2t,
i
=
C0S
tan--
()
Now we need to find Co and So. Using equation 6.35 and substituting for Bm from
equation 6.28 we get:
CO= (
0
IW x (3 1 cosJ+2#
so= ()IW
=4
2
sin 6) x
x (,3 sin6J- 23 2 cosJ) x
2
175
P
4
Where:
~k4(1 + k2)4 + 2k2(1 + k2) cos(6) + 1
_k(1 - k 2)4 + 2k2(1 - k2) cos(J) + 1
#2= tan- 1 [k2(1
+ k2)2
+ cot(e5) 1 --
tan-1
k2(1 -k k2)2-
-
+cot1
)
Cot(01_+
sin(J)
and
Ra
k2=-
rm
l=R 0 -Ri
Ra=
2Ra
Ro+Ri
2
VPo
The induced EMF in one conductor is given by:
(6.39)
- at
The total induced EMF in k-th phase stator winding by the gap field is given by :
Ek =
/2p2at
t
Cos 0 -(k - 1)-27
I
~mI
dp
(6.40)
or
Ek = -WbNKwb sin wbt -
1W
(k - 1) - - $1]
m
It is more convenient to work in the complex domain.
(6.41)
Thus we can write the
complex induced emf:
_E-k=
-jWbNK,<be
"(--)
Where
k
2ir
=(k - 1)-2m
176
(6.42)
Therefore the voltage to be impressed to the stator winding is given by:
(6.43)
= .juNK,<be-
-k
Apparent Rotor and Gap Currents
The stator current can be resolved into a load and exciting component from equation
The exciting current creates the required air gap flux
6.9 as shown in figure 6-5.
and its losses, while the load component produces a MMF that corresponds to rotor
current. Indeed, the total current is stored in inductor or rotor as magnetic energy
or dissipate in ohmic resistor of coil or eddy current in the rotor. Thus we can write:
(6.44)
I = Ig + Ir
The current in the k-th phase winding is given:
Ik = I cos(wbt -
tV)k)
(6.45)
The complex current in the k-th phase winding is:
(6.46)
Ik - Ie(Wbt-Ok)
(6.28) into eq.(6.23) and then
To resolve the stator current we substitute eq.
divide both sides by mV and 2noticing that W
=
2K-N,
Pir
we can split the two parts
into two individual time varying components. We get:
rp2 g9trBm
r cos(wbt Krp29 yor
I = mK,,N
i
I,=
r
irr B
mK N y
cos(wbt
-
pO
-
-
po)
to + 6)
The complex expression of these apparent currents are:
177
(6.47)
(6.48)
_
2
T
l~
gtrBm
(WtIkP0
(wt--kk-Po)
g
-k
por
-gmKN
(6.49)
(6.50)
ej(Wt--*0)
1rBmejs
-k - mK
Ny
Since the phase difference between I and Ir is 6 and also the impedance for I is
only reactance, thus we can draw the phasor diagram shown in figure 6-11.
------------I/
I
Figure 6-11: Phasor diagram adapted from [26].
From this phasor diagram we can deduce:
ik
I_,
Z=
= Ik + I_
(6.51)
_Ik -- Iejv1
(6.52)
in(#1j)eid
= I sin(6)
(6.53)
I cos($1)
-
Ir cos(6)
_'= WbNKw-beji
178
(6.54)
(6.55)
Impedances
Having induced EMF and apparent currents we can find the apparent impedances by
using Ohm's law. The gap impedance is given by:
M = j
(6.56)
=f
;-g
Therefore:
2mK.2N 2 pl Ra
M = mao 2
(6.57)
+ 43) sin(6)
aeo = (#2
16k k 2 ,8 2
(6.58)
Z,
k
(6.59)
er sin ()
(6.60)
7rg
P
where
and for rotor we have:
=
R, +jLh,
which gives:
Rr
Wb
Lh, =
mKl N 2 iiy,
2
X27rRa2
a
mKr 2
2
r
cos(O )
(6.61)
Where:
a=
(#2 + 42)6.2
k 2 [8#1 - 16#2 cot(6)]
V, = 27rrtl
179
(6.63)
6.3.4
Dynamic Equations of Motion
In order to incorporate the rotor dynamic equation in to motor analysis we can use
conservation of angular momentum.
Jdor
Te - Td = J
p dt
(6.64)
The above equation is solved associated with the following initial condition:
Wr = 0
We also need to include the lag angle created by hysteresis. Knowing that lag angle
6 is the phase difference between magnetized pole on rotor and rotating magnetic field
we can find the time rate of change of 6 as follows:
WbWr
p
- d6
dt
The above differential equation is bounded by
property. Recall that the
6
max
(6.65)
6max
which is function of material
is function of frequency therefore we will make a lookup
table in our SIMULINK block diagram discussed in the next section. Thus if the lag
angle is more than its maximum value then it is reset to value assigned by lookup
table.
6.4
Circuit Analysis
According to figure 6-12 we can write the following relations for impedances. In this
figure the subscript s denotes stator quantities and r for rotor. The Lhr is apparent
inductance due to rotor hysteresis. The Re stands for rotor eddy current resistance
and s is the slip defined by:
S = 1 - Wr
Wb
180
(6.66)
R
Li
Zr
Figure 6-12: Hysteresis motor equivalent circuit.
Z,5= R, + jwL,
(6.67)
Zg = jwM
(6.68)
= Rr + jwLhr
(6.69)
Ze
Re/s
(6.70)
1
(6.71)
Zrh
=
Then the rotor equivalent impedance is:
Zr
Zrh
Ze
Then the total impedance of motor:
1
Z=Zs+
Zg
In turn we can find the associated current:
181
(6.72)
Z,
V
(6.73)
Z3I3
Z9
(6.74)
VZSIS
(6.75)
is =
19-V
Ir
6.5
-
Zr
Simulations and Experimental Verification
In this section we present simulation and experimental results for the two motors that
we built, and which were explained in chapter 3. First, we present the SIMULINK
block diagram that we developed for simulations and then we show the results for
both small and large setup.
6.5.1
Simulations
We developed a SIMULINK block diagram along with MATLAB code based on equivalent circuit of hysteresis motor presented in this chapter for hysteresis motor modeling. The basic parameters are calculated in the MATLAB code shown in figure 6-13
and transferred into SIMULINK block shown in figure 6-14. In this block diagram
there are two sub-blocks to calculate air-gap and rotor impedances that are shown in
figure 6-15and 6-16.
182
clear all
clc
close all
P=2;
m=3;
wb=35*(2*pi);
tr=14/1000;
r_o=40/1000;
r_i=18/1000;
g=0.8/1000;
Kw=0.85;
N=8*40;
J=4.2e-4;
Re=1.2;
R_s=5;
L_s=4.2e-3;
V=6;
mu0=4e-7*pi;
mur=145;
mu=mu0*mu r;
phi=30*(pi/180);
sai=cos (phi);
R_a=(ro+ri)/2;
1=r o-r i;
% Number of pole
% Number of phase
% Stator frequency[rad/sec]
% Disk thickness
% Outer radius[m]
% inner radius[m]
% gap[m]
% Winding factor
% Number of turns per phase
% Rotor moment of inertia[Kg.m^2]
% Eddy current resistance[ohm]
% Stator resistance{ohm]
% Stator leackage inductance [H]
% Stator voltage[V]
% Average radius
V_r=pi*(r_ oA2-ri^2)*tr; % Rotor volume
A=le-5;
B=8e-8;
% Mutual inductance (airgap inductance)
MO=2*m*Kw^"2*N^2*mu0*Ra*l/(pi*g);
R_r=(wb*m*Kw^2*N^2*Vr*muo)/(pi^2*R_aA2); % Apparent rotor resistance[ohm]
L_hr=(m*KwA2*N^2*Vr*mu0)/(pi^2*R-a^2); % Apparent hysteresis inductance[H]
rm=sqrt (g*tr);
k2=l/(2*Ra);
Figure 6-13: MATLAB script for initial parameters.
183
00
X
X
Piorque
P1
Adesi
opeea
orque Torquel
-
I
deltal
power______________________________
Power
X
T
p
P/2
Product1
pole
gap_current
Complex to
Magnitude1
|u
1r
Complex to
Magnitude
Figure 6-14: Hysteresis motor SIMULINK block diagram.
rotor-current
Eddyres
Rotor
Resistance
delta
Add
sin
Real-Imag to
Complex
00
C.J1
Cos
Figure 6-15: Rotor equivalent impedance SIMULINK block diagram.
D2
w
Mur
Complex
const
Product2
M0
Figure 6-16: Air-gap equivalent impedance SIMULINK block diagram.
M
Z_g
In a different attempt we also developed a general MATLAB code (Appendix C) to
simulate hysteresis motor based on linearized model presented in this chapter. In this
code the ODE45 solver (Runge-Kutta fourth-fifth order) is used to solve differential
equations (6.64) and (6.65).
6.5.2
Small motor
Figure 6-17 shows the stator of small motor along with load cells that are capable of
measuring force in three dimension.
Figure 6-17: Hysteresis motor stator.
Table 6.1 gives the small motor parameters that we built and tested. Before we
plug in these data into MATLAB code we need to find the drag toque.
