A Configurable 3-Phase Machine for Laboratory Instruction
by
Eric Gregory Tung
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2006
@ Eric Gregory Tung, MMVI. All rights reserved.
The author hereby grants to MIT permission to reproduce and distribute publicly
paper and electronic copies of this thesis document in whole or in part.
Author...........
De
ent of Electrical Engineering and Computer Science
June 2, 2006
....................
Certified by.
Steven B. Leeb
Professor of Electrical Engineering
Thesis Supervisor
Certified by......
(1-1
Accepted by...........K
........................
James L. Kirtley, Jr.
Professor of Electrical Engineering
Thesis Supervisor
............
Arthur C. Smith
Chairman, Department Committee on Graduate Students
MASSACHUSETTS INSMTUTE,
OF TECHNOLOGY
AUG 1 4 2006
LIBRARIES
BARKER
2
A Configurable 3-Phase Machine for Laboratory Instruction
by
Eric Gregory Tung
Submitted to the Department of Electrical Engineering and Computer Science
on June 2, 2006, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
In order to learn about and work effectively with electromechanical systems, many students need exposure to these systems before completing their education. This thesis work
introduces two novel teaching aids for laboratory classes. The first is a 3-phase axial-flux
machine which can be configured as a permanent-magnet or induction machine with moderate effort for teaching about power electronics. The second is an introductory robot which
demonstrates and controls electromagnetic actuators for teaching an introductory freshman
class.
Thesis Supervisor: Steven B. Leeb
Title: Professor of Electrical Engineering
Thesis Supervisor: James L. Kirtley, Jr.
Title: Professor of Electrical Engineering
3
4
Acknowledgments
I would like to thank Professors Leeb and Kirtley for their assistance, advice, patience,
and guidance over the course of this thesis. This thesis was made possible by essential
funding from the Cambridge-MIT Institute and the Grainger Foundation. The material
herein would not have been possible without the prior work of fellow graduate students
Andrew Thomas and Mariano Alvira. Warit Wichakool, Greg Belote, and Clinton Buie
helped design important subsystems, and along with Alex Crumlin and Candace Wilson,
helped with construction. Finally, I'd like to thank all my family and friends for their
support.
5
6
Contents
1
2
Introduction
15
1.1
3-Phase Machine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Robot . . . . .......
......
..
. . . . . . . . . . . . . . . . . . . . . . .
3-Phase Machine Mechanical Description
2.1
Introduction ....................
2.2
Preliminary Calculations ..........
2.3
The Design Process .............
2.4
Peripherals ....
2.5
2.6
.................
2.4.1
Torque Arm .............
2.4.2
Position Encoder . . . . . . . . . .
2.4.3
Motor controller
2.4.4
Drive Motor . . . . . . . . . . . . .
2.4.5
Tri-Totem Board . . . . . . . . . .
U sage
. . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
2.5.1
Permanent Magnet . . . . . . . . .
2.5.2
Induction . . . . . . . . . . . . . .
2.5.3
Generator . . . . . . . . . . . . . .
2.5.4
M otor . . . . . . . . . . . . . . . .
Future Improvements . . . . . . . . . . . .
2.6.1
Radial Wall Compliance
2.6.2
Mounting Grid . . . . .
. . . . . . . . . . . . . . . . . . . . . .
2.6.3
Standardized Screw Size
. . . . . . . . . . . . . . . . . . . . . .
2.6.4
Fully Symmetric Rotors
35
7
3
3-Phase Induction Machine Electrical Description
37
3.1
Introduction.
37
3.2
3-phase Induction Model . . . . . . . . . . . . . . .
. . .
37
3.3
Improved Single Phase Model . . . . . . . . . . . .
. . .
39
3.4
Analytical Analysis . . . . . . . . . . . . . . . . . .
40
3.4.1
Armature winding resistance
. . . . . . . .
41
3.4.2
Distributed Winding . . . . . . . . . . . . .
42
3.4.3
Magnetizing Inductance . . . . . . . . . . .
43
3.4.4
Leakage Inductance
. . . . . . . . . . . . .
48
3.4.5
Rotor Resistance . . . . . . . . . . . . . . .
53
3.5
4
5
......
................
Measurement and Comparison
. . . . . . . . . . .
56
3-Phase Permanent Magnet Machine Electrical Description
65
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
65
4.2
3-phase Permanent Magnet Model
. . . . . . . . . . . . . . . . . . . . . . .
65
4.3
Sinusoid vs. Tri-totem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.4
Measurement
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
Robot Mechanical Description
75
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.2
The Design Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.3
5.4
5.2.1
Card Rack
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
5.2.2
Wheel Base . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
5.3.1
Circuit Prototyping: Green Bc )ard . . . . . . . . . . . . . . . . . . .
79
5.3.2
Microcontroller: Basic ATOM . . . . . . . . . . . . . . . . . . . . . .
80
5.3.3
Movement: Top Board . . . . . . . . . . . . . . . . . . . . . . . . . .
81
5.3.4
Navigation: LC-resonant wire . . . . . . . . . . . . . . . . . . . . . .
83
5.3.5
Data Collection: Hobo . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5.3.6
Data Collection: Current Sens o rs . . . . . . . . . . . . . . . . . . . .
85
Future Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
Wheel Traction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
Peripherals
5.4.1
8
6
Assessment and Conclusion
87
6.1
3-Phase Induction Machine Objectives .....
6.2
3-Phase Permanent Magnet Machine Objectives . . . . . . . . . . . . . . . .
88
6.3
Robot Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6.4
Checkoff Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
.....................
A 3-Phase Machine Mechanical Drawings
87
93
B Datasheets
107
C Controller Parts and Schematics
113
C.1
C.2
Position Encoder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
113
C .1.1
B ase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
C.1.2
Sensor Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114
C.1.3
Encoder W heel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
115
M otor Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
D Vendor Contact Information
121
E Armature Construction Instructions
125
E. 1
Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
125
E.2
W ind Segm ents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126
E.3
Connect Segments
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
E.4
3-phase Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
128
E .5
Flux Check . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
129
F Permanent Magnet Machine Assembly Instructions
131
G Induction Machine Assembly Instructions
137
H Induction Machine Analysis Scripts
143
H.1 Amplitude and Phase Extraction: fa3.m . . . . . . . . . . . . . . . . . . . .
143
H.2 Error From Sinusoid: err.m . . . . . . . . . . . . . . . . . . . . . . . . . . .
144
H.3 Deriving Parameters From Data: fanal.m
144
H.4 Error From Data: derr.m ........
. . . . . . . . . . . . . . . . . . .
............................
H.5 Deriving Parameters From Dimensions: tmot.m .....
9
145
................
146
H.6 Comparing: fresp.m
I
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Permanent Magnet Machine Analysis Scripts
I.1
..................................
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
152
Motor Constant Identification Via Voltage-speed:
gen.voltage-speed.m
1.6
152
Error From Torque-Current Line:
torque-current.err.m
1.5
152
Motor Constant Identification Via Torque-Current:
torque.current.m
1.4
151
Error From Speed-Voltage Line:
speed-voltage.err.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
151
Motor Constant Identification Via Speed-Voltage:
speed-voltage.m ..........
1.2
149
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
153
Error From Generator Voltage-Speed Line:
gen-voltage-speed-err.m
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
153
J Robot Mechanical Drawings
155
K Robot Controller Parts and Schematics
169
K .1
Top Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
K .2 2 kHz Generator
170
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
173
K .3 LC Sensor Board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175
10
List of Figures
1-1
Photograph of the 3-phase teaching machine . . . . . . . . . . . . . . . . . .
16
1-2
Photograph of the robot . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2-1
Wireframe view of the final motor design showing various features
. . . . .
20
2-2
Important dimensions of the stator . . . . . . . . . . . . . . . . . . . . . . .
22
2-3
Unrolled section of stator showing armature winding pattern
23
2-4
Shaded detail view showing features of stator/rotors in the permanent mag-
. . . . . . . .
net configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
2-5
Combination guide for winding segments and stator support . . . . . . . . .
26
2-6
Detail view of bearings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
2-7
Torque Arm. Holes allow attachment of a spring scale. . . . . . . . . . . . .
27
2-8
Encoder wheel: outer track is CLOCK, inner track is RESET . . . . . . . .
28
2-9
Schematic of encoder circuit, designed by Warit Wichakool
. . . . . . . . .
28
. . . . . . . . . . . . . . . . . . . . . . . . . . .
29
. .
30
2-12 Detail view of drive motor peripheral. Timing belt not shown . . . . . . . .
31
. . . . . . . .
32
. . . . . . . . . . . . . . .
34
2-10 Motor controller schematic
2-11 Example waveforms generated at various points in the motor controller
2-13 3-phase totem controller, based on design by Mariano Alvira
2-14 Axial misalignment of the top and bottom cases
. . . . . . . .
36
. . . . . . . . . . . . .
38
2-15 Complementary rotors: (a) asymmetric; (b): fully symmetric
3-1
Simple model of a 3-phase locked induction machine
3-2
Physical arrangement of a single phase inductor vs. balanced 3-phase inductors 38
3-3
Improved 3-phase model for the locked induction machine . . . . . . . . . .
39
3-4
Single phase of the improved model from the viewpoint of the voltage source
40
3-5
Cross section of stator, showing two turns of armature winding decomposed
41
11
3-6
Progression from single wire to distributed winding . . . . . . . . . . . . . .
42
3-7
Electrical winding angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3-8
Typical solenoid
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
44
3-9
Simplified cross-sectional sketch of magnetic field . . . . . . . . . . . . . . .
45
3-10 Simplified cross-sectional sketch of rotor and stator . . . . . . . . . . . . . .
45
3-11 Sketch of flattened, simplified rotor and stator, part of one phase . . . . . .
47
3-12 Sketch of flattened coordinate system of interest, only currents on top face
are show n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
3-13 Dimensions of interest, one phase highlighted . . . . . . . . . . . . . . . . .
49
3-14 Contour for flux due to rotor current . . . . . . . . . . . . . . . . . . . . . .
54
3-15 Sample data resulting from f a3. m showing voltage and current for a single
phase at 300 H z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
3-16 Sample best-fit model generated by f anal. m and data from locked medium
rotors ...........
.........
..............
.....
......
3-17 Sample predicted 5A-limited torque-speed curve generated by tmot.m
3-18 Sample predicted 12V-limited torque-speed curve generated by tmot
.
59
.
.
61
.m .
.
62
3-19 Sample comparison between measured data and calculated parameters generated by
fresp.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
. . . . . . . . . . . . . .
66
4-1
3-phase model for the permanent magnet machine
4-2
Model for a single phase of the permanent magnet machine
. . . . . . . . .
66
4-3
Simplified cross-sectional sketch of magnetic field with magnets added . . .
67
4-4
Model of the brushless 3-phase PM machine . . . . . . . . . . . . . . . . . .
69
4-5
Sketch of 3-phase current from tri-totem vs. rotational angle
. . . . . . . .
70
4-6
Generated Voltage vs. Drive Speed [rad/sec] . . . . . . . . . . . . . . . . . .
71
4-7
Maximum torque vs. DC current . . . . . . . . . . . . . . . . . . . . . . . .
72
4-8
Rotational speed vs. DC voltage
. . . . . . . . . . . . . . . . . . . . . . . .
73
5-1
Wireframe view of the final robot design . . . . . . . . . . . . . . . . . . . .
76
5-2
Sketch of final card rack design . . . . . . . . . . . . . . . . . . . . . . . . .
77
5-3
Sketch of wheel lock
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5-4
Two motor configurations, configuration (a) also shows a possible battery
placem ent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
79
5-5
Picture of green board, designed by Mariano Alvira . . . . . . . . . . . . . .
80
5-6
Schematic of top board . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
5-7
Schematic of 2 kHz generator . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5-8
Schematic of LC sensor
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
84
5-9
Picture of current sensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
85
6-1
Sample evaluation sheet for the 3-phase induction machine
89
6-2
Sample evaluation sheet for the 3-phase permanent magnet machine
6-3
Sample evaluation sheet for the robot
. . . . . . . . .
. . . .
90
. . . . . . . . . . . . . . . . . . . . .
91
E-1
Winding guide; one side marked for winding direction consistency . . . . . .
126
E-2
Half-wound segment showing direction of winding . . . . . . . . . . . . . . .
126
E-3
Tape keeps the segment from shifting as you wind other segments . . . . . .
127
E-4 Two finished, taped segments . . . . . . . . . . . . . . . . . . . . . . . . . .
127
E-5
128
Connections between phase segments; distance is exaggerated for clarity . .
13
14
Chapter 1
Introduction
People are increasingly dependent on a web of electrical and electronic devices. One end of
this web is the initial power generation from various mechanical sources. Electrical energy
is also used to power essential electromechanical actuators. In order to learn about and
work effectively with these systems, many students need exposure to electromechanical
systems before completing their education. This thesis work introduces two novel teaching
aids for laboratory classes, one focusing on 3-phase power electronics, the other focusing
on an introductory robotics application for demonstrating and controlling electromagnetic
actuators.
1.1
3-Phase Machine
In applications where power density is important, such as power generation or industrial
machinery, 3-phase machines are preferred over single-phase machines. Students rarely have
experience with 3-phase devices. In Chapter 2 we describe the design of a 3-phase axial-flux
machine for laboratory instruction. The machine is configurable as a motor or generator
and as an induction or permanent magnet synchronous machine. Chapter 3 describes the
electrical characteristics of the 3-phase machine when configured as an induction motor.
Chapter 4 describes the electrical characteristics of the 3-phase machine when configured as
a permanent magnet synchronous motor. This teaching machine was successfully used in
a power electronics class at MIT to teach about 3-phase power and power electronic motor
drivers.
15
Figure 1-1: Photograph of the 3-phase teaching machine
1.2
Robot
Robotics is a rapidly growing field.
From small vacuum cleaners to large autonomous
vehicles, robots are an increasingly visible application of technology. At the same time,
robots can be excellent, fun vehicles for education as shown by the success of classes such as
2.007, 6.270, and MASLab at MIT and national competitions such as FIRST and RoboCup.
Robotic products excite students and are terrific platforms for understanding the use of electromagnetic actuators. In Chapter 5 we describe the design of an expandable robot suitable
as a platform for an introductory-level class. This robot was successfully used in a freshman
seminar at MIT to teach about simple circuits with the motivation of environmental data
collection.
Finally, Chapter 6 concludes with assessment techniques used in class with the 3-phase
machine and the robot.
16
Figure 1-2: Photograph of the robot
17
18
Chapter 2
3-Phase Machine Mechanical
Description
2.1
Introduction
This chapter describes the mechanical design of a 3-phase electric machine for teaching
about motors and generators, including both induction and permanent magnet synchronous
machines.
This chapter also includes details about peripherals that are to be used in
conjunction with the teaching machine in Section 2.4. Section 2.5 provides an overview of
use of the machine, and finally, overall performance is discussed in Section 2.6.
The teaching machine was intended to be used as a teaching aid in a large class on
power electronics, so we wished to construct on the order of 100 machines. This is a large
number from the perspective of an individual constructing machines, so we also wished
the design to be manufacturable by local shops. A design goal was to make the machine
able to be assembled in a variety of configurations using only simple tools. Cost was a
concern because of the number of machines desired and because educational budgets are
limited. All of these factors influenced the design process, which is described in more detail
in Sections 2.2 and 2.3.
A complete set of mechanical drawings and a list of parts can be found in Appendix A.
19
Top Case
Drive Motor
CD Encoder Wheel
Encoder
Figure 2-1: Wireframe view of the final motor design showing various features
2.2
Preliminary Calculations
We decided to aim for a machine capable of electromechanically converting 100W and
powered by a 24V peak source. These parameters were determined based on constraints of
existing lab kits (which are capable of 24V and about 100W) and safety (by choosing a low
voltage, students would not be at risk of shock injury). The machine will be a 3-pole pair,
3-phase machine and operate around 60Hz.
To calculate the current the machine would have to handle, divide the peak voltage by
x72 to get RMS voltage and by another v3 to get the line-to-neutral voltage because this
is a 3-phase machine. Then the RMS line-to-neutral is
24V~ = 9.798V
v 2/3
20
Each phase must handle a third of the power in the machine. Thus the current in each
phase averages
100W
3 9.798V = 3.402A
Thin wire is preferred for armature windings because it permits a smaller air gap,
Both wire resistance and heat
which is especially important for the induction machine.
dissipation influence the minimum wire size; heat is initially assumed the limiting factor
and the resulting resistance will be checked to ensure it is an acceptable value from the
perspective of machine performance.
To check that heat dissipation is not a problem, use the rule of thumb that if the current
density is less than 2. 106A, the wire will remain an acceptable temperature. The necessary
cross-sectional area is then
3.402A
3.402A
2 .106A
= 1.70 . 10 6 m 2
This cross-sectional area corresponds to AWG 15 (dia = 1.45 mm, area = 1.65 mm 2 ).
Knowing that the performance of an induction machine is highly dependent on the air gap
between rotor and stator, this wire size seemed larger than comfortable. Since the 2. 1064
was a rule of thumb and some warming is acceptable, AWG 18 (dia
=
1.024 mm, area
=
0.823 mm 2 ) was used instead. This results in a current density of 4.13 - 106k.
For reasons of cost and availability, the T107/65/25-3F3, a toroidal ferrite core from
Ferroxcube (see Appendx B for datasheets), was chosen for the stator. From the materials
datasheet, the core saturates at a field of
0.4T around the machine operating conditions.
By Faraday's law, the number of turns over an entire phase necessary to generate this field
can be calculated.
N
V
B o Acrossw
9.798V
0.4T . 525mm 2 . 377a
-=123.75
sec
where
Bo
Acros s
W
Magnetic flux density
C ross-sectional
area of stator
Rotational speed
21
0.4T
525 mm 2
377
La
sec
W
Ri R~o
Figure 2-2: Important dimensions of the stator
Ri
Inner radius of stator
3.175 cm
R,
Outer radius of stator
5.398 cm
w
Width of stator
2.54 cm
p
Number of pole-pairs
3
Since a loop around the stator takes approximately 10 cm, the windings require about
12 m of wire per phase. Since there are 3-pole pairs per phase, and each pole pair requires
two winding segments, there are 6 segments per phase. The number of turns per segment,
Na, is then
N
Na
6
20.6
Finally, resistance in the wire must be verified to ensure it is not a major power sink. In
the worst case, all power is dissipated in the windings and none is left to couple the rotor
and stator.
V
I
9.798V
- R=2.880
=
3.402A
_
Thus the wire resistance must be much less than 2.88Q. Assuming the winding is made of
copper wire, and remembering that the DC resistance of a wire is
R =
pL
Aw
-1.68
-10- 8Qm*.12m
525mm 2
22
=3.84 - 10-4Q
The resistance of the wire is likely not a problem for developing a machine with acceptable
performance.
2.3
The Design Process
One early decision that had to be made was if the machine should use an axial or radial
flux pattern.
A desire for easy re-configurability suggested an axial flux pattern, since
the gap between rotor and stator could be easily varied by adjusting the axial position of
the rotors. A similar exercise with a radial flux pattern machine would require either a
variable-diameter armature or variable-diameter stator, both of which seemed too complex
to be worthwhile or which would require many parts which differed in only small aspects.
The stator was wound in a pattern like that indicated in Figure 2-3. In the figure, one
phase is highlighted while the other two are faded out for clarity. Bold arrows represent the
direction of the magnetic field, light arrows show the direction of current in the windings.
Connections are exaggerated for clarity.
Winding Segment
.Phase
A
Pase B
-
-
~Phase C
Figure 2-3: Unrolled section of stator showing armature winding pattern
For a permanent magnet machine, the rotors should consist of strong magnets backed
by a magnetically conductive material such as steel. For an induction machine, the rotors
should consist of a conductive yet magnetically transparent material such as copper backed
by a magnetically conductive material such as steel. To save on materials and storage space
the rotors can be combined by noting, approximately, it does not matter what is on the side
facing away from the stator because the rotor steel will guide the magnetic field. Therefore
23
it is possible to have a single magnet-steel-copper rotor that is flipped depending on the
intended machine type. Other rotor types are possible, such as for a doubly-fed induction
machine, but we will not discuss them.
The rotors must be supported as they turn, and must also have some way of transferring
mechanical energy out (in the case of a motor) or in (as with a generator). This is easily
solved with the use of a keyed shaft; the shaft supports the rotors while the key links torque
on the rotors and torque on the shafts. The dimensions of the rotors and desired magnets
were derived from the stator; the outer diameter of the rotor was selected to be slightly
larger than the outer diameter of the stator. The magnets were to be placed in a circle
approximately the same size as the stator, limiting the maximum width to 27r(R 02+Ri)/2
2p
The magnets were sourced from a surplus site, and were chosen to be a reasonable form
factor that was available in both ceramic and rare-earth (NdFeB) materials. This allows
for experiments with differing-strength magnets. More details about the magnets may be
found in Appendix A.
A case is necessary to contain and support the machine. One design idea consisted of a
base plate with two walls for support of the keyed shaft, but this was rejected on the basis
of inadequate (mechanical) safety. Another potential design was a cylinder with removable
endcaps; this would be much safer during operation but reconfiguration would be difficult.
The final design is a mix of these ideas - a case that easily opens for access to the stator
and rotors, but is closed during operation. For simplicity of construction, a rectilinear case
design was chosen instead of a cylindrical case design. For simplicity of operation, the top
case is hinged instead of being completely removable so that the top cannot be lost. For
future expansion, the case is longer than it needed to be so that additional rotors or devices
could be placed inside the case; expansion ledges were also added so that additional devices
could be mounted outside the case. Large windows on the case permit students to see the
machine while it is working, and small holes in the windows enable wires (such as for the
stator windings) could exit or enter the case.