In this section we try to find the drag toque experimentally. For this purpose,
we measure the rotor deceleration after we unplug the motor at different speeds. By
multiplying this deceleration and rotor moment of inertia we can find the total drag
187
Table 6.1: Small Motor Parameters
2
3
6V
35Hz
Number of poles (P)
Number of phase (m)
Voltage (V)
Motor base frequency (Wb)
Rotor outer radius (R 0 )
Rotor inner radius (Ri)
Rotor thickness (t,)
Winding factor (Kw)
40mm
18mm
14mm
Rotor moment of inertia (J)
0.85
4.2 x 10 4 Kg.m 2
Gap (g)
0.8 mm
Stator resistance (R.)
Leakage inductance(L.)
Number of coil per phase (N)
5Q
2.5mH
8 x 40
torque shown in Figure6-18.
x 10,
T
a)
:3
0Y
CD
cc
0
500
1000
1500
Angular speed[rpm]
2000
2500
Figure 6-18: Drag torque measured experimentally.
In order to include the drag torque into our SIMULINK block diagram we have
fitted a curve as a function of rotor angular speed on these data. Equation 6.76 gives
the fitted curve for small motor.
Td = 1 x 10- 5 + 7 x 10- 8w 2
188
(6.76)
Figure 6-19(a) shows the experimental and simulation results of rotor speed vs
time. In this figure we have only considered the hysteresis torque and neglected the
eddy current effect. On the basis of the nearly straight line speed curve the acceleration (or torque) is almost constant and less than the experimental torque. In figure
6-19(b) the eddy current has been added to simulation. Surprisingly the acceleration
at low speed is still less than the experimental results. We could empirically fit the
model on experimental data by decreasing the eddy current resistance and increasing
the drag torque. However a more detailed model including transient effects may be
able to capture this behavior.
Figure 6-20(a) shows the simulation results for the torque generated by the motor
without considering eddy currents. Unlike general definition of hysteresis motor we
can see that torque is not quite constant due to fact that hysteresis loop itself is
frequency dependent.
Thus if we only consider the hysteresis torque, it increases
slightly as rotor gets synchronized. Figure 6-20(b) shows the motor torque including
eddy current effect. As we expected there is a significant boost in torque at low speed.
We can also observe larger damping of oscillation (hunting)when we include the eddy
current effect.
We can also see these effects on torque versus speed curves shown in figure 6-21(a)
and 6-21(b).
Figure 6-24(a) and 6-24(b) shows a zoomed-in view of torque and lag angle behavior as a function of speed as rotor gets synchronized.
Figure 6-28 shows the stator terminal voltage and current at start up. Therefore
we can measure the power factor as well. The total input power is given by:
Pin = mVmsIrms cos(O,)
The current was measured with a sense resistor
Rsense =
(6.77)
0.1Q. According to figure
6-28 we find Pin = 2.1watt.
Figure 6-27 shows the simulated input power. The difference between measured
and simulated results is due to the initial eddy current torque. We also measured
189
E 2000
4-
1500-
1000 -0
5-
10
0
10
20
40
30
60
50
70
Time(sec)
(a) Neglecting eddy current.
2500
Experimental
Model
-
2000-
-
-1
1500
0.
15000
(D
CA
0
5
10
210
1'5
25
3
35
40
Time(sec)
(b) Including eddy current.
Figure 6-19: Simulation vs experimental results for hysteresis motor.
starting torque with mounted load cells shown in figure 6-17. The measured torque
was Tta,,t = 0.01N.m.
Figure 6-29 shows the hunting phenomena that was measured. The logarithmic
decay of this graph gives the damping of the system.
190
3'
x103
2.5
2
2
0-
H-
0
40
20
60
Time(sec)
100
80
120
(a) Torque neglecting eddy current.
x103
5
4
E
:3
2
1
0
10
20
40
30
Time(sec)
50
60
(b) Torque including eddy current.
Figure 6-20: Simulation results for torque.
191
70
x 103
3
2.51
E
2
0
1.5
-
1
500
0
-
-
1000
1500
Angular velocity(rpm)
2000
2500
(a) Torque versus speed neglecting eddy current.
x
5
103
4
a)3
0
2
1
0
500
1000
1500
Angular velocity(rpm)
2000
2500
(b) Torque versus speed including eddy current.
Figure 6-21: Simulation results for torque v.s. speed, with and without eddy currents.
192
6050
a)40-a)
C" 0-
CO
20100
40
20
60
Time(sec)
100
80
120
(a) Lag angle neglecting eddy current.
50
45
40-
30CU
252015
0
10
20
40
30
Time(sec)
50
60
70
(b) Lag angle including eddy current.
Figure 6-22: Simulation results for lag angle, with and without eddy currents.
193
60
50
I
T 40
i 30
Ca
IJ
20 F
10C
0
500
1500
1000
Angular velocity(rpm)
2000
2500
(a) Lag angle neglecting eddy current.
5
50
7
45
a)
a)
CO
-o
40-
3530
25
-
20[
10
I
0
500
I
1500
1000
Angular velocity(rpm)
2000
2500
(b) Lag angle including eddy current.
Figure 6-23:
currents.
Simulation results for lag angle v.s.
194
speed, with and without eddy
45
40 1-1
CD
35 -
aU 30
25 -j
2015
2090
2095
2105
2100
Angular velocity(rpm)
2110
(a) Lag angle versus speed including eddy current.
x 10-3
31
2.5
2
0
1.5
2090
2095
2105
2100
Angular velocity(rpm)
2110
(b) Torque versus speed including eddy current.
Figure 6-24: Simulation results for torque and lag angle v.s. speed, zoomed in near
synchronous speed.
195
0.7
0.6 0.5C',
a 0.4 =3
o
-
0.3
0.2 0.1 0
0
20
40
60
Time(sec)
80
100
120
(a) Output power neglecting eddy current.
0.7
0.6
0.5
Ca,
m 0.4
0
0.3
CL
o
0.2
0.1
0r
0
10
20
40
30
Time(sec)
50
60
(b) Output power including eddy current.
Figure 6-25: Simulation results for output power.
196
70
0.7
-
0.6 0.5 0.4 0.3 o
0.20.1
00
500
1000
1500
Angular velocity(rpm)
2000
2500
(a) Output power neglecting eddy current.
0.7
0.6
0.5
0.4
0
0.3
b
0.2
0.1
0
500
1000
1500
Angular velocity(rpm)
2000
2500
(b) Output power including eddy current.
Figure 6-26: Simulation results for output power.
197
1.5
1.4 1.3 a)
3:
1.2 0
0.9
0
500
1000
1500
Angular velocity(rpm)
2000
2500
Figure 6-27: Small motor input power versus time.
Figure 6-28: Terminal voltage and current versus time.
198
2120
2100
E
2080 o2060--
2040-2020-
2000
0
'
5
'
10
'
15
'
'
20
25
time[sec]
'
30
'
35
'
40
45
Figure 6-29: Hunting phenomena (Experimental results).
199
6.5.3
Large Motor
Figure 6-30 shows the stator for the large setup. In addition to the size difference,
the large setup has a four pole motor as opposed to the two pole motor for the small
setup. Table 6.2 gives the large motor parameters.
Figure 6-30: 36 slot stator made out of soft composite materials used for large setup.
As we can see in Figure 6-31, the simulation result has a good agreement with
experiment for the large setup as well. Therefore we believe that the hysteresis motor
modeling presented in this chapter can be used for designing such a class of induction
motors with a reasonable accuracy.
200
Table 6.2: Large Motor Parameters
Number of poles (P)
4
Number of phase (m)
3
Voltage (V,)
Motor
Rotor
Rotor
Rotor
base frequency (Wb)
outer radius (R,)
inner radius (Ri)
thickness (t,)
Winding factor (K)
Rotor moment of inertia (J)
Gap (g)
Stator resistance (R.)
Leakage inductance(L,)
Number of coil per phase (N)
8V
40Hz
60mm
28mm
9mm
0.85
1.4 x 10-3Kg.m 2
0.9 mm
4.6Q
3.3mH
12 x 40
1400
1200
1000
L)
a
800
U)
600
400
200
0 f-
0
5
10
15
Time(sec)
20
Figure 6-31: Large motor angular speed versus time.
201
25
55
45
,40-
aD
-o-
30co
25
20
15
0
5
10
15
Time(sec)
20
25
30
Figure 6-32: Large motor lag angle.
55
45 35zm 40aD
C)
-0
-2
35-
~,30
252015
0
p
200
II
400
a____
600
800
1000
Angular velocity(rpm)
1200
Figure 6-33: Large motor lag angle.
202
1400
0.025
0.02
0.015
2
.
a)
0.01
0
01
0
200
400
600
800
1000
Angular velocity(rpm)
1200
1400
Figure 6-34: Large motor torque versus angular velocity.
x
10~3
10
9
8
7
0
F--
6
5
4
1192 1194 1196 1198 1200 1202 1204 1206 1208
Angular velocity(rpm)
Figure 6-35: Large motor torque v.s. speed zoomed in near synchronous speed.
203
40 -
-U 3530C
>'25-
20III11
1190
15
1195
1200
1205
Angular velocity(rpm)
1210
Figure 6-36: Large motor lag angle versus speed zoomed in near synchronous speed.
2
1.5
CD
0
0-
0
0.5
0'
0
5
10
15
Time(sec)
20
25
Figure 6-37: Large motor output power.
204
30
2
1.5 CD
0
0.5
0
0
200
400
600
800
1000
Angular velocity(rpm)
1200
Figure 6-38: Large motor output power.
205
1400
6.5.4
Torque Constant and Back EMF
We can calculate the back EMF from equation 6.43. The back EMF can also be measured under open circuit condition. Figure 6-39 shows the measured and simulated
back EMF of the motor which gives the K,
=
4.2 x
0.4
10
-
0.3
-3V.sec
rad~
Model
Experiment
0.2 0.1
>
0-0.1
-0.2
-
-0.3 -0.41
0
002
0.0
01
O0
Time(seconds)
0.1
012
0.14
Figure 6-39: Measured and simulated back EMF.
We can also approximate the experimental results shown in figure 6-40 with a first
order system. Recall equation 6.64 we can find the speed current transfer function
as:
oKt
- = __(6.78)
I
Js+B
From the first order approximation we can find the Kt
=
0.01N.
Unlike the
DC machine the K, and Kt are not the same. This is because there are two source
of torque production in hysteresis motor (mutual and reluctance) as oppose to DC
motor having only mutual torque. The Kt is a good practical parameter for motor
specification.
6.6
Motor Design Considerations
In this section we first try to find the power density relation and then establishing a
criteria for optimal motor design. We also discuss the limits for using soft composite
206
250
200 -
150 -
220
10
a 100
s+
1
E
50
0
0
10
20
30
40
50
60
time[sec]
Figure 6-40: Torque constant.
materials for stator.
6.6.1
Optimal Motor
This section is presented based on a paper published by Ishikawa [27]. By examining
equations 6.28 one can realize that magnetic flux density in the gap is function of
rotor radius (r). Therefore the motor size can directly affect the motor power density.
In order to find the maximum power density (i.e. the optimal motor size) we need to
find the associated rotor radius with maximum flux density.
Figure 6-41 shows the magnetic flux density vs rotor radius according to equation
6.28. Taking derivative respect to r will provide the radius associated with maximum
B.
&Bm
m
ar
= 0
(6.79)
gives the radius (rm):
gtr
/p0
rm =
Thus the maximum flux density is:
207
(6.80)
Bm(r)
r
rm
Figure 6-41: Magnetic flux as a function of radius.
(6.81)
PI O
Bmax = mIW
2gtr(1 + cos(6))
max
In general the motor output power is given by:
mV
PO =
2
Z,
(6.82)
sin(o)
From equivalent circuit we know that:
V = IZt
Replacing for total impedance we get:
2ppot'
g (ki +-I
)
lwKwNmIW
(6.83)
+ 2 cos()
By replacing equation (6.83) into equation(6.82) we get:
2
Po sin(6)wN (mIW)
2gtr (k2 + 1 + 2 cos(j))
The power density per unit volume is given:
208
Vrao
ar
(6.84)
P,,
Vr
Noticing that ki =
powN(mIW) 2 sin(o) ao
2gtr (k2 + ' + 2 cos(6) a,
(685)
one can see that the power density is a function of average
radius. By examining the equation (6.85) we find that power density takes it maximum value for k1 = 1 or Ra = rm. Therefore the optimal average rotor radius is
equal to rm.
We can also deduce from equation 6.85 that the power density is proportional to
angular velocity. Thus in order to have a high power density machine, we need a
high speed system. As we discussed in previous chapters, hysteresis motors are very
suitable for high speed application because of its rotor configuration. Therefore the
hysteresis motor is a good choice for high power density motor.
6.7
Material Considerations
In this section we try to compare the materials that we chose to use in this project.
We used D2 steel and Maraging steel for the rotor, as they have a large hysteresis
loop. Soft magnetic composite and laminated grain oriented silicon steel were also
used for the stator. Each of these materials has its own cons and pros that will be
discussed here.
6.7.1
Rotor Materials
Figure 6-42 compares the B-H curve of the two materials used for the rotor. The D2
steel has larger permeability and higher Bm while its coercivity is less than Maraging
steel. If we decompose the total torque provided by motor into eddy current torque
(Te) and hysteresis torque (Th), where we have:
T = Te + Th
The eddy current torque is function of material resistance and rotor configuration.
209
In this case since both rotors have the same geometry and almost identical
0.8
+
o
Maraging
D2
------------
0.2 -------------
8m
-------
-------
r------ r--------
0.6 ----------------------------0.4 -----------------
-----------
r------r-
----
---- -
--
- -----------
----------------------------- - -----
L------- -- ------- -----0 ---------------
----------- - ------
-1
-5000 -4000 -3000 -2000 -1000
2000 3000
0
1000
H(Nm)
4000 5000
Figure 6-42: Maraging and D2 steel B-H curves.
resistance; thus the eddy current torque is equal. Therefore we can conclude that
the torque difference between the two materials is due to the hysteresis component.
The hysteresis torque relation can be derived from equivalent circuit discussed in
this chapter. The hysteresis output power is given by:
(6.86)
Pr = mI7 R,
Thus the hysteresis power loss is:
(6.87)
PI =SPr
The net hysteresis output power is then given by:
Po = m1R2(1
-
s)
(6.88)
Hysteresis torque is:
Pho
210
(6.89)
Where:
r
S)W
(6.90)
p
If we substitute for Ir, Rr from equations 6.60 and 6.50 then we find torque:
Th
=
BmV.,sin(S)
(6.91)
We also know that the area enclosed by a hysteresis loop is given by:
H.dB
(6.92)
m sin(S)
(6.93)
-Sh Vr
(6.94)
Sh =
or
Sh = 7r
B2
Thus the hysteresis torque is:
Th =
2ir
The mechanical work done by hysteresis torque (ThdO) is equal to the magnetic
energy of the system.
ThdO = -dWm
(6.95)
The magnetic energy of the system is proportional to the area enclosed by BH curve. Having said that, one would expect larger hysteresis torque for D2 steel
compared to Maraging steel, by comparing the B-H curves in figure 6-42. In order to
find out the two rotor materials behavior, we conducted a test for both materials by
keeping all other parameters the same. The test condition is similar to what given
in table 6.1 except for the gap that was increased to g = 1mm. Figure 6-43 presents
the speed vs time behavior for both materials.
As we expected the D2 steel rotor reaches to synchronous speed in shorter time.
211
2500
1500 -
-
1000
500 -
0
0
I
I
I
|
|
I
5
10
15
20
25
30
35
40
45
Time (seconds)
Figure 6-43: Maraging and D2 steel speed comparison.
The initial acceleration where the eddy current torque is dominant is almost the
same for both rotors. However as the rotor speeds up the the eddy current torque
contribution becomes smaller and this torque vanishes at synchronous speed. Under
this circumstance the hysteresis torque which is proportional the the area enclosed
by the B-H curve, is dominant. This phenomena is clearly observed in figure 6-43.
The motor peak current measured at the terminal for D2 and Maraging steel was
ID2 = 0.6A and IMaraging = 0.7A respectively. Since we used the same stator for both
rotors, therefore the apparent rotor reactance is larger for Maraging steel. This could
be easily understood by examining equation 6.61.
Even though it seems the larger B-H curve provides better performances, there are
other parameters which need to be considered before material selection. For instance,
if the rotor is meant to run at very high speed without any reinforcement material,
then the strength of rotor becomes critical.
In this case Maraging steel is a very
suitable option. However if we are allowed to reinforce the rotor by means of over
wrapping high strength materials such as carbon fibre, then we need to consider other
factors such as cost. For high speed flywheels where we just need to keep the rotor
at very high speed, a low cost large hysteresis material such as D2 steel looks very
212
attractive, in conjunction with carbon fiber over wrapping.
6.7.2
Stator materials
One of the drawbacks for using SMC material is their relatively low permeability and
saturation point. In this section we try to find out how these limitations would affect
the motor. In general, we consider the core permeability is much larger than the air
gap so we can assume that the magnetic field intensity is zero in the core. For SMC
that we used the relative permeability is about 400. Noticing that magnetic path is
much larger than the air gap it is necessary to find out if the above aforementioned
assumption is true. We can visualize the magnetic path for one phase of motor simply
like one shown in figure 6-44. In this figure the
1
R
is the magnetic path length for
rotor and 1c is the total path length for the core.
A
A
I
I
ICI
I
I
c
-
Magnetic
-14Path
Figure 6-44: Magnetic path.
The magnetic flux is simply obtained by:
NI
Rc + R R + Rg
6-6
Since the rotor permeability of rotor is much larger than the air and stator we can
find the magnetic flux density for this circuit from Ampere's law as below:
213
NI
B = pto
1cr+
I'r
(6.97)
29
For small setup the 1c = 100mm therefore the (
= 0.25 which is order of mag-
nitude of the air gap. Thus, despite of all advantages of SMC, it seems that this
material is only suitable for high frequency i.e. high speed motors and not for low
speed applications.
6.8
Finite Element Analysis
This section introduces the finite element method (FEM) that we intended to use
as a tool for solution of magnetostatic problems.
In this method, the solution is
discretized into simple geometric shapes called finite elements. For each element, a
stiffness matrix (Ke)is computed such that the material properties and applied loads
(f) are related to the values at the nodes of the element by equation:
[Ke][x]
=
[f]
(6.98)
In the above equation the [Ke] is the element stiffness matrix, [x] is the unknown
nodal values and [f] is the force vector applying on the element. It should be noted
that the stiffness matrix depends on both geometry and material properties. Once we
have nodal equations then we can calculate the global equations by simply inserting
the elements of of stiffness matrices into a global stiffness matrix [K9] and using
equation (6.99).