Winding the stators is a significant labor because of the number of machines to be built.
Construction can be simplified by using some sort of guide to keep winding segments in
place. The guides do not have to be removable as long as the winding guides are cheap,
simple to produce, and will not interfere with the normal operation of the machine.
A
simple guide is a circumferentially ridged ring around the stator. Given that the ring would
24
Permanent Magnets
Steel Disk
Cop er Disk
_Spacer
01
Lock Collar
Mountin Hole
LII'
Figure 2-4: Shaded detail view showing features of stator/rotors in the permanent magnet
configuration
already be secured relative to the stator in order to hold windings in place, the utility of
the ring can be extended to support the stator by securing the ring relative to the case.
Figure 2-5 shows a front view of the final winding guide. The winding guide has spaces for
18 winding segments because this is a 3-phase machine, and each phase has 3 pole-pairs and
thus requires 6 winding segments. The relatively complex shape of the winding guide would
be tedious to conventionally make on a mill, so instead it was shaped by laser cutting. For
aesthetic reasons and manufacturing speed, the guide material was chosen to be transparent
acrylic. Instructions for the construction of wound stators can be found in Appendix E.
The shaft must be supported and kept it in place. With the permanent magnet machine,
there can be large axial forces on the rotors if they become too close to the stator, so there
must be some way of ensuring the rotors are axially stationary. The keyed rotors are secured
to the shaft using shaft collars and a spacer between them. This axially links the rotors
to the shaft. The shaft itself then rests on a pair of ball bearings, allowing the shaft to
25
Guide
Rid e
Stator
SUpports
Figure 2-5: Combination guide for winding segments and stator support
easily rotate, and on the outside of the case there is a thrust bearing and a shaft collar.
The outside shaft collars keep the shaft (and therefore the rotors) from moving axially,
while the thrust bearing enables rotation. Theoretically only the outside shaft collar and
thrust bearing on one side of the machine is needed, but for simplicity of operation, they
are included on both sides since the user will probably not know without testing which side
requires them. Figure 2-6 shows a detail view of the final bearing mechanism. More details
about the bearings may be found in Appendix A.
2.4
Peripherals
Several peripheral devices were designed to be used in conjunction with the machine. These
peripherals enable the machine to be used in a variety of ways and a variety of experiments.
2.4.1
Torque Arm
The torque arm, shown in Figure 2-7 enables the student to measure the stall torque of the
machine, or quantify the torque applied to the machine. The torque arm fastens to the shaft
by a screw that compresses the "gripping fingers". A spring scale can then be attached to
one of the holes along the arm. The torque arm is only intended for use while the shaft is
not moving; safety concerns about a potentially rapidly rotating arm caused us to make the
arm long enough to prevent a full rotation of the shaft while properly attached. Figure 2-7
shows a picture of the torque arm. A mechanical drawing may be found in Appendix A.
26
Thrust Bearin
Ball Bearing
Lock Collar
Figure 2-6: Detail view of bearings
Urimizginug
E
2
0
0Z1
10 cm
2.5 cm
Figure 2-7: Torque Arm. Holes allow attachment of a spring scale.
2.4.2
Position Encoder
The position encoder, designed by Warit Wichakool, enables the machine user to determine
the relative position, and thus velocity or acceleration, of the shaft. While a permanent
magnet motor can be run open-loop, it is unstable and so the loop must be closed. Knowing
the frequency of rotation is also important for other calculations, such as determining slip
in an induction machine.
Position sensing is accomplished by an encoder wheel (Figure 2-8) attached to the shaft
which which is read by an encoder board (Figure 2-9).
27
The encoder wheel consists of
two rings of alternating white and black regions; one ring alternates every time the phase
advances one winding; the other ring acts as a marker for the phase. The encoder board
simply reads the brightness of the wheel and buffers the sensor signal to solid logic levels.
Layouts and a part list may be found in Appendix C.1.
Figure 2-8: Encoder wheel: outer track is CLOCK, inner track is RESET
+5v
25v
iCypsi
8k
33k
14
5V+
158k3.k
k
Phctodicde/transistor
6
10
c
+
A38
4+5.
'5k
d
k
7
Out
74HC14
lo
.8
0k
U
?4HCk
'
3.3k
V
18047kCbypass
15k
-D
686k
4+5v
L4358
- 4
15
3.3k
1ek
Photodiode/transistr
I
>nl
689k
P
t
k
t
74HC14 U
108k
,
I
I
+5V
.
.
r----
3.3k
Figure 2-9: Schematic of encoder circuit, designed by Warit Wichakool
28
2.4.3
Motor controller
The motor controller generates a 3-phase control signal for use by the totem board from a set
of square waves. The input may come from the position encoder (in the case of a permanent
magnet motor) or from an onboard generator (in the case of an induction motor) and may
be selected by S1. The onboard generation of square waves can be accomplished in a variety
of ways. The controller shown in Figure 2-10 uses a Schmitt trigger. Board layout and a
parts list can be found in Appendix C.2.
-R1
3.3k
3-
i
R2
S
10k IC5D
Z5
4.3k
C2
74HC14
VCC
74HCI, N
OND
CVA
JP1
VCC
GNO
Si
To Encoder
-[i
74ALSW0N
R7
VC1B
2k
4
CLOCK
R4
LM1N510
E
AL0N
ICC
VCTRDIV16
S5CT=0
IC7
GND
2k
GND
GNO11
PWM
RSTs
74ALS0
M1
IC3A
1D
J2
G33CT=15
~O8
:1 N48
14
A COUNT
IC2A
8
[2
[41
BCUNT
74LSOBN
O
6A:r
1
COUN
1C2B
74LSOON
GND 74S-138N
Figure 2-10: Motor controller schematic
Figure 2-11 shows some sample waveforms generated by the controller based on input
from the position encoder described in Section 2.4.2. The 74163 converts the CLOCK and
!RESET signals into a binary count on the COUNT lines, with A COUNT being the least
significant bit and C COUNT being the most significant. The 74LS138 then demultiplexes
the COUNT signals into 6 different control lines, which are taken pairwise and put through
a 74LS00 to generate the preliminary control signals. At this point, the low-side control
signals are correct, but one more stage of processing is needed for the high side. As noted in
Section 2.4.5, the high side control signals must not have a duty cycle of 1 for the tritotem
board to function properly. In Figure 2-10, a relatively high frequency PWM signal is
29
generated, with the duty cycle controllable by potentiometer R5. This PWM signal is then
ANDed with the preliminary high side control signals via a 74LS08 to form the final high
side control signals.
Res et
Clock In
A Count
B Count
C Count
Low A
Low B
Low C
PWM
High A
High B
High C
Figure 2-11: Example waveforms generated at various points in the motor controller
2.4.4
Drive Motor
The drive motor, shown in Figure 2-12, enables the student to use the machine as a generator
by providing motive power, or to control the slip rate of the induction machine. The motor
is attached to the base via a mounting bracket; the motor is coupled to the shaft via timing
pulleys and a timing belt. A mechanical drawing of the mounting bracket may be found in
30
Appendix A.
M
jutig Bracket
Drive Motor
Timing Pulleys
Thumbscrews
Figure 2-12: Detail view of drive motor peripheral. Timing belt not shown.
2.4.5
Tri-Totem Board
The totem board, designed by Mariano Alvira [1], enables the student to provides 3-phase
power to excite the armature windings. It receives a 3-phase control signal from the finger
connectors, then amplifies the signals via a IR2125 gate driver and a 40N10 NFET. The
3-phase control signal may originate from the motor controller or any other source. Note
that the totem board does not have any safety interlocks on the control signals; for example,
it is possible for both high and low switches in the same totem to be active at the same
time, thus shorting Vdd and ground. It is up to the user to ensure such undesirable states
do not occur. If correctly built, the motor controller mentioned in Section 2.4.3 ensures
proper operation. Also note that the IR2125 cannot maintain a duty cycle of 1 when used
as a high side gate driver; a path to ground must be present with a minimum frequency of
approximately 400 Hz when the high side is active.
31
%C-
z:
MEEM
r
Low C
High VC%
CIts
Low
43
C 0
B2
High B
<r
.a E
0
4113
2
0
Iw
Low Am
High A e
V
-I
Gi
0
c~
o
0
Q
rr
U
01
M0
Cn
3
-4
0
I_
Figure 2-13: 3-phase totem controller, based on design by Mariano Alvira
2.5
Usage
This section briefly describes how to configure the teaching machine as a motor or a generator and as a induction machine or a permanent magnet machine.
2.5.1
Permanent Magnet
When used as a permanent magnet machine, the rotors should be oriented such that the
magnets are facing the stator. In this configuration, the copper disk serves no purpose and
may be omitted if desired.
There is a large attractive force between permanent magnets and the stator; a temporary
spacing material, such as cardboard, should be used when assembling the PM machine. If
this spacing material is not used, the rotors will attach themselves to the stator; separation
is not easy and risks damaging the armature. The insulation on the windings may be
abraded, leading to reduced performance if the windings short. More detailed instructions
for PM machine assembly can be found in Appendix F.
32
2.5.2
Induction
When used as an induction machine, the rotors should be oriented such that the copper
side is facing the stator. In this configuration, the magnets serve no purpose and may be
omitted if desired.
The torque produced by the induction motor depends heavily on the gap between the
rotor and the armature.
While this should be as small as possible to maximize power
transfer to the rotor, contact between rotor and armature could lead to the insulation on
the windings being damaged and possibly shorting segments or phases across the rotor.
Thus it is especially important to make sure the gap is large enough and the lock collars
snug enough that this does not happen. More detailed instructions for induction machine
assembly can be found in Appendix G.
2.5.3
Generator
When used as a generator, some prime mover must be connected to the shaft. We use the
drive motor from Section 2.4.4. Assemble the the induction machine or permanent magnet
machine as detailed above, then attach the large timing pulley to the right side of the shaft
via the set screw. Next, loosely attach the motor bracket to the base via thumbscrews (but
do not yet tighten). Figure 2-12 shows a partially connected drive motor at this stage of
construction. Loop the timing belt over the two timing pulleys, then tighten the belt by
moving the motor bracket. Finally, tighten the thumbscrews.
2.5.4
Motor
When used as a motor, there must be a source of 3-phase excitation to the stator. For
class experiments, the motor controller described in Section 2.4.3 and the tri-totem board
described in Section 2.4.5 are used, optionally using the position encoder from Section 2.4.2
if operating as a permanent magnet motor. An alternate choice is a 3-phase generator such
as the HP-6834B.
2.6
Future Improvements
There are some improvements that could be made to the machine that could be addressed
for future revisions. These include additional compliance for radial walls, finished expansion
33
ledges, and fully symmetric rotors.
2.6.1
Radial Wall Compliance
The most important change is to include additional compliance on the radial walls. Due
to imprecision in assembly of the wooden frame, the top and bottom cases do not always
align, as in Figure 2-14. This misalignment causes the ball bearing to try to tilt in order to
fit into the cavity, thus increasing friction on the shaft. Similarly, if the cases are misaligned
in the other axes, the ball bearing may be compressed and unable to rotate freely. If the
frame is misaligned, there will be resistance to closing the case. This problem has proven
to be a minor one in the first run of prototype machines. Tolerances on the machining of
the walls generally permits good machine operation.
Figure 2-14: Axial misalignment of the top and bottom cases
A solution that we successfully used was to expand the top cavity in both the axial and
the radial dimensions, thus accepting some misalignment. If the cavity in the bottom case
were expanded instead, the ball bearing would be able to shift and would not provide stable
support. Other solutions which would have to be done at the time of initial construction
include making the case out of a more uniform material such as metal or the addition of
alignment pins.
34
2.6.2
Mounting Grid
Another important change is adding additional mounting holes on the base during construction. Our machines were intended to have a smooth finish, with mounting holes being added
as needed. While a smooth surface is visually appealing, we found it was time-consuming to
add mounting holes. However, the necessary mounting holes are complete and are likely sufficient for foreseeable tasks. A regular hole pattern on both the expansion ledges and inside
the case would be easier to add during construction; properly designed mounting brackets
would then allow the use of currently unimagined devices. Note the mechanical drawing of
the base in Appendix A reflects the additional holes added after initial construction.
2.6.3
Standardized Screw Size
An improvement that goes along with the mounting grid is the standardization of mounting
screw size. The position encoder (Section 2.4.2) and motor bracket (Section 2.4.4) do not
use the same size screws because of a lack of foresight. Redesigning one or both of these
in concert with implementing a mounting grid would reduce the number of different parts
necessary to use the teaching machine.
2.6.4
Fully Symmetric Rotors
The final change we recommend regards the magnet pattern on the rotors. We intended
the rotors to be symmetric to reduce the variety of parts; however, as originally designed,
the rotors are not fully symmetric. Note from Figure 2-3 that the direction of the magnetic
field on one side of the stator is opposite the direction of the field on the other side. Thus
the magnets on the rotors should oppose each other, i.e. north opposite north and south
opposite south. We might arrange the magnets for the PM machine as in the top part of
Figure 2-15a. When the rotor is flipped for the other side of the stator, half of the magnets
oppose their own direction and are as desired, but the other half are across from the other
direction, and so the interaction of those magnets and the stator cancel instead of doing
useful work. We did not discover this problem until the disks had been ordered, and so we
worked around the problem by using two different patterns such that the magnets present
the same pole to the corresponding magnet on the opposite disk as in Figure 2-15a. The
magnet arrangement in Figure 2-15b would have enabled the use of only one type of rotor.
35
S
S
S
S
N
N
(a)
(b)
S
S
S
S
0
S
N
N
Figure 2-15: Complementary rotors: (a) asymmetric; (b): fully symmetric
36
Chapter 3
3-Phase Induction Machine
Electrical Description
3.1
Introduction
This chapter describes the electrical performance of the teaching machine when configured
as an induction motor. An important measure of a motor's performance is the torquespeed characteristic. In order to predict this characteristic, the induction machine must
be modeled.
Sections 3.2 and 3.3 propose a model for the 3-phase induction machine.
Section 3.4 derives model parameters from physical parameters and electromagnetic theory.
Finally, the model is verified with measurements of the system and a torque-speed curve
for the model is calculated in Section 3.5.
3.2
3-phase Induction Model
A high-level model of the 3-phase induction motor can be thought of as in Figure 3-1. A
3-phase source drives the stator. The stator presents a 3-phase inductor, in parallel with an
ideal 3-phase transformer whose secondary links a rotor resistance. The ideal transformer
is actually the interaction between the stator and the rotor via magnetic fields, the 3-phase
inductor is the magnetizing inductance, and the rotor resistance is a function of slip, i.e.
the difference between the shaft rotational frequency and electrical frequency.
This model should be verified with measurements of the physical machine, but a single
phase of the motor cannot always be isolated when taking measurements. However, it is
37
Rotor
3-phase source
Vb(t) = Re{vote e
}
-'
Stator
V(t)
-- -- --
Rotor Resistance
Re{vewte"3 }
Figure 3-1: Simple model of a 3-phase locked induction machine
sometimes convenient to analyze a single phase machine. In order to reconcile measurement
with analysis, the difference between a 3-phase system and a single-phase system must be
considered.
+
ia
La
V (t)
+
vs.
L1
Lt
60 -:
Sck
Figure 3-2: Physical arrangement of a single phase inductor vs. balanced 3-phase inductors
Suppose there is a balanced Y-arrangement of inductors as in Figure 3-2 and the resulting
magnetic flux in the coils during sinusoidal steady state is to be determined. The 3-phase
case is similar to a single-phase case, but there is additional flux contribution from the
other phases. Call these Lab for the mutual inductance of the A and B phases, and Lac
for the mutual inductance of the A and C phases. Because the arrangement is balanced,
by symmetry Lab = Lac.
Again by symmetry, flux components not in the direction of
the flux due to phase A from phases B and C cancel. From the geometry of the setup,
Lab ~
-
cos(60 0 )ILbI = - cos(60 0 )La. By Kirchoff's current law, Ia = -(Ib+Ic). Assuming
sinusoidal steady state operation, the phase voltage complex amplitudes (V, Vb, V) can be
38
described in terms of the current phasors (Ia, Ib, I,) and winding reactances (Xa, Xb, Xc):
Xa =
wLa
Xab =
WLab
Xac =
WLac
and
Va = jXa - Ia + jXab - lb + jXac - Ic = jXa - Ia + jXab - (-1a)
Factoring out Ia,
Va = j(Xa - Xab) - la
Using Xab
=
Va =j(Xa+L)
Xa3
22
a =j(3Xa)-Ia
So the field in a stator phase due to a 3-phase arrangement of inductors is approximately
3 that of the field due to a single-phase arrangement given equal inductors and balanced
3-phase excitation in current.
3.3
Improved Single Phase Model
A more accurate model of the machine reflects some resistance in the stator windings, Rarm,
and some magnetic flux that fails to couple the stator and rotor, Lleak. When the rotor is
locked, the induction machine can be modeled as in Figure 3-3 [3].
R.a,,
~
Leaic
Rrot
La|
Figure 3-3: Improved 3-phase model for the locked induction machine
In a balanced three-phase machine, central nodes are always at OV. When considering a
39
single phase of the machine, La acquires an additional factor of 1 as argued in Section 3.2.
3L
La
Lmag
A single phase then looks like Figure 3-4. For clarity, the impedance can be broken down
+
R arm
Lleak
t_:
-S
-a
Figure 3-4: Single phase of the improved model from the viewpoint of the voltage source
into series and parallel components
Z, (jw) = Rarm + jW Lleak
ZPOW) =
jwLmagRrot
jwLmag + Rrot
The impedance of the entire single phase is then
+ .jc4 Lmag Rrot
Zphase(UW) = Zs + Zp = Rarm + j(A Lleak
jwLmag + Rrot
The variable Rrot now and henceforth will represent the reflected rotor resistance.
The
following section derives expressions for each parameter based on physical dimensions of
the induction machine.
3.4
Analytical Analysis
This section derives analytical expressions for the model parameters Rarm, Lmag, Rrot, and
Lleak using physical dimensions of the induction machine and electromagnetic theory. A
matlab function to evaluate the expressions for these parameters, tmot.m, is included in
Appendix H.5. The derivations in this section are largely based on class notes for 6.131 [6].
40
3.4.1
Armature winding resistance
Ignoring skin effects (which may be important) the resistance of the armature winding Rarm
is approximately the resistance of the wire. The windings on the armature vary in depth
and position due to individual construction; for simplicity, we assume a uniform two layer
winding throughout the armature. We define the following variables
Rarm
Resistance of armature winding,
1
Conductivity of copper, 5.9 - 10 7 (Q)-
a-
Cross-sectional area of wire, 8.229
A,
.10-
1
7
m2
rwire
Radius of wire, 5.12. 10-4 m
1
Length of armature wire winding, Nalturn
Na
Number of turns per phase, 120
lturn
Length of wire per turn around the stator
r.
Stator
-1-
I
__
4 Rwre
-. 4<-
_+
I
Figure 3-5: Cross section of stator, showing two turns of armature winding decomposed
Figure 3-5 shows 2 turns of wire around the stator, which can be thought of a four
segments of length w, four segments of length R, - Ri, and the corners. By inspection,
21
turn = 4w + 4(Ro - R ) +
2
7rrjire + 2 lr3 rwire
and the total length of the windings is
1 = Na - 2 - (w + RO - Ri + r2rwire) = 120 - 2 - 5.0841 cm = 12.20 m
41
The resistance of the windings is then
Rarm
-
12.20 m
5.9 -107 1O- 8.229 - 10-7M2
-
0.251Q
Note this underpredicts wire length (and therefore underpredicts Rarm) since inter-segment
wire has been ignored. Skin effect ([4], chapter 10.7) and increased resistance due to any
heating, both of which would also increase Rarm, have also been ignored.
3.4.2
Distributed Winding
We will briefly consider the impact of distributed windings. The following sections assume
that winding segments consist of a single wire loop; in reality, winding segments are composed of about 20 loops which are spread out over some non-negligable distance. This has
the effect of "smearing out" the field. The ratio between the 1-conductor case and reality
is called the winding factor, k,.
0
..------.-
Magnetic field
Wire
Tifff
Stator
...
Sum of field
contributions
0/2
Figure 3-6: Progression from single wire to distributed winding
Figure 3-6 shows the progression from a single wire to distributed winding. The total
current in each stage remains constant. For simplicity, only the fundamental of the magnetic
field is considered. The vectors in the top row represent the contribution of each wire to
the sinusoidal magnetic field, the bottom row shows the sum of the contributions. In the
limit of an infinite number of wires, the winding factor is the ratio of the chord length to
the arc length. The chord length is 2r sin(u), the arc length is r6O,
42
and so the winding
)
factor is s"
2
2nt
O= 27c/6
Figure 3-7: Electrical winding angle
As Figure 3-7 shows, the magnetic field due to the armature windings does a complete
N-S-N (27r) pattern over 2 segments of the same phase. In between are 4 other winding
segments from the other phases, so a single winding segment occupies an electrical angle
of
3..
7r
0, = electrical angle = 3
so
km = winding factor for distributed winding
=
sin(-)
2
0.954
2
Note this winding factor is for the limit case - i.e., there are an infinite number of wires.
This is an approximation, so the true winding factor is in fact somewhere between 0.954
and 1.