[K9][X] = [F]
(6.99)
Where [X] and [F] are global unknown nodal values and global forces respectively.
There are two most common solution type in magnetic problems: magnetic scalar
potential and magnetic vector potential. The magnetic scalar potential is used when
there is no external current source; however the vector magnetic potential is suitable
214
for application where the external current source exist.
For analysis both electric
motor and magnetic bearing we use vector magnetic potential solution as there is
always current source drive.
6.8.1
ANSYS Simulation
In this project we used ANSYS to simulate the hysteresis motor. A three phase two
pole motor was constructed for the small setup.Figure 6-45 shows the absolute value
of magnetic flux density in the rotor. The stator flux distribution is also shown in
figure 6-46.
B[tesla]
6. 7558e-001
6. 3406e-001
5. 9254e-001
5. 5102e-001
5, 0950e-001
4. 6798e-001
4. 2646e-001
g3.
8494e-001
3. 4342e-001
3 . M190-001
2. 6038e-001
2. 1886e-001
1. 7734e-001
1. 3582e-001
9. 4295e-002
5. 2775e-002
1. 1255e-002
Tim e
=1A416s
Speed =22-9723&m
Posilon=-97.366729deg
Figure 6-45: Absolute magnetic Flux density in rotor.
Even though the magnetic flux density and other magnetic properties looks fine,
we were not able to get the right torque. This is because ANSYS software is only
able to solve the nonlinear behavior of hysteresis along path 1 shown in figure 6-47.
However for hysteresis motor we need to solve the governing equations along path 2
and 3 as well. To the best of our knowledge no other FEA package is able to simulate
215
Figure 6-46: Absolute magnetic Flux density in stator.
the whole hysteresis loop and thus hysteresis motor simulation is not feasible at this
time.
As we can see from FEA analysis in figure the magnetic flux in the rotor is the
same as regular stator discussed in chapter 6. Therefore the only difference in this
model is the mutual inductance between rotor and stator. This mutual inductance is
much smaller for our segmented stator compared to regular stator; however it can be
simply maximized by changing the tooth configuration. In this thesis we had limited
manufacturing resources to make this type stator and we just showed the proof of
concept.
216
B
2
H
3
Figure 6-47: Hysteresis modeling in ANSYS. Only path 1 can be modeled.
217
Chapter 7
Hysteresis and Squirrel Cage
Motor Comparison
In this chapter, we compare hysteresis motor performance with squirrel cage induction motors, which are widely used in industry. Both hysteresis and squirrel cage
motors lie in induction machines category with a similar stator and winding configuration. The only difference between two motors is their rotor. As we mentioned
before the hysteresis motor has a solid smooth rotor made out of materials with large
hysteresis. On the other hand, the squirrel cage rotor consists of a number of slotted
steel laminations that create the magnetic path. Slots are filled with low resistance
material such as copper or aluminum with two end conductor rings at each end for
eddy current circulation. It should be noted that, because of the rotor material and
configuration; hysteresis motor can operate synchronous while the squirrel cage is an
asynchronous machine.
We built a hysteresis and an induction squirrel cage motor of identical sizes. We
replaced the rotor of a duplicate commercial induction machine with a hysteresis
material. We ran IEEE standard tests to compare these two motor performances.
Among various efficiency test methods suggested by IEEE we use EE1 and F1 tests
in this thesis.
218
7.1
Induction Motor Equivalent Circuit
In general, an equivalent circuit can be used to simulate induction machine, either
with a fixed frequency power supply or a variable frequency drive. Test methods F
and F1 in IEEE Std12 are based on such an equivalent circuit. A detailed analysis
of an induction machine is given in [17] and [28].
However in this section we just
present the equivalent circuit model along with torque and power analysis by use of
Thevenin's theorem.
XS
V
Rr /s
Figure 7-1: Induction motor equivalent circuit.
Figure 7-1 shows the equivalent circuit of a single phase of an induction motor
with following symbols:
* V: Phase voltage in V
* s: Slip
" R,: Rotor resistance
* X,: Rotor leakage reactance
* XM: Magnetizing reactance
* R,: Stator resistance
" X,: Stator leakage reactance
" I,: Rotor apparent current
219
e I:
Air-gap apparent current
We assume the motor is running under balanced three phase system and the rotor
speed is varying slowly. Under these conditions, we can analyze the circuit under
steady state conditions.
The rotor impedance is given by:
Zr= jXr +
R
s
(7.1)
If we ignore the power loss in the air gap, we can simply find the impedance seen
by air-gap given by 7.2:
Z
jXZ,Z
XM+=
(7.2)
Thus we can find the total impedance:
Zt = jXs + R + Z,
(7.3)
The stator current is
It
= -
Zt
(7.4)
The apparent rotor current is given by:
j Xr+ E+ JXM
(7.5)
The real power that passed through the air-gap for an m phase machine is
2Rr
Pag = mlr
(7.6)
The power dissipated in the rotor resistance is
Pri =
P
220
gS
(7.7)
Therefore the net output power is
(7.8)
Pm= P(1 - s)
The electric input power to the motor is:
Pin = Pag + 3I2R5
(7.9)
Pet = Pag- Pfw
(7.10)
The output power is:
Where Pfw is the power loss due to friction and windage. We can find efficiency,
power factor and torque respectively:
Pout
(7.11)
P
PF=Pi
(7.12)
T PW"r
7.2
(7.13)
Radial Hysteresis Motor Equivalent Circuit
In this section we present the equivalent circuit for radial flux hysteresis motors.
Figure 7-2 shows the equivalent circuit for radial flux hysteresis motors which is very
similar to induction motors except for the rotor impedance.
The rotor apparent
resistance is the equivalent of eddy current and hysteresis resistance given by:
S
Rr
(7.14)
As we can see from figure 7-2 the equivalent circuit for radial flux motor is exactly
221
Xhr
V
Q1
"g
XA
X
Figure 7-2: Radial hysteresis motor equivalent circuit.
the same as axial flux type. However since the geometry is different, the equivalent
parameter will be different. The equivalent parameter derivation is the same as what
we discussed in chapter 6. Figure 7-3 shows the motor geometry and magnetic path
for a radial flux motor. If we follow the same procedure that we did for axial flux
motor in chapter 6, we can find the equivalent parameters given below:
Figure 7-3: Radial hysteresis motor magnetic path.
XM
=
2mKN
2
pornl,
mVP 21g
Xhr
Rh =
=
K 2VN
mK2VN
2
7 rM
222
2
p
(7.15)
cos(J)
(7.16)
sin(6)
(7.17)
Where in the above equations:
0
rm: Rotor average radius
* 1,: Rotor length
l19: Air gap
* V: Rotor volume
* p: Rotor permeability
* K,: Winding factor
* m: Number of phase
Figure 7-4 and 7-5 show the SIMULINK block diagram, we developed for radial
flux motor. We will use this block diagram to simulate the radial flux hysteresis
motor.
223
M
f-b
Const
Product1
pole
Ig
gapcurrent
Figure 7-4: Radial hysteresis motor SIMULINK block diagram.
wr
Rotor
Add
M-F4
delta
Product4
Rotor
Hysteresis
Figure 7-5: Rotor apparent impedance.
225
Real-Imag to
Complex
Z-r
7.3
IEEE Standard Test Procedure for Polyphase
Induction Motors and Generators
In this section we present IEEE standard test for polyphase induction machine known
as IEEE Std 112. This procedure consists of basic requirements for conducting tests,
no load and locked rotor tests and test with load, for determination of efficiency.
There are various efficiency test methods suggested by IEEE. In this thesis we use
EE1 and F1 tests; hence we present general requirements that needed for these two
tests. For further information reader can refer to IEEE Std 112 given in [25].
7.3.1
Definitions
In this section we present general definitions and terms used in this chapter. It should
be noted that all voltage and currents are root mean square (rms).
Voltage: In polyphase machine it is very common to measure line-to-line voltage V11.
Therefore if we need line to neutral voltage we can divide it by v/5.
Current: The arithmetic average of line current will be used in calculating motor
performance.
Resistance: The DC resistance between two phases R should be measured at operating temperature. Then we can calculate per phase resistance R 1 :
R
2
Slip: The slip speed is the difference in angular velocity between synchronous and
measured speed.
slip speed=w, - wr
where: w,: is the synchronous speed
w,: is the measured speed
226
The slip per unit is defined:
Ws
Wr
-
WS
(7.18)
No-load losses: The motor input power at no-load condition is the total losses in
the motor including: stator loss, friction and windage loss and core loss.
Stator loss: The stator loss PSIR for a three phase motor in watts is given by:
1.5RI 2 = 3R 1 1 2
PSIR =
(7.19)
Mechanical power: The motor output power in watts at each load is given by:
p
_w,T
9.549
(7.20)
Where:
Wr : is the measured speed in rpm
T: is the shaft torque in N.M
Power factor: Power factor PF is given by:
P-n
PF= -Pi
v/35Vul
(7.21)
Where:
Pin is the input power in watt.
Efficiency: In general the efficiency is the ratio of output power to input power.
However a commonly defined form of efficiency is given by:
Pin - losses
(7.22)
Pn
Total harmonic distortion: The total harmonic distortion (THD) is given by:
227
THD =l
E1
(7.23)
Where:
E: is the total root mean square of voltage.