3.4.3
Magnetizing Inductance
While some of the electrical power input to the stator is lost in the armature windings due
to wire resistance, a (hopefully) larger amount will couple to the rotors and be converted
to mechanical force. The force coupling the stator and rotor is due to the interaction of
the time-varying magnetic field with the rotors;these distributed fields are represented in a
lumped model called magnetizing inductance, or Lmag. This subsection assumes that only
the axial sides of the stator couple to the rotors, and that on these sides there is no flux
loss due to leakage or end effects.
The familiar constitutive law for an inductor is v
43
=
= L!.
The variable L is a
convenient term derived from various parameters of the inductor which are usually constant,
and so are lumped together. A magnetic flux 4D links the elements of the magnetic circuit
in the machine:
A(area)
N turns
Ur
(permeability)
Figure 3-8: Typical solenoid
Li
=
A
=
N4)= total flux linkage of inductor
L
Inductance
i
Current through inductor
(D
Magnetic flux through inductor, pop,HA
Po
permeability of free space, 47r . 10-7 Hy/m
Pr
relative permeability of core
Hx
magnetic field
A
Cross-sectional area encompassed by inductor
A value for Lmag can be determined given an expression for the magnetic flux through the
inductor, 4.
Figure 3-9 shows a simplified sketch of the rotor and stator. For simplicity,
instead of being a ring and disc, it is shown flattened into a bar and plate. Additionally,
the single coil represents N turns. For now, only one winding segment will be considered,
so
N =
Na
number of turns in a single phase segment = N
_total
~p
turns in a single phase
nur
of sgents
number of segments
Figure 3-9 shows only part of a single winding phase, and does not show the second rotor;
everything is applicable to the second rotor by symmetry. Assuming flux remains in the x-y
plane, 4) = A = 5y due to continuity across the indicated control volume and symmetry.
44
/
Rot.
I-A / A
1,F
IF
IF
'Stator
)I
I 1
1
IF
NN'%
vp 11 NJ
11 IN 11
X
19 T T T T t T T T T T Q9 I 111)
Figure 3-9: Simplified cross-sectional sketch of magnetic field
Consider Ampere's law around the path indicated in Figure 3-10. The magnetic flux density
/
/
Rotor
Stator
Ly
No
X
U
Figure 3-10: Simplified cross-sectional sketch of rotor and stator
in the rotor and stator is presumably very small compared to the flux density in the air, so
the only contribution to the path integral comes from the two segments of the path in air.
H - dl = 2g|Hy| = Ni
45
Rearranging, the magnitude of the axial field is then
J~j-Ni
2g
Note that the air gap given is expected to vary widely between motors, and even between
different assemblies of the same motor. A typical value would be 0.9 cm, which is the
air gap from the measurements in Section 3.5.
Since segments of a phase are wound in
opposite directions, the direction of the magnetic field in the stator alternates and therefore
pushes outward between phase segments as in Figure 2-3. Since the phase segments have
some width, the winding correction factor k, as described in the previous section must be
included. Only the fundamental of the field is considered, so the magnitude increases by A.
Returning to the case of a toroidal stator, the field is sinusoidal in 0 with period 2r . The
fundamental of this field is then
Hy =
Ni 4
4k, sin(pO)
2g 7r
Figure 3-11 shows two views of the same flattened, simplified rotor and stator. The dashed
line (2D view) and grey region (3D view) represent the same area of interest. Integrating
the magnetic field over this area yields the magnetic flux, 4Y. For the corresponding case
for the un-flattened rotor and stator, the outer integration and substitution of Hy, yields
D
-k
Y =k
0
fo
J
Ro
Ri
2 - R.
poHyjr - dr -d=
kj
2
2o
P
fo i
Ni 4
2g7ksin(pO)d6
k
Pulling out constants and carrying out the remaining integral then gives
4
y=
4 Nk2 i(R 0 2 - R 2)
-O
7r
4g
l
.
s4g)d
4
7r
-1p0
Nkw2 i(R2 - R 2 ) 2
4g
p
This is the flux due to a single loop of wire, but the model is in terms of the inductance of an
entire phase. Flux was assumed to remain in the x-y plane, so 4D, and 4
y. A factor of two
must be introduced because the symmetric second rotor side has thus far been neglected.
The flux due to a single phase segment containing N turns is then
2N4 x = A = 2N
4
y = 2-p10
7r
N 2k
2
i(R0 2 - R 2 )
4
N 2k
= 2-p02
2gp
7r
46
2
i(R0 2 - R 2 )
2gp
J/7
/~_
Rotor
_~
2D View
-
/ I7/,(7/
3D View
Figure 3-11: Sketch of flattened, simplified rotor and stator, part of one phase
Remembering that N = E, the inductance due to a single phase is then
L=
2
A
p-
=
Na2 kw2 (Ro2
22
4
-po
Finally, to compute Lmag from LI, there is a factor of
Ri 2 )
j
following the argument in Sec-
tion 3.2. The magnetizing inductance in the single-phase machine model corresponding to
the 3-phase machine operating in balanced conditions is
3
Lmag = 22
-
34
*- - u;
1
47
Na2 kw 2 (Ro2 - R, 2 )
2g2
Lmag is then approximately
3 4
2 7r
_7 Hy 1202 -0.9542 . (0.0539752 - 320.03175 2 )m 2 = 3.855.102- 0.008636m m
Hy
This expression is expected to be higher than reality because all flux was assumed to remain
in the x-y plane and all flux was assumed to cross the air gap.
3.4.4
Leakage Inductance
The magnetic field due to the stator currents does not completely couple with the rotors;
some of the magnetic fields link only the stator or only the rotor. These distributed fields
are represented by the lumped model leakage inductance Lieak. This subsection assumes
that the radial faces of the stator are the only components that are of interest in terms of
leakage. Fields from the axial faces are ignored because they were previously assumed to
completely couple with the rotors. End effects are also ignored, i.e. the fields are treated as
if they were generated by a wire of infinite length. Figure 3-12 shows the coordinate system
used for this subsection, while Figure 3-13 shows the important dimensions. We define the
following variables
RWO
Outer radius of windings
Ro + 4Rwire
Rw
Inner radius of windings
Ro -
Rs
Shaft radius
4
Rwire
0.635 cm
Figure 3-12: Sketch of flattened coordinate system of interest, only currents on top face are
shown
48
*Stator
noesfo 668 [].Sicet
The
e
mer
of the
ein
m
e
i
ifrntebudr
fil srltdtocreti h wnig y pr'sla.
mageti
V*xFB =4
pei
Temgeifilcaalobwrteintrothvecto
al A.
Be = V x
Themagnetice
ae
can 3 alobeitensos
pee interst ofeasvae poeilA.e.
49
Substituting,
V x (V x A) = poJ= V(V -A) - V 2 A
Using the Coulomb gage
V-A=0
then
V 2A = -poJ
Assuming that current only flows in the y-direction, this becomes
1 8
BA1
(rA
a2
+ r2
02
Ay = -po J
Assuming AY and J, both have angular dependence eiP0, i.e. vary sinusoidally,
10
r OrI
( r 0A,\)
p2
r2AY
J
r2AY= -PO JY
Or)
and the general solution for the vector potential is
AY = A+rP + A-r-P - Po0Jr 2
4- p
The regions of interest are
0
RWO < r < oo
wo
Ro
wi
Rwi
c
r < Rwo
r < Ri
r < Rwi
Rshaft
Note that region "o" interacts only with region "wo"; similarly, region "wi" interacts only
with region "c". Also note the constants A+ and A_ are different for each region. Finally
note there is no current in regions "o" or "c", so in those regions the solution for the vector
potential is just
AY = A+rP + Ar~P
From B = V x A,
1 0A
r 06
50
and
OA~
Or
B 0 =-
ar
First consider the outer regions, which compose the vector potential AY0 . As r
-+
oo,
BO -+ 0 since from very far away, the stator looks like a point. Therefore
lim BO = 0 = lim -pA
r-.oo
r-*oo
o+ rP
1
+ pAO-r-P-1
which implies Ao+ = 0. Another boundary condition is that at r = R, BO
0, so
-pAwo+RoP~ 1 + pAwo-Ro-P- 1 + 2poJRo=
0
4 - p2
Solving this for AwO+ in terms of Awo_,
A
=
2 + 21oJy Ro-p+2
Aw 0 Rp(4 - p 2)
The circumferential component of the magnetic field, BO, must be continuous at r
Rw0 .
This implies
Substituting~
in4 p~wo-
+~P-1+
oJRWo
A - is then - p2
--
=
+pAo
-P-1
= p AoRwo-P-1
Substituting in for Awo+, A,_ is then
Ao_ = Awo (1 -
)+
ROo2
RWO-
2
)Rup+2
RoP-2)w
21 " (I
p(4 - p2)
-
_2
The final boundary condition is that Br must also be continuous, even at r
=
Rw,. This
implies
Awo+RwoP
+ AwoRwo-P- 1 - yoJyR2"'
= Ao+RwOP- 1 + AoRwo-P~ 1 = AoRwo-P-1
4 - p2
Substituting in for Awo+ and A_,
A
=
4
2
p( - p )
(1
2Rw
p
2
51
-
0
-
RO
)R
o
Substituting into the expression for A,
AWO+t =
in terms of Aw 0 ,
2RwP-
p
2
-o V2 (3
p(4 - P2
2
I
Ro-2
RWOp-2
For convenience, instead of using Aw,- and Awo+, divide these numbers by poJ, to get
P(4 -1 p2 ) (1+- 2
A -
2RwOP- 2
RO
RoP-2
)~
RwoP-2
-
and
"P
p(4
2Rwop-2
p
2
p2) (3
RoP-2_
1
R.,P-
The leakage flux of the entire winding is then
4wkwNa
Aleako
B
=
2
RAY(r)rdr
JR.
-
Substituting in AYO
Aleako
S4wkwNa
2
2
Awo+?rP+
RWO - Ro
+ Awo-r-P+l
-
Po Jy3dr
2_
R0
Integrating and slightly rearranging,
Aleako
_-
e oJy4wkwNa
2 _p
2
A
(Rop+2
V-WOP
-
p
2
RO+
+ 2
)
( RwO-P+2 - Ro -p+2)
+Aw""
-p + 2
Since J'i =
sr
and Lle
4
=lea
0
=
R(-p4
RO
4(4 -p2) )
the leakage inductance due to the outer surface
is
2
=4
2
po4wkw Na
7L (Rwo2
-Ro2)2
(RwoP+
(AwOP
2
RoP+ 2 )
-
p+ 2
(RO~-j+
2
-
-p+
Ro-p+
2
2
)
Rwo 44(4 -p
RO4
2
) )
This expression for the leakage inductance is mathematically awkward, however it is convenient for computer evaluation. The Matlab script tmot.m in Appendix H.5 contains this
equation and was used to evaluate
Lleako.