E:
7.3.2
is the root mean square of the fundamental of the voltage.
Basic Requirements
There are several basic needs before we can conduct the IEEE standard tests. In this
section we review these requirements briefly.
Voltage unbalance
The voltage unbalance between phases should not exceed 0.5%. Typically, modern
variable frequency drive meet this requirement.
Frequency
The frequency should stay within ±0.5% of the desired frequency for the test. Any
rapid frequency variation not only affects the motor performance but also it causes
some error in measured data.
Total harmonic distortion
Typically, a switching power supply is used for AC motors. The total harmonic
distortion of the power supply should not exceed 0.05 during tests. The THD is
decreased by increasing the switching frequency.
7.3.3
No-load Test
The no-load test provides the motor information associated with exciting current
and no-load losses. This test is performed by running motor at rated voltage and
frequency with no load connected to the motor shaft. After running the motor at
228
rated voltage and rated frequency for a relatively long time we measure line voltage
line current and the electric power input to the motor. The measured input power
represents the total losses in the motor at no-load condition. The no-load loss consists
of the core loss, stator loss and friction and windage loss. We need to find each of
these losses for later calculations.
Friction and Windage loss
The friction and windage loss, Pfp, may be determined from the power versus voltage
squared curve under no-load condition.
The y axis of this graph is obtained by
subtracting the stator loss (RI) from total input power. We use linear regression to
fit a curve to the measured power at different voltages. The extension of this curve
to zero voltage, provides the friction and windage losses.
Core loss
The core loss, Ph, can be obtained using (7.24) at each test voltage. We can find the
core loss at any desired voltage from a plot of core loss versus voltage.
Ph = Po - Pm - RI
7.3.4
2
(7.24)
Locked Rotor Test
A locked rotor test is used to find the information associated with leakage impedances.
Since the rotor of squirrel cage motor is symmetric, therefore the rotor impedance
is the same regardless of rotor position. Even though the locked rotor test can be
implemented at rated voltage, the guiding principle suggests that the test can be
performed at reduced voltage such that the current is approximately the same as
operating condition. IEEE Std112 suggests the rotor locked test frequency of 25% of
rated frequency.
The locked rotor test needs the following measurements when the rotor is locked
for maximum 5 seconds as well as no load test measurements.
229
e V,: The line to neutral voltage [V]
"
Iir: The line current [A]
" Pie: The total input power [watt]
" fi,: The frequency of the test [Hz]
The calculations procedure is given below:
1. Calculate reactive power at no load(Qo).
Qo =
(7.25)
mVI - P2
2. Calculate rotor locked reactive power (Qir).
Ql, =
(7.26)
mVriIi -P
3. Calculate the magnetizing reactance XM
X gle
mV2
0
Qo-(mI2XS)
(7.27)
+
Xs_
2
Xu
4. Calculate the stator leakage reactance at test frequency:
XS
Qi
[x
+ X,
[x +
mI2,21
X,
+
I
(7.28)
XM.
5. Calculate the stator leakage reactance at rated frequency
X,
=
frX,
fir
(7.29)
6. Use equation 7.27, 7.28 and 7.29 to find Xs and XM. However we need another
relation to find these quantities. The distribution of rotor and stator leakage reactance has relatively little effect on motor performance and analyses. Therefore
230
IEEE Std112 suggests empirical distribution for different motor classes given in
table 7.1. If the motor class is unknown, it is recommended to assume X, and
X, are equal.
Table 7.1: IEEE Std112 empirical leakage distribution for induction machines
Motor Class Description
A
Normal starting torque, Normal starting current
1
B
Normal starting torque, low starting current
0.67
C
High starting torque, low starting current
0.43
D
High starting torque, high slip
1
In order to solve equations 7.27, 7.28 and 7.29, we perform the following steps:
a) Solve equation 7.27 for XM, assuming a value for X, and js
b) Solve equation 7.28 for X,, using the same value of
X.
c) Solve equation 7.29.
d) Solve equation 7.27, using X, from equation 7.29 and the ratio of
Xs
XM
from
equation 7.29.
e) Continue iteration until the values of X, and XM are converged within 0.1%
7. Calculate rotor reactance at tested frequency
X,
(X-)
(7.30)
8. Calculate rotor reactance at rated frequency
XT
=
-Xi
fir
9. Calculate core loss conductance in siemens
231
(7.31)
Cfe =
h (1+
mV2
X(7.32)
X(.32
10. Calculate rotor resistance
Rr = (
7.3.5
- R.9)
(1 +
-
(X)
2 (XsiCte)
(7.33)
Efficiency Test Methods
In general, efficiency of an electric machine is the ratio of output power to total input
power. The output power can be either measured directly or calculated if we know
the total losses. Thus, we can determine the efficiency if two of the three variables
(input power, output power, losses) are know.
IEEE has various methods for efficiency test given below:
1. Method A: Input and output power is measured directly
2. Method B: Direct measurement of input and output with segregation of losses
and indirect measurement of stray-load loss
3. Method B1: It is the same as method B under assumed temperature condition
4. Method C: Duplicate machines
5. Method E: Under load input power measurement with indirect measurement of
stray load losses
6. Method El:Under load input power measurement and assumed stray load losses
7. Method F: Equivalent circuit with stray load loss measurement
8. Method Fl: Equivalent circuit with assumed stray load loss
9. Method C/F: Equivalent circuit calibrated with Method C
10. Method E/F: Equivalent circuit calibrated with Method E
232
11. Method E1/F1 Equivalent circuit calibrated with Method E with assumed stray
load loss
Each of the above aforementioned methods are discussed in detail in [25]. In this
thesis we chose El and E/F tests.
7.3.6
E or El Efficiency Test Method
We discussed the equivalent circuit method in the previous section. In this section we
present the E efficiency test method. For this test we need a variable load attached
to motor shaft. For this purpose we connected an AC servo motor as variable load to
the induction motor shown in figure 7-6. The AC servo motor is set to current mode
so we can control the load torque.
Figure 7-6: Test bed for efficiency test.
In this method we measure the total input power and losses under load. The
output power is determined by subtracting total losses from the input power. Once
we have the input and output power we are able to determine the motor efficiency as
well as output torque. We also need the no-load and locked rotor test results.
It should be noted that hysteresis motor is a synchronous machine while squirrel
cage motor is asynchronous motor. Therefore we decided to compare motors at the
233
same slip rate. The rated slip rate for squirrel cage motor is 11% thus we tested
the hysteresis motor at the same slip for comparison. At this slip rate the hysteresis
motor torque is produced by means of both eddy current and hysteresis.
We can use IEEE STD112 E2 form to calculate the motor parameters shown in
figure 7-7.
234
IEEE
Std 112-2004
IEEE STANDARD TEST PROCEDURE FOR
9.11 Form E2-Method E-E1 calculations
Ohms @ (2)
Cold Stator Winding Resistance Between Terminals (1)
Specified Stator Temperature, t, _(3)_
*C From 6.7.1.1
*C in a 25 *C Ambient, From 3.3.2 c)
(Test)(Standard) Stray-Load Loss, (P'sL) = _
Item
Phase
Serial No.
Model No.
Rating
Frame
Synchronous r/min
Time Rating
Type
Design
Frequency
Volts
Degrees C Temperature Rise
Description (Motoring)(Generating)
(4)
*in W @, I'2, _(5)
A
Source or Calculation
6
Ambient Temperature, in 'C
From test of 6.7.1.2
7
tStator Winding Temp, t,, in 'C
From each point of test 6.7.1.2
8
Frequency, in Hz
Line frequency
9
Synchronous Speed, in r/min
= 120 x (8) / number of poles
10
Observed Speed, in r/min
From each point of test 6.7.1.2
11
Observed Slip, in p.u.
=[(9)- (10)] / (9)
12
Corrected Slip, in p.u.
(10) corrected per 5.3.2
13
Corrected Speed, in r/min
[1 - (12)] x (9)
14
Line-to-Line Voltage, in V
From each point of test 6.7.1.2
15
Line Current, in A
From each point of test 6.7.1.2
16
Stator Power, in W
From each point of test 6.7.1.2
17
Core Loss, in W
From 6.7.2.3
18
Winding resistance corrected to t,
Correct (1) per 5.2.1
2
19
Stator 1 R Loss, in W, at t,
= 1.5 x (15) 2 x (18)
20
Power Across the Gap, in W
= (16) - (17) - (19)
2
21
Rotor 1 R Loss, in W
= (12) x (20)
22
Friction and Windage Loss, in W
= From 6.7.2.2
23
Rotor Current, in A
From Equation 23 using (15) and 10
24
Stray-Load Loss, in W
See 5.7.2.5 for Method E or 5.7.4 for Method El
25
Total Loss, in W
= (17) + (19) + (21) + (22) + (24)
26
Shaft Power, in W
For motor: = (16) - (25)
For generator: = (16) + (25)
27
Efficiency, in %
For motor: = 100 x (26) / (16)
For generator: = 100 x (16) / (26)
28
Power Factor, in %
= 100 x (16) / [1.732 x (14) x (15)]
t,= temperature of stator winding as determined from stator resistance or temperature detectors during test.
Parentheses,
( ),normally used with equation numbers are not used here to
avoid confusion with the form item numbers.
Figure 7-7: E2 form for efficiency calculation [25].