52
Now consider the inner surface. The difference is that the boundary conditions are that
BO = 0 at r = R, and r
=
Ri and that Br and BO are continuous at Res. After quite a bit
of math,
A w- -
A 4[oJ
2p(4 - p 2 )
( (2
- p)Rwip+2 RjP-2
\ s2p - R 2p
(2 + p)Rs2PRip-2
+ (R8 2p - R 2p)RwiP-2
2
4R4R p2P
Rp+2
R .32p - Ris2p ) R
and
p[o Jy
2p(4 - p 2 )
( (2 - p)Rwip+ RjP-2 + (R(2 +p -p)R,2PRi
p-2
R 2p)RwiP-2
2
2
\ 82p _ R2p
4R
R 92p
2p
R
_
For convenience, instead of using Awi_ and Awi+, divide these numbers by
2
A
((2-p)Rwip+2Rip-2
2p(4 - p 2 ) \
R
_ R 2p
-2p
1
2
(2 + p)RS PR p2 - R 2p)RwiP-2
(Rs P
2p
+4)
poJV to get
4R, 2
2
R8 p
1
- RT 2p
Rp+2
R
and
1
2p(4 - p 2 )
(2 -
2
p)Rwip+
R8 2P - R,
Ri p2
2
p
(2 + p)R 8 2PRiP-2
+ (Rs2P - R 2p)Rwip-2
4R2p
1
+ 4)
R,2P - Rj2P
RP-2
The limits of integration are now from Rwi to Ri, so
T.-4
Lleaki =
ipt4w
2
Na2
. (R~ 2
N
't (RS 2 - Rwi 2 )2
Ap( Rip+2 - Rwi p+2)
p+ 2
(Ri-p+ 2 - Rwi-p+ 2 )
-p+
2
Ri4 - R4
4(4 - p2)
The total leakage inductance is therefore
Lleak = LIeaki ± Lleak, = 1.57 x 10-4 + 1.50 x 10-4 = 3.07 x 10- 4 Hy
Note this expression may be higher than reality because end effects were ignored, or lower
than reality because there is actually some contribution from the axial faces.
3.4.5
Rotor Resistance
The varying magnetic field due to the stator induces a current in the rotor. The parallel
component Zp of the induction model forms a RL circuit and thus has an associated time
53
constant
TR = L"9.
Since Lmag is known from Section 3.4.3, identifying
Tr
will lead to
Rrot.
Assume that since the stator supports currents in a sinusoidal pattern in space, the rotor
will exhibit this pattern as well. Then for some KR, the rotor surface current is
K,
= KRcos(p)
KR
r-o~G
and the electric field is
cos(p9)
rowcu
KR
Er =KRr
c-wCu
where
wcU
Conductivity of copper
5.9. 107(Qh)-l
Thickness of copper
1.5875. 10-3 m
The rotor surface current also contributes to the magnetic field. Consider an arbitrarily
narrow contour as in Figure 3-14. The current inside the contour is
Rotor'
Stator
r 0
)
)
Figure 3-14: Contour for flux due to rotor current
I
0
9+dO
KRrrdG= -
sin(p)
p
=
KR
(sin(pO + pd6) - sin(pO))
p
10
which can be expanded to
KR (cos(pG) sin(pdO) + sin(pO) cos(pd9)
- sin(pO))
p
54
Using the small-angle approximations sin(pd9) = pd9 and cos(pd9) = 1, the current inside
the contour is
KR
K (cos(pO)pd9 + sin(pO) - sin(pO)) = KR cos(pO)dO
By Ampere's law, this current is related to the magnetic field around the contour. As
before, assume the only contribution occurs in free space, so
JB
-ds = gHy, 0 (0) - gHy(0+ dO) = KR cos(pO)dO
Rearranging,
Hyot (0) - Hyot (0 + dO)
dHyrot
dO
dO
KR
g cos(pO)
-
Integrating to get an expression for the magnetic field in terms of KR,
Hyrot =
J
KR
K
cos(pO)dO
9
=
KR
-
p9
sin(pO)
By superposition, the total magnetic field in the gap is re-estimated to be
KR
= Ni 4
2g 7r
+ - sin(pO) =
pg
Ni 4
2g ir
+
KR\
-
sin(pO)
pg )
By Faraday's law,
V xE=
at
We are interested in the r-component of electric field and the magnetic flux density in the
y direction, so the relevant equation is
1 &E,
r
aH'
0
oa
Substituting in Er and Hy1 and evaluating the partial derivative,
2-
r CwC
p1
KR sin(pO) = po
; L .1k, ++ K)sin(pO)
sinP9
(2)
d (Ni4
dt
ot
2g
Canceling terms and pulling out constants,
-
di
N4
p
KR =,o 2g --w0
dt
x
r2 awcu
55
po dKR
pg dt
7r
KR\
+ -)
sin(pO)
P9
Rearranging,
porwcu dK
p2
+ KR =
N4
2g ir
dt
This identifies the rotor time constant
R
r 2 awcu di
dt
p
r:
,=tor 2owcu
2
p~g
Since this is a parallel RL circuit, r = L"ag or Rrot = Lmag
rot
34
2 7r
Na2 kw 2 (R 022 - R, 2 )
p 2g
2gp
[or 2acu
, 5o
3 4 Na2 kw 2 (Ro2 - ji2)
2 7r
2awcur2
This expression for Rr0 t is a function of r, with the values of r are between Ri and RO, so
Rrot should fall in the range
0.134Q =
3 4 Na2 kw 2 (RO2 2- R, 2 )
3 4 Na2 kw 2 (RO2 2- R, 2 )
< Rrot < -0.386Q
2 7r
2a-wcuRo2
-- 7r
2-wcuR12
Since it would be much easier to work with a constant Rrot, we arbitrarily make the simplification r = RO2'j, which means
3 4 4Na2 k 2( RO2 - R, 2)
2 7 2owcu(R, + Ri) 2
2rot
3 4 Na2 kw 2 2(Ro - Rj)
2 7r uwcu(RO + Ri)
so
34
12020.9542
2(0.053975 - 0.03175)
= 0.139Q
Rrot = -2 7r 5.9 - 1071.5875 - 10-3 0.053975 + 0.03175
3.5
Measurement and Comparison
In order to determine approximate parameters of this machine, the impedance of the locked
rotor at a variety of frequencies was measured.
The best-fit model parameters to the
measured data were then calculated.
In order to measure the impedance of the locked rotor, the rotor must first be locked in
place. This was accomplished by shorting the leads of the drive motor together and connecting the drive motor to the shaft of the induction machine as described in Section 2.5.3.
The stall torque of the induction motor is not enough to overcome the static resistance of
the drive motor, thus keeping the shaft and rotors stationary. The induction motor was
56
driven with a 3-phase sinusoidal generator, the HP-6834B. Voltage and current of one of
the phases was measured on a Tektronics TDS-2014 oscilloscope, using the oscilloscope's
floppy drive to save the data to a CSV file. The voltage-current measurement was repeated
for a variety of frequencies in order to collect data about the frequency response.
Matlab was then used to find a best-fit sinusoid to the voltage and to the current (f a3,
Appendix H.1) for each frequency. A typical session might go as follows
>> is
derr.m
TEK0000.CSV
TEK00005.CSV
TEK0O10 .CSV
TEK00015.CSV
err.m
TEK0000l.CSV
TEK00006.CSV
TEK0001l.CSV
TEK00016.CSV
fa3.m
TEK00002.CSV
TEK00007.CSV
TEK00012.CSV
TEK00017.CSV
f anal .m TEK00003. CSV
TEK00008.CSV
TEK00013. CSV
TEK00018. CSV
fresp.m TEK00004.CSV
TEK00009.CSV
TEK00014.CSV
TEK00019.CSV
tmot.m
>> fa3
I: Frequency = -299.893 Hz, Amplitude =
3.291 A (Peak), Phase =
0.881
V: Frequency =
300.866 Hz, Amplitude =
2.282 V (Peak), Phase =
-0.035
Phase shift
-0.916 rad ( -52.506 deg)
=
In this case there are 20 datasets, corresponding to 10 datapoints where the frequency
ranges from 45 Hz to 1 kHz. The directory also contains f a3 and related functions. The
transcript shows f a3 being run on the 300Hz datapoint, resulting in Figure 3-15.
24
(D
0
>2-
oo
\
-2
0
2
/
.
4
Seconds
1
6
8
10
x 10~
Figure 3-15: Sample data resulting from f a3. m showing voltage and current for a single
phase at 300 Hz
Note that there are no arguments to f a3, filenames of datasets to be used are hardcoded
57
into the file. In order to get the results of different datasets, one must edit the filenames
in f a3. m. Another thing to note is that data is sometimes returned in undesirable forms
- in this example, current was reported with a negative frequency. Although this is mathematically accurate, it does not make physical sense and the user should correct for this
before proceeding to the next step. The error function err.m attempts to fit the dataset to
a function of the form A cos(27rft
+ph). Noting that cos(9) = cos(-9), the example best-fit
line for the current can be restated as
3.291 cos(27r - -299.893 - t + 0.881) = 3.291 cos(27r - 299.893 -t - 0.881)
= 0.846. Another possible confusion would
and the true phase shift would be -0.035+0.881
be if f a3 reported a negative amplitude; this can be fixed by noting - cos(9) = cos(9 +,7r).
By applying f a3 to each dataset and possibly correcting the frequency and amplitude to
both be positive, the magnitude and phase for each datapoint can be extracted. This data
is then entered into f anal (Appendix H.3) and derr (Appendix H.4) to find the best-fit
parameters to the model Zphase(JW)
=
Rarm +
jwLieai
>> f anal
Frequency Response Analysis of Induction Motor
Armature Resistance = 0.311976 ohms
Rotor Resistance
= 0.152495 ohms
Magnetizing Inductance = 0.000211059 Hy
Leakage Inductance
= 0.000251927 Hy
58
+
L
+
Frequency Response Analysis of Induction Motor: Data Points and Best-Fit Line
100. -
C,,
E
0
10
-
10
102
10 3
102
103
1.5
(n
1
C
Ca
cc 0.5 -
0
101
Hz
Figure 3-16: Sample best-fit model generated by f anal. m and data from locked medium
rotors
59
The model parameters can also be derived from the geometry of the induction machine.
The script tmot .m (Appendix H.5) evaluates the expressions for Lmag, Rarm, Rrot, and Lleak
developed in the previous section and plots the voltage-limited and current-limited torquespeed curves based on the analytical (dashed red) and measured (solid blue) parameters.
>> tmot
Toy Induction Motor Analysis
Outer Radius = 2.125 in
Inner Radius = 1.25 in
Axial Length = 1 in
Magnetic Gap = 0.34 in
Rotor Cond
= 0.0625 in
Stator Cond
= 0.0403 in
Turns/Coil
=
20
Pole Number
=
6
Armature Resistance = 0.251307 ohms
Rotor Resistance
= 0.138837 ohms
Magnetizing Inductance = 0.000386261 Hy
Leakage Inductance
= 0.000306848 Hy
60
Toy Motor: Current Limited Predictions (Solid = from measured, Dashed = from analytical)
0.081
1
1
1
E 0.06
z
0.04
-'
N
~~
0
-
-
0.02
-N
0'
200
400
600
800
1000
1200
1.95
UO
0)
1.9
N
N
E 1.85
-a
N
N
N
C
1.8
1.75
0
200
400
600
800
1000
1200
Speed, RPM
Figure 3-17: Sample predicted 5A-limited torque-speed curve generated by tmot . m
The final step is to compare the extracted parameters from f anal with the modeled
parameters produced by tmot. This comparison is done by fresp (Appendix H.6).
>> fresp
Expected data based on extrapolated response
f = 45
f = 48.4
f = 75
IzI = 0.340936
IzI = 0.348715
IzI
f = 100
IzI
f = 200
IzI
f = 300
IzI
f = 500
IzI
0.399481
=
=
=
angle(z) = 26.9126 deg
angle(z) = 27.6596 deg
angle(z) = 32.0063 deg
0.436363
angle(z) = 35.4135 deg
0.567954
angle(z) = 48.0792 deg
0.716579
angle(z) = 57.4917 deg
1.05427
angle(z) = 68.3944 deg
61
Toy Motor: Voltage Limited Predictions (Solid = from measured, Dashed = from analytical)
3
1
1
1
1
1
1
E
I 2F
z
=3
0r
L_
1
VN.
I
0
'N
ffi
~
200
400
200
400
-
600
800
1000
1200
800
1000
1200
14 -
33
a)
0
32
31
I
0
-
-
-
-
600
Speed, RPM
Figure 3-18: Sample predicted 12V-limited torque-speed curve generated by tmot
.m
This table summarizes the induction machine parameters as derived from geometry
(from Section 3.4, shown in "Analytical" column) and from the voltage-current measurements described in this section (shown in "Measured" column).
Parameter
Rarm
Analytical
Rarm > 0.251Q
Measured
0.312 Q
Ligak
Lleak
3.07 - 10-4 Hy
2.52 10-4 Hy
Lmag
Lmag < 3.86 -10~ 4 Hy
2.11. 10-4 Hy
Probable error source(s)
inefficient winding, skin effect,
resistance in solder joints
ignored end effects, ignored
contribution from axial faces
ignored end effects, assumed
all flux
Rrot
0.134Q < Rrot 5 0.386Q
0.152 Q
crossed
gap
arbitrary simplification of resistance
Table 3.1: Summary of predicted and modeled teaching machine parameters
62
Induction Motor Frequency Response: Measured data and analytical prediction
10.2
CO
0
10 -0.5
10
-
102
103
0
01
1-
2
-
40 ....
.....
Hz
Figure 3-19: Sample comparison between measured data and calculated parameters generated by fresp.m
63
64
Chapter 4
3-Phase Permanent Magnet
Machine Electrical Description
4.1
Introduction
This chapter describes the electrical performance of the teaching machine when configured as
a 3-phase permanent magnet machine. We build on the analysis of the induction machine
in Chapter 3 to model the behavior of the permanent magnet machine. Analysis of the
permanent magnet machine is much simpler than the induction machine because we will
show in Sections 4.2 and 4.3 that it behaves similarly to a normal DC machine.
The
derivations in Sections 4.2 and 4.3 are largely based on class notes for 6.131 [5]. In Section 4.4
we measure a sample permanent magnet machine to give an idea of the motor constant of
the teaching machine when assembled as a permanent magnet machine.
4.2
3-phase Permanent Magnet Model
A high-level view of the 3-phase permanent magnet motor can be thought of as shown
in Figure 4-1: A 3-phase source connected to a 3-phase speed-dependent voltage by the
armature resistance and the magnetizing inductance. Note this model ignores the leakage
inductance introduced in Section 3.3. We already understand the armature resistance and
magnetizing inductance from Sections 3.4.1 and 3.4.3, so consider the speed-dependant
voltage source.
As in Chapter 3, it is simpler to think about an individual phase instead of the complete
65
Ram
Lma.,
~
Ea
Figure 4-1: 3-phase model for the permanent magnet machine
3-phase machine. Assume the phases are balanced and so a single phase can be modeled as
in Figure 4-2.
=VwLmag
+a
arm
+a
Figure 4-2: Model for a single phase of the permanent magnet machine
In the case of the induction machine, the magnetic field linking the rotor and stator
was due to current in the windings; as argued in Section 3.2, this was !Li.
In the case
of the permanent magnet machine, this linkage is still present but there is an additional
contribution from the magnets on the rotors, A(Om) as in Figure 4-3. The field due to
the magnets is a function of the distribution of the magnets around the rotor and will be
periodic with period
g.p
For some function Ao representing the distribution of the additional
flux due to the permanent magnets, the flux linked by a single phase is then
3
2
Aa = -Li
+ AO(6m)
For some constant AO representing the magnitude of the additional field, the field due to
66
qtor
_j
L_
a
V
V
V
I
V
I
I
Stator
)I
III
uTttTt TtTTTQI IlIi I)
Figure 4-3: Simplified cross-sectional sketch of magnetic field with magnets added
the permanent magnets can be approximated as a sinusoid
AO(0m) = AO cos(jpOm - 00)
We define the following variables
W
Electrical frequency
Wm
Mechanical frequency
Om
Rotor position
00
Rotor offset
Now assume the machine is running at synchronous speed so that wm
rewritten as
Aa(t) =R
{
=
Flux can be
L( I + Aoe-o)Oeiwt}
The terminal voltage va can be expressed in terms of the peak-peak voltage V and is
a = R {Vaew t } = dAa + Rarmia = R
jW
( LjIa + Aoe--Oo
eiwt + Rarmlaejwt
In Figure 4-2, electrical power is dissipated in two places: a resistive loss in the windings
and a conversion to mechanical movement in Ea. The magnetizing inductance of course
67
does not dissipate power on average. Average power is
1
< Pa >=I-R{VaIa}
2
so substituting in Va,
< Pa >=
R
jW3 L.a2 + jwAoIae~9 " + Rarmla2
Pulling out terms and dropping imaginary components,
< Pa >= 1 R jwAoIae-jO
2
2
+ Rarm a
2
2
+ Rarm a
R jwAoIa(cos(0) - j sin(0))I
sno)
=2
2j~\O\;
Further simplification yields
1
Rarm a 2
1
=-wAoI sin(Oo)+
<Pa >= -R{jwAoIacos(o) + wAoIasin(Oo))} +
2
2
2
Rarm a 2
2
2
The power dissipated in all three phases is then
Ptotal = 3 < Pa >= 3wAoIa sin(Oo) +
2
2
Rarm a 2
We identify the second component as the power dissipated in the armature winding, and
knowing that no power is dissipated due to the magnetizing inductance, the remaining term
must be the power transferred to mechanical motion, or Pmech. Mechanical power can also
be written as the product of torque and rotational speed, so
Pmech = Twm = 3 wAOIa sin(OO) = 3 pwmAoIa sin(OO)
22
Assuming the controller is tuned to maximize torque, i.e. sin 0 = 1, then torque is
3
2
T = -pAoIa
The magnitude of the "back voltage" of a single phase, lEal, is
JEal = wAo
68
and the average back voltage on a single phase is
< Ea >
3
3
3
wAo cos OdO = -wAo = -pLmAo
_JZ6
ir
-
ir
Pure sinusoidal 3-phase sources may not be readily available, and the controller suggested in Sections 2.4.3 and 2.4.5 does not produce pure sinusoidal waves. The effect of a
nonpure input has on torque must be determined.
4.3
Sinusoid vs. Tri-totem
Instead of powering the machine with pure 3-phase sine waves, consider the setup shown in
Figure 4-4. The gates of the tritotem are controlled by the motor controller (Section 2.4.3)
and position encoder (Section 2.4.2).
772
ol
~ Tritotem
/
Ec
/IC
E
Figure 4-4: Model of the brushless 3-phase PM machine
This setup produces a sine wave approximation that ideally looks something like Figure 4-5, although this figure omits the high frequency switching necessary to keep the top
totems active.
Switching events happen every E radians, so if the current from the tri-totem board is
Io, the fundamental 11 is
I=
4
)
=
-sin(_)IO=
?r
3
69
4
7r
_/_
2
2
0
2n
41c
3
33
Figure 4-5: Sketch of 3-phase current from tri-totem vs. rotational angle
Again assuming the controller is tuned such that sin0o = 1, substituting I1 for
Ia in the
expression for the torque due to a sinusoidal drive voltage results in
T
=
-pAo -Io
2
7r
27
sin O = pA 0
I0
=
K1o
where some terms have been collected to form a motor coefficient K
K = pAO 3v3
7r
When calculating the effective back voltage for the three phase case, the line-line voltage
should be considered instead of the voltage of just a single phase. This is an increase by a
factor of 0
, and the back voltage increases accordingly. The final back voltage is therefore
<
Eback >=
3V/5
3--pWmAo = Kwm
7r
The motor coefficient K behaves similarly to the motor coefficient of a normal DC motor.
4.4
Measurement
The permanent magnet machine was configured as a generator by attaching a drive motor as
described in Section 2.4.4. The resulting voltage was measured from the A terminal to the
B terminal. The data and best-fit line is shown in Figure 4-6 (from gen.voltage-speed.m,
70
Appendix 1.5). The equation for the best fit line was
VAB = 0.198w + 7.90 x 10-4
Note that VAB is the peak-peak line-line voltage, but the motor constant was developed
using the peak line-line voltage.
Therefore the motor constant from the voltage-speed
measurement, K,,, should be
Kvs
8
I
I
0.198
21 = 0.099
2
I
I
I
I
I
7
6
5
0)
4
im
3
2
1
0
I
0
5
I
10
15
20
I
I
I
25
30
35
40
Rotational speed [rad/sec]
Figure 4-6: Generated Voltage vs. Drive Speed [rad/sec]
We then removed the drive motor, configuring the permanent magnet machine as a
motor. The maximum stall torque was measured at various DC currents with a spring
scale and the torque arm described in Section 2.4.1. Current was injected into the A phase
and grounded the B phase. This caused the shaft to rotate to a position approximately 30
71
degrees from the position of maximum torque, so the torque arm was attached in such a
way that maximum torque is produced when the torque arm is vertical. This prevents the
mass of the arm from affecting the torque measurement. Finally, using a spring scale, the
maximum force was recorded and distance from the shaft required to turn the shaft, which
allows the calculation of torque.
The data and best-fit line is shown in Figure 4-7 (from torque-current . m, Appendix I).
The equation for the best fit line was
T = 0.0980i - 0.00244
Because the measurements were made at DC rather than from a sinusoidal input, the
0.3
-
0.25
x
0.2
x
0
F- 0.151
E
'"
E
0.11
0.05
0
0
0.5
1.5
1
2
2.5
3
Applied DC Current
Figure 4-7: Maximum torque vs. DC current
factor of 1A
0.098 * 1.10
= 1.10 introduced in Section 4.3 has not yet been accounted for, so Kt
=
0.0108.