235
7.4
IEEE Standard Test Results
In this section, we present the experimental and analytical results for both squirrel
cage and hysteresis motor. Table 7.2 gives the motor specification. It should be noted
that we replaced the rotor of a commercial induction machine with D2 steel.
Table 7.2: Induction Motor Parameters
4
Number of poles (P)
Number of phase (m)
3
Rated voltage
200 V
Rated frequency
60 Hz
Rotor outer diameter
47.5mm
Rotor inner diameter
12mm
Rotor length
60mm
Gap
0.8mm
Stator resistance
39Q
Number of coil per phase (N)
8 x 80
7.4.1
Squirrel Cage Motor Results
In this section we present the squirrel cage motor test results for no-load, loaded and
locked rotor tests. We will use the no-load and locked rotor test results for equivalent
circuit method. Later we will use all three test results to determine motors efficiency.
We will compare our test results with manufacturer catalogue data given in table 7.3.
Table 7.3: Manufacturer data
Rated frequency
Rated current
Rated voltage
Rated speed
Rated torque
Starting torque
Output power
for squirrel cage motor
60 Hz
0.8 A
200 V
1600 rpm
0.57 N.m
0.7 N.m
90 watt
No-load test results
Table 7.4 gives the measured quantities for no-load test:
236
Table 7.4: Measured no-load
Line to line voltage (V)
Line current (I0)
Total input power (PO)
Figure 7-8 shows the P
test variables
200V
0.45A
26 watt
- RI2 versus squared voltage for squirrel cage motor.
By extending the curve to zero voltage we find the friction and windage loss to be
3.2 watt.
20
15
10
(N
C
6
0L
0
1
2
Voltage squared [V2 ]
3
4
x 104
Figure 7-8: Friction and windage loss for squirrel cage motor.
Once we have the friction and windage loss, we can find the core loss at different
voltage by using equation 7.24. Figure 7-9 shows the core loss for the squirrel cage
motor.
Locked rotor test results
Table 7.5 gives the measured variables for locked rotor test:
We can now calculate the motor parameters given in table 7.6 according to section
7.3.4.
237
15
S10
-
0
0
'
50
100
Voltage [V]
150
200
Figure 7-9: Core loss versus voltage for squirrel cage motor.
Table 7.5: Measured locked rotor test variables
Line to line voltage (Vir)
55V
Line current (I;r)
0.65A
Total input power (Pr)
54 watt
Tested frequency (fir)
15 Hz
Equivalent circuit results
Once we have all parameters shown in figure 7-1 from no-load and locked rotor test
results; we can analyze the equivalent circuit. We developed a MATLAB code presented in Appendix D based on analysis discussed in section 7.1.
Figure 7-10 and
7-11 show the motor torque and output power versus speed respectively. As we can
see the rated torque and output power are very close to the manufacturer's data.
Efficiency test results
We performed El efficiency test method for squirrel cage motor described in section
7.3.6. Table 7.7 shows the test results which is very close to motor specification from
manufacturer given in table 7.3.
Overall, there is a good agreement between the two IEEE standard test that we
performed for squirrel cage motor and the manufacturer's data.
238
Table 7.6: Calculated motor parameters
No-load reactive power (Qo)
136VA
Locked rotor reactive power (Qir) 24.3VA
Air gap reactance (XM)
256 Q
Stator leakage reactance (X,)
26.6 Q
62.2 Q
Rotor leakage reactance (X,)
Rotor apparent resistance (Rr)
35.6 Q
1
0.8
E
0.6
0 0.4
0.2
0
0
500
1000
rpm
1500
2000
Figure 7-10: Squirrel cage motor output torque.
140
120
a> 100
-O 80
0
60
C.
0
40
20
OK
0
500
1000
rpm
1500
2000
Figure 7-11: Squirrel cage motor output power.
7.4.2
Hysteresis Motor Results
In this section we present the hysteresis motor test results for no-load, loaded and
locked rotor tests. We will use the no-load and locked rotor test results for equivalent
239
Table 7.7: EEl test results for squirrel cage motor at rated frequency
Observed slip
0.11
Rated voltage
Line current
Rated frequency
Stator power input
Core loss
Stator RI 2 loss
Power across the gap
Stator resistance
Rotor RI 2 loss
Friction and windage loss
Rotor current
Stray load loss
Total loss
Shaft power
Output torque
200 V
0.8 A
60 Hz
158 watt
15 watt
37.4 watt
105.6 watt
39Q
11.7 watt
3.2 watt
0.66 A
1.6 watt
68.9 watt
89 watt
0.54 N.m
Efficiency
0.563
Power factor
0.57
circuit method. Later we will use all three test results to determine motor efficiency.
No-load test results
Figure 7-12 shows the P,, - RI
2
versus squared voltage for hysteresis motor.
By
extending the curve to zero voltage we find the friction and windage loss to be 4.2 watt.
Once we have the friction and windage loss, we can find the core loss at different
voltage by using equation 7.24. Figure 7-13 shows the core loss for hysteresis motor
at different voltage.
Locked rotor test results
Table 7.8 gives the measured variables for locked rotor test:
Table 7.8: Measured locked rotor test variables
Line to line voltage (V,,)
58V
Line current (Ir)
Total input power (Pr)
Tested frequency (fir)
0.65A
42 watt
15 Hz
240
20
15 ca
3N
10 -
5-
00
1
2
Voltage squared [V2]
3
4
x 104
Figure 7-12: Friction and windage loss for hysteresis motor.
15
10
CU
0
-j
0
0
5
0
50
100
Voltage [V]
150
200
Figure 7-13: Core loss versus voltage for hysteresis motor.
We can now calculate the motor parameters given in table 7.9 according to section
7.3.4.
241
Table 7.9: Calculated motor parameters
204VA
No-load reactive power (Qo)
Locked rotor reactive power (Qir) 54VA
195 Q
Air gap reactance (XM)
77 Q
Stator leakage reactance (X,)
181 Q
reactance
(Xr)
Rotor leakage
45 Q
Rotor apparent resistance (Rr)
Equivalent circuit results
We used the SIMULINK block diagram shown in figure 7-4 to solve the equivalent
circuit. Figures 7-14 and 7-15 show the torque and output power of hysteresis motor
versus speed respectively.
0.4 0.35 -
E 0.3 7a;
0.2
-
0.2
0
0.1
-
0.1
0.05 0
500
1000
rpm
1500
2000
Figure 7-14: Torque versus speed hysteresis motor.
Efficiency test results
We also performed El efficiency test method for hysteresis motor described in section
7.3.6. Table 7.10 shows the results for radial hysteresis motor.
242
50
40
a)
0
a-
30
20
010
0V
0
500
1000
rpm
1500
2000
Figure 7-15: Output power versus speed hysteresis motor.
Table 7.10: EE1 test results for hysteresis motor at 11% slip
Observed slip
0.11
Rated voltage
Line current
Rated frequency
Stator power input
Core loss
Stator R 2 loss
Power across the gap
Stator resistance
Rotor RI 2 loss
Friction and windage loss
Rotor current
Stray load loss
Total loss
Shaft power
Output torque
220 V
0.64 A
60 Hz
101 watt
14 watt
24 watt
62 watt
39Q
6.8 watt
4 watt
0.44 A
1.6 watt
50 watt
49 watt
0.3 N.m
Efficiency
0.5
Power factor
0.4
243
7.4.3
Comparison
Table 7.11 compares the performance of the motors. As we can see from this table, the
squirrel cage motor has higher efficiency, power factor and torque density. However
since the hysteresis motor maximum eligible speed is more than squirrel cage motor,
thus it has significantly larger power density.
Table 7.11: Motors comparison at 11% slip
Performance
Squirrel Cage Hysteresis
Efficiency (%)
Power factor
(%)
Torque density [KPa]
Max. peripheral velocity
[7]
Power density [ w3]t
57
50
57
40
6.4
3.5
220
450
59
81
It should be noted that the maximum achievable rotor peripheral velocity in table
7.11 are calculated only based on rotor material. However due to stress concentration
that occurs in squirrel cage rotor slots, the maximum peripheral velocity for this
motor is less and hence the power density is less.
244
Chapter 8
Conclusion and Future Works
8.1
Conclusion
Self-bearing motors and active magnetic bearings increasingly find application in
industry, thanks to their unique features: contact free support, compact configuration,
maintenance free, suitable to work in vacuum, and etc.
All of these outstanding
characteristics make self-bearing motors very attractive for high speed application.
Introducing hysteresis self-bearing motor, was the starting point of the present
thesis. We designed, built and tested a hysteresis self-bearing motor with multiple
winding first. Later, we introduced the segmented stator design and accordingly a
novel control scheme for induction self-bearing motors. We tested the proposed idea
successfully for the hysteresis self-bearing motor. On a different attempt, we presented
a novel single axis self-bearing motor with zero power levitated configuration. After
we did the first order analysis We built and tested it for the proof of concept.
We started theoretical analysis by developing a SIMULINK model to simulate
nonlinear behavior of suspension forces and moments of hysteresis self-bearing motor
including loop widening due to frequency variation. Among existing hysteresis models
we chose the Chua model because we found it to be very straightforward and also the
ease of including loop widening. We used Chua model to create a SIMULINK block
that models an electromagnetic actuator based on fundamental laws of electromagnetism. We studied nonlinear behavior of suspension forces and moments by using
245
this model. We verified the simulation results experimentally.
We also derived hysteresis motors governing equations from fundamental laws
of electromagnetism.
Our major contribution to this part was including hysteresis
frequency dependency. We verified our model with two different motor sizes and pole
configurations.
In summary, our primary contributions are summarized as follow:
" Introducing Hysteresis self-bearing motor.
" Segmented stator-single winding method for induction self-bearing motors.
" Single axis self-bearing motor design.
" Nonlinear hysteresis analysis for hysteresis self-bearing motors.
" Hysteresis motor modeling including hysteresis frequency dependency.
8.2
Future Works
In the following section we propose some areas for possible future research based on
this thesis work.
8.2.1
Hybrid Motors
As we know, hysteresis motors are a synchronous induction machine with unique
characteristics.
Typically the rotor is made out of hard magnetic materials which
has large resistance compared to aluminum or copper. Therefore the eddy current
torque of hysteresis motor is not as much as the squirrel cage motor where the cage
is typically made out of low resistance material like aluminum. Thus it seems that a
combination of hysteresis and squirrel cage motors can lead to a better asynchronous
performance.
Meanwhile, a combination of hysteresis and reluctance torque seems to be plausible. One can investigate the effect of adding geometrical features on rotor. In other
246
words one can replace the material of a synchronous reluctance motor with hysteresis
material.
8.2.2
Segmented Stator
In this thesis we built a segmented stator for a hysteresis motor. This type of stator
can be used for any induction machine with its own benefits especially for axial flux
type motor. Due to time and manufacturing limits, we used U core actuators for our
design. However the pole face configuration can be optimized for specific applications.
In this thesis we just proposed the idea and made the proof of the concept. However
there is a lots of room for design and analysis of such a stator configuration for
different induction machines, and in particular for axial flux type one.
8.2.3
Single Axis Self-bearing Motor
In this thesis we did the first order analysis for a practical type of self-bearing motor
with only one axis being actively controlled. For the proof of concept we built and
tested a vertical type of the proposed idea. This is an area that needs extensive study
and work for different applications. A detailed analysis is needed for real application
design. The same concept can be implemented for other types of electric motors.
8.2.4
Hysteresis Active Magnetic Bearing
As we mentioned throughout the thesis, one of the major advantage of hysteresis
motor is its solid and smooth rotor which can be made out of high strength steel. On
the other hand, active magnetic bearing's rotor typically is made out of laminated soft
silicon steel to minimize the eddy current and hysteresis loss. Therefore hysteresis selfbearing motors can be considered as an alternative for magnetic bearings especially for
high speed systems. Typically the rotor of industrial machines such as turbines and
compressors are made out high strength steels. Thus a hysteresis magnetic bearing
seems to be a practical research topic.
247
8.2.5
Linear Hysteresis Self-bearing Motor
Considering the fact that the rotor of hysteresis motor is just a simple chunk of steel,
thus makes it attractive for some linear application such as clean room transportation.
Unlike permanent magnet linear motors, the stator can be significantly smaller than
rotor(moving part). The out-gasing is significantly less than other linear motors that
have permanent magnets.
8.2.6
Smart Drive
In this thesis we used voltage-frequency ratio (V) method to control the motor speed.
Induction motors performances are highly improved by using smart drives with sensorless vector control. This area seems to be untouched for hysteresis motors, and needs
to be investigated.
8.2.7
Nonlinear Hysteresis Motor Analysis
In chapter 5 we used the Chua nonlinear hysteresis model to simulate the suspension
forces and moments. However in chapter 6 we used a linearized hysteresis model for
motoring analysis. We suggest the nonlinear hysteresis model for motoring analysis
as well.
8.2.8
Finite Element Method For Hysteresis Motor Analysis
To the best of our knowledge, by the time we finished this thesis there is no FEA
package that can model the entire hysteresis loop. Apparently this is an area that
needs further investigation.
8.2.9
Transient Analysis Of Hysteresis Motor
In this thesis we used equivalent lumped circuit model to simulate hysteresis motor.
The underlying assumption for such a model is sinusoidal steady-state condition. This
method is widely used for motor design; however for transient analysis, a space vector
248
analysis is required. The transient analysis is very important for optimal drive design,
and should be investigated further.
249
Appendix A
Electromagnetic actuator
This chapter describes the basic fundamentals of electromagnetic actuator design. It
also deals with some important characteristics of these actuators such as eddy current
loss, leakage inductances, force and core saturation.
A.0.10
Electromagnet
Figure A-1 shows an electromagnet used to suspend an I-shaped core. The current i,
generates the magneto motive force MMF=Ni. The reluctance of magnetic circuit is
defined:
A
Ni
1
p
p.A
Wb_
In this configuration
R9 =
9
(A.1)
pLo.w.d
Since the permeability for iron core is much larger than air so we can neglect the
reluctance due to iron path.The equivalent electrical circuit gives us:
Ni
Ni powd
2Rg
2 ' g
(A.2)
Thus the magnetic flux density is:
B = po.
250
Ni
2g
(A.3)
Ni g
Ni4
I.
-+
Figure A-1: U core electromagnetic actuator and equivalent circuit.
We could also use ampere law for this circuit as follows:
SH.dl = Ni
B = po. Ni
2g
(A.4)
(A.5)
Flux linkage is defined as number of coil turns N multiplied by flux passing through
the coil i.e. A = N#.
N 2 i poWd
2
A.0.11
g
(A.6)
Electromagnetic Force
The magnetic co-energy stored in a magnetic system such as in A-1 is obtained by:
=
A di
/i0J
(A.7)
In the case of homogenous field in the air gap, the co-energy stored in the volume
of the air gap, V = 2gA is:
251
, 1
2
1
2
Wm = -BHV = -BHA(2g)
(A.8)
The magnetic force produced by an electromagnet can be found by:
8W'
Ox
B 2A
p
(A.9)
By substituting equation (A.5) in equation(A.9) we get:
K (
(A.10)
1
r=-p1 oN2 wd
4
(A.11)
fm =
Where:
The surface force density P in
[2]
on a magnetic material with a field normal
to the material surface is given by
B2
P = B(A.12)
2pio
where (B = ti)
is the magnetic flux density and and [to = 47r x 10-,
and
[to is the permeability of free space. It is reasonable to pick an upper operating flux
density of Bmax = 1.5T , although this can be increased to about 2.2 T with the use
of Cobalt alloys. These values result in an attractive force density of about 106 (
)
(about 150 psi) acting on the magnetic material exposed to the flux.
For B = 1T and Area = 1cm 2 the magnetic force of fm
=
40N = 4Kgf can be
produced.
A.0.12
Inductance
In the field of design and analysis of electromagnetic actuator as well as electric machines; it is very important to understand the concept of self and mutual inductance.
Faraday's law (equation A.14) tells us that a changing magnetic flux will induce an
252
emf in a coil.
J
E.ds = -
(A.13)
B.da
Thus:
dp
dt
dA
dt
(A.14)
ii
is
Figure A-2: Mutual inductance.
Picture two coils next to each other, on a core such as one shown in figure A-2.
The current that passes through each coil is i 1 and i 2 . The magnetic flux can be
easily obtained as follows:
(A.15)
p = (N~i + N 2i2 ) / O
9
The flux linked to each coil is then given by:
A, = N
)i
1
+ N1 N 2 (
i) 2
(A.16)
A = N2 ( 2
4)
i2 + N 1 N 2
(
1oA)
In the above equations we define the self inductances L 1 and L 2 and Mutual
inductance L 1 2
=
L21=
M as below:
253
L1 = N12
(0A)
L2 = N2 (2uOA
\
(A.17)
9/
L12 = L21 =N1N
2
( poA)
\ 9/
So we get:
A1 = Liii + L 12 i 2
A2
(A.18)
= L 2 i 2 + L 21 i 1
The nominal inductance of electromagnetic U core circuit shown in figure A-1 is
defined as:
Lapprox -N 2
2
*/i 0Wd
(A. 19)
g
Indeed Lap,,x overestimates the actual inductance because we have neglected
iron core losses and flux leakage. Electromagnetic inductance is in fact inversion of
dynamic effect of magnetic field.
According to the law of inductance, the induced voltage V in a coil with N turns
equals:
e=Nd =L
dt
dt
(A.20)
If the copper resistance is neglected then the output voltage of the power amplifier
generates a current slope in the coil according to this formula. Obviously the smaller
inductance is the faster current rises. The time constant in an inductor i.e. time the
current reaches to 63% of its final value is L/R. In motors most of the flux produced
by stator crosses the gap. However there are small percentage of flux that doesn't
cross the gap and links to stator windings which is called stator leakage inductance.
254
A.0.13
Eddy current
If a magnetic field with frequency w is applied to a solid core that has a conductivity
of - and permeability of p then skin depth is:
(A.21)
2
6
with o- = 0, the skin depth is infinite and we have the same field flux throughout
the core at every frequency of field oscillation.
However, with a finite
- the field
begins to drop off spatially within the core with the form
H, = ReH,(x)ejwt
Where z is in the horizontal axis and H2(x)
=
(A.22)
Ce-
. Due to the finite conduc-
tivity, eddy currents flow within the magnetic material and essentially block out the
external H-field. As the frequency of field oscillation increases, more of the field is
blocked out, and the amount of field diffusion into the material decreases. A more
detailed discussion on skin depth and eddy currents is given in [21].
The most important issue with eddy current happens at high frequency where
it shields the flux out of the core material, thereby reducing the force capabilities,
frequency response, and power efficiency of the actuator.
In general soft magnetic materials have low conductivity and thus large skin depth.
Ferrite has very low conductivity and is widely used in high frequency devices. However, ferrite materials saturate at about 0.3-0.4 T. Soft composite materials are also
very suitable for high frequency application. Laminated silicon steel is widely used
in transformers and stator for electric motors. Silicon steel is laminated in order to
reduce eddy currents at the power line frequency. For the power line application,
lamination thicknesses on the order of 1 mm are acceptable. However, we can calculate that we need lamination thicknesses about 50 pm in order to work efficiently at
1 kHz for high speed motors.
255
A.0.14
Saturation
Saturation occurs in all iron-core electromagnets. This is a situation where the magnetization does change as we increase the magnetic field intensity (H). Magnetic
saturation is one the major limits for electric machines force density. Soft magnetic
steels used in transformers and electric motors have saturation of about 2 Tesla. Magnetic saturation puts a limit on the specific load capacity for magnetic bearings. For
common available materials the maximum specific load for AMB is about O.4Mpa
which is about quarter of typical mechanical conventional bearing.
256
Appendix B
MATLAB Codes
Appendix D presents MATLAB codes that we developed for this thesis. In the first
section we present the code that we developed for axial flux hysteresis motor modeling. In the second section, the code that was used for induction motor modeling is
presented.
B.O.15
Axial flux hysteresis motor
1
clear all
2
clc
3
close all
4
5
global
J
6
P=2;
Number of pole
7
m=3;
Number of phase
8
w-b=35* (2*pi);
Stator frequency[rad/sec]
9
delta_max=38*(pi/180);
B-H lag angle [rad]
10
tr=14/1000;
Disk thickness
11
r_o= 40/1000;
Outer radius [m]
12
r_i=18/1000;
inner radius[m]
13
g=0.8/1000;
gap[m]
257
Kw=0.8;
% Winding factor
N=8*40;
% Number of turns per phase
J=4.2e-4;
% Rotor moment of inertia[Kg.m^2]
R~e=9;
% Eddy current resistance[ohm]
R~s=5;
% Stator resistance{ohm]
L_s=5e-3;
% Stator leackage inductance
V=6;
% Stator voltage[V]
[H]
mu0=4e-7*pi;
mur=1000;
mu=mu0*mur;
p=P/2;
W=2*Kw*N/p/pi;
r-avg=(r-o+r_i)/2;
% Average radius
r r=r avg;
r-m=p*sqrt(mu*g*tr/mu0);
l=ro-r_i;
kl=rr/r-m;
k2=1/(2*r-r);
V_r=pi*(r_o^2-r_i^2)*tr;
% Rotor volume
ti(1) = 0;
% defining time span for ODE45
dspan=.l;
tspan=[ti(l)
% timing resolution
dspan];
wi(l) = 0;
% omega
(initial speed)
deltai(1)=delta-max;
A=5e-8;
B=9e-9;
time=500;
41
42
for i=l:time
43
numbl=kl^4*(l+k2)^4+2*kl^2*(1+k2)^2*cos(deltai(i))+1;
258
44
denbl=kl^4*(1-k2)^4+2*kl^2*(1-k2)^2*cos(deltai(i))+l;
45
betal=log(numbl/denbl);
46
beta2=(1/(kl^2*(l-k2)^2+cos(deltai(i)))-1/(kl^2*(1+k2)^2+cos(deltai(i))))
47
alphar=(beta1^2+4*beta2^2)/(k2*(8*betal-16*beta2*cot(deltai(i))));
48
alphaz=(beta1^2+4*beta2^2)*sin(deltai(i))/(16*kl^2*k2*beta2);
49
M=2*m*Kw^2*N^2*muO*rr*l*alphaz/(p^2*pi*g);
50
s(i)=(w-b-wi(i))/w-b;
51
R-r=(w-b*m*Kw^2*N^2*Vr*mu)*alphar*sin(deltai(i))/(pi^2*r_r^2);
52
L_hr=(m*Kw^2*N^2*Vr*mu)*alphar*cos(deltai(i))/(pi^2*r_r^2);
53
Zg(i)=j*w-b*M;
54
Zh (i)=Rr+j*wb*Lhr;
55
Ze(i)=Re/s(i);
56
Z-r i)=1 /(1 / Z-h(i) +1 /Z-e
57
Z_s(i)=Rs+j*w-b*L-s;
58
Z(i)=Z-s(i)+1/(1/Z-g(i)+1/Z-r(i));
59
I_S(i)=V/Z(i);
60
I_g(i)=(V-Zs(i)*I_s(i))/Zg(i);
61
I-r(i)=(V-Z-s(i)*I-s(i))/Z-r(i);
62
Te(i)=1.5*M*abs(I_r(i))*abs(I_g(i))*sin(deltai(i));
63
T_L(i)=A+B*abs(wi(i))*wi(i);
64
T(i)=Te(i)-TL(i);
65
Tvar=T(i);
i))
66
[t,w,Tvar] = ode45(@omega,tspan,wi(i), [],Tvar);%ODE45 call
67
sw=length(w);
68
wi(i+1)=w(Sw);
69
wd=wi(i+1);
70
[td,delta,wd] = ode45(@del_f,tspan,deltai(i), [],wd);%0DE45 call
71
delt=length (delta);
72
deltai
73
if deltai(i+l)>deltamax
(i+1) =delta (delt);
259
(i+1)=deltamax;
deltai
end
if deltai(i+1)<-deltamax
deltai
(i+1) =-delta-max;
end
ti (i+1) =ti
(i) +dspan;
tspan=[ti(i) ti(i+1)];
% back emf calculation
It=abs
(Ir (i) )+abs (Ig (i));
C=m/2*IKt*W*(betal*cos(deltai(i))+2*beta2*sin(deltai(i)))*p*mu/4;
S=m/2*I-t*W*(betal*sin(deltai(i))-2*beta2*cos(deltai(i)))*p*mu/4;
Fi=sqrt (S^2+C^2) *2*tr;
epsi(i)=wi(i)*Kw*N*Fi;
end
figure;
plot (ti,wi*60/ (2*pi)
,
'red');
xlabel ('time [secI')
ylabel('omega [rpm]')
title('Speed');
hold
plot (n (1: sn)
60*v (1:sn) ,'o' ) ;
legend('Model','Experiment')
figure;
plot(ti,deltai*180/pi);
xlabel ('time [sec]')
ylabel('delta
title
[deg]')
('Delata');
260
104
105
figure;
106
plot(ti(2:end),Te);
107
xlabel('time[sec]')
108
ylabel(' Torque
109
title('Torque');
[N.m]')
110
il
figure;
112
plot (ti (2:end),abs (I_r));
113
xlabel('time[sec]')
114
ylabel('Current
115
title('Rotor current');
[A]')
116
117
figure;
118
plot(ti(2:end),abs(I_s));
119
xlabel('time[sec]')
120
ylabel('current[A]')
121
title('Stator current');
122
261
B.O.16
Induction Motor
clear all
close all
clc
P=4;
%Number of pole
m=3;
%Number of phase
f=60;
%Stator frequency[Hz]
p=P/2;
%pair of poles
Vs=220/sqrt(3) %Vln
R1=39/2;
%Statro resistance per phase
R2=35.6;
%Rotor apparent
resistance
X1=26.7;
%Stator leakage
reactance
X2=62.2;
%Rotor leakage reactance
Xm=256.8;
%air gap reactance
Pw=3.1;
Ns=2*pi*f/p;
omegas=2*pi*f/p;
for n=1:200
s(n)=n/200;
N(n)=Ns*(1-s(n));
Rr(n)=R2/s(n);
Zr(n)=j*X2+Rr(n);
Za(n)=(j*Xm*Zr(n))/(j*Xm+Zr(n));
Zt(n)=R1+j*X1+Za(n);
262
29
Is (n) =Vs/Zt (n);
30
12(n)=Is(n)*j*Xm/(j*X2+R2/s(n)+j*Xm);
31
32
Pag(n)=m*abs(12(n))^2*Rr(n);
33
Ps(n)=Pag(n)*s(n);
34
Pm(n)=Pag(n)*(1-s(n));
35
Pin(n)=Pag(n)+3*R1*abs(Is(n))^2;
36
Pout(n)=Pag(n)-Pw;
37
Tout(n)=Pout(n)/omegas;
38
eff(n)=Pm(n)/Pin(n);
39
PF(n)=Pin(n)/(3*abs(Vs)*abs(Is(n)));
40
%Rotor Power dissipated
%Mechanical power
% Net output power
%Net output torque
end
41
42
figure;
43
plot (N*60/2/pi, Tout);
44
xlabel('rpm');
45
ylabel('Output
46
figure;
47
plot(N*60/2/pi,Pm);
48
xlabel('rpm');
49
ylabel('Output Power[watt]')
50
figure;
51
plot(N*60/2/pi,eff);
52
xlabel('rpm');
53
ylabel('Efficiency')
54
figure;
55
plot(N*60/2/pi,Pin);
56
xlabel('rpm');
57
ylabel('Input
58
figure;
Torque[N.m]')
Power
[watt]')
263
59
plot(N*60/2/pi,PF);
60
xlabel('rpm');
61
ylabel
('PF')
62
264
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