72
=
Rotational speed of the permanent magnet motor was measured at various DC voltages
when driven by the controller/tritotem from Sections 2.4.3 and 2.4.5. The data and best-fit
line is shown in Figure 4-8 (from speed-voltage
.m,
Appendix I). The equation for the best
fit line was
wm =
6
.75vdd - 0.782
Rearranging, this is
Vdd = 0.1 4 8wm + 0.116
Even when the motor had been running for some time, it was still drawing current, indi-
70
60
50
CO)
40
-L
I.-
CO
0
30
20
10
0
0
1
2
3
4
5
6
7
8
Applied DC voltage, 50% PWM
9
10
11
Figure 4-8: Rotational speed vs. DC voltage
cating the motor was unable to reach no-load speed. Since there was current in the circuit,
some power would be lost in the armature resistance, explaining the non-negligable intercept. Also note the high-side voltage was a PWM of 50%, so although there is a linear
73
relationship between applied DC voltage and rotational speed, the motor constant from
this measurement is not expected to be identical to the ones from the voltage-speed or
torque-current measurements.
74
Chapter 5
Robot Mechanical Description
5.1
Introduction
This chapter describes the mechanical design of an expandable teaching robot which was
designed for an introductory robotics course.
This chapter also includes details about
peripherals to be used with the robot to abstract away behavior or provide teaching opportunities. The goals of the introductory robotics course was to teach simple circuit building
techniques, to teach about simple autonomous behavior, and to teach simple mechanical
skills with the motivation of environmental data collection. The initial projected class size
was approximately 30 students. A complete set of mechanical drawings and a list of parts
can be found in Appendix J.
5.2
The Design Process
Design of the robot was done in two phases. The first phase was the design of the card rack,
a device to contain the controller and circuits of the robot. After the design of the card
rack had mostly stabilized, the wheel base was considered which anchors the mechanical
portions of the robot. By using a modular design, the card rack (and perhaps the wheel
base) can be reused for other projects.
5.2.1
Card Rack
We wanted to reuse the card system designed for 6.131 by Mariano Alvira [1], so this
dictated a fixed width and depth for the card rack. We wanted a system where cards could
75
Card Rack
Cards
0
41
0Mtr
Cgst-rs
Figure 5-1: Wireframe view of the final robot design
be easily added and removed, so the sliding rail approach used in Alvira's design for a power
electronics lab kit [1] was used. Since we wanted good mechanical support for the cards,
the cage must be closed like a ring rather than open like a "U".
The major decision for the card rack was to decide the capacity. Rails were placed on
a 1-inch spacing. We potentially wanted to use the tritotem cards to drive the mechanical
systems, but the height of the drive FETs causes tritotem cards to take up more than one
slot. Similarly, using any TO-220 size parts (such as a 7805 voltage regulator) on the green
board (described in Section 5.3) causes them to take up more than one slot. In light of this,
two slots per card were alloted. An alternative was to space the slots further apart, but if
tall cards were not used this would be a waste of space. The robot might require a card
for the robot controller, a card for the mechanical interfacing, and an open space for future
expansion. This implies 6 rails, which roughly determines the height of the card rack.
All boards share a common 26-pin card-edge interface. The tab on each card fits into a
card edge connector which in turn connects to a ribbon cable, providing an inter-card bus.
The design implication of this is that there must be enough space to comfortably attach
and detach the card connector.
The design of the robot was influenced by lessons learned from the teaching machine,
and a hole grid was desired for expansion. The grid was arbitrarily chosen to be 1/4-inch
76
holes on a 1-inch spacing. The hole grid was considered for all four sides of the card rack,
but it was decided that covering the sides was not necessary since any attachment on the
sides would be problematic due to the rails and potential cards.
Figure 5-2: Sketch of final card rack design
5.2.2
Wheel Base
At this point, attention shifted to the wheel base. The card rack design was not yet finalized
because details of the wheel base might change aspects of the card rack. There are two major
components left in the robot: a power source and a mechanical system. A mobile robot
was desired because we believe students would be more interested in a mobile robot than
a stationary robot such as an arm. Although designs where the power supply container
and the wheel support were distinct modules were considered, a combined design offered
simplicity of construction and a wider range of power and motor choices since they share
space.
As with the teaching motor described in Chapter 2, price was a concern. DC motors
were used for simplicity, so the first major choice was choosing such a motor to use from
various surplus sites. Desired features included operating at 12V, reasonable speed and
torque, and convenient mounting options. A high-speed motor would require gearing down
77
before connecting to wheels, so it would be nice to choose a motor with a built-in gearbox so
the motor could directly drive the wheels. Direct drive meant that the simplest movement
strategy would be two coaxial motors and wheels, with some number of casters for stability.
Once the motors were determined, appropriate wheels must be selected. The major
criteria for the wheels were an appropriate bore size for mounting and a reasonable diameter
for movement. The wheels ultimately chosen matched the size criteria but lacked a locking
mechanism. Early tests showed that after light use and with moderate load, the wheels
would slip. Greg Belote devised the wheel lock shown in Figure 5-3.
The notched hole
in the center connects the shaft and the wheel lock, while the holes in the side allow for
attachment to the wheel. Note that this solution does involve modifying the wheels by
drilling appropriate mounting holes.
S0o
Figure 5-3: Sketch of wheel lock
Choosing the drive wheels gave a rough idea of the size of the robot. The wheel size
determined the mounting height of the motors, which could go under or to the side of the
card rack. For stability, the robot should have a low center of gravity, so the motors were
placed to the side of the card rack. The need for side supports and a place to mount the
card rack implied a tray-like design. This gave a rough idea of the height the base would
be off the ground, so appropriate casters could be chosen. Larger casters improve mobility
because they will not get stuck as easily, but it is not clear a larger ground clearance is
desired. Because of this, the final wheel base design can accept either of two caster sizes.
The larger size has a ground clearance of approximately 2.5 inches, the smaller caster size
has a ground clearance of 1.68 inches. In order to keep the robot level, this necessitated two
mounting positions for the drive motors. The motors themselves can be mounted in either
a vertical or horizontal orientation with the smaller casters. Figure 5-4 shows two possible
motor configurations; larger images may be found in Appendix J. For reasons that will be
described in Section 5.3.4, the configuration in 5-4b was chosen. The use of different motors
78
will necessitate the redesign and placement of the motor mounting holes.
1"In,
o oiV.-o'
o H
oo.-
o
...........
g-odoooodoaccooo
Ism Z 8*h.S
'.s
41'.6"
(b)
6mmmmr4 o
1.
2
Figure 5-4: Two motor configurations, configuration (a) also shows a possible battery placement
The remaining decision was battery position. There were three choices of 12V lead acid
batteries readily available, each with different dimensions.
Although only one type was
needed, the tray-like shape meant the difference in sizes was negligible enough that all three
battery sizes could be supported.
The wheel base shares the 1/4-inch holes on a 1-inch grid pattern from the card rack,
so the wheel base and the card rack may be easily connected with a few short screws. The
hole pattern turned out to be extremely useful; it became standard practice to use zip ties
to hold the batteries in place by threading the ties through convenient holes. The holes
were also used to attach the top board (Section 5.3.3) and LC sensor board (Section 5.3.4),
neither of which was imagined during the design of the wheel base. Mechanical drawings
for the final design of the card rack and wheel base can be found in Appendix J.
5.3
5.3.1
Peripherals
Circuit Prototyping: Green Board
The green board, designed by Mariano Alvira in [1] and shown in Figure 5-5, provides two
prototyping breadboards and screw terminals for the inter-card bus connections. Most of
the circuit work for students is to be done on this board.
79
Figure 5-5: Picture of green board, designed by Mariano Alvira
5.3.2
Microcontroller: Basic ATOM
While simple behavior is possible with discrete logic, we felt students would be more involved
if given the chance to program the robot. Since this was intended to be an introductorylevel course, prior programming knowledge could not be assumed, so the robot must be
programmable using a very simple language.
The Basic ATOM is a PIC-based microcontroller with various onboard peripherals.
This microcontroller was chosen because it can be programmed in-circuit in a variant of
BASIC. It has onboard PWM and A/D conversion, both very useful features. One thing
it lacks is adequate space for data storage, so it is suggested that a serial EEPROM be
added if students are to collect data using the microprocessor. A datasheet can be found
in Appendix B.
Programming information may be found in
on two analog sensors might go as follows
; Define constants
pwm-period con 1024
desired-duty con 512
max-diff con 500
gain con 5
80
[2]. A sample program for navigation based
right-wheel con
1
left-wheel con 0
right-adc con AXI
left-adc con AXO
; Declare variables
right-speed var word
left-speed var word
right-sensor var word
left-sensor var word
sensor-diff var sword
feedback-error var sword
loop:
Read in left and right sensor values
adin right-adc, 2, AD-RON, right-sensor
adin left-adc, 2, AD-RON, left-sensor
Determine the error
sensor-diff = right-sensor - left-sensor
feedback-error = gain * sensor-diff
; Error should saturate at max-diff so we don't overflow the next step
feedback-error = feedback-error max max-diff
feedback-error = feedback-error min -max-diff
; Calculate new motor speeds
right-speed = desired-duty - feedback-error
left-speed = desired-duty + feedback-error
; Set the motor speeds
hpwm right-wheel, pwm-period, right-speed
hpwm left-wheel, pwm-period, left-speed
; Pause for 10ms
pause 10
goto loop
5.3.3
Movement: Top Board
Early testing of the robot showed that various ad hoc connectors for power, motors, sensors, and communication were fragile and prone to error after weeks of use. This concern
was magnified because the intended audience did not necessarily have prior knowledge of
81
electronics and might not be able to recognize problems should they arise. This problem
was solved by introducing a circuit board onto which pre-chosen connectors are mounted.
By having actual connectors instead of wires in a breadboard, reliability was no longer
a concern. The circuit board had the additional advantage of abstracting away parts of
the robot so more time is available for other topics. Finally, the top board provides an
opportunity for simple soldering lessons if desired.
If this card were placed inside the card rack, students would not have easy access to
the connectors, so this card was designed to be fastened to the top of the card rack. One
consequence of putting the board on top is that connectors can be and are placed near the
edges of the board - while an unpopulated top board might be placed in the card rack, a
populated board will not fit.
12+1V
ai
LM7805
In
Out
.33uF
+
+12V rail
+2783v~
railI
+12V Pail
CoM
0.uF
MR10
130
Right
2
-2V
1vrailI
1sk
,7
In A
Q
GMD
n)
Out
4~~~
2 =~-lo
L.
5
6
7
0
0
14
15
L
IC-
C
LL
2
212223-
0
*12u
rai
a~l~
Su
t
n
+12v rail
4N1
+12v
rail
I
Left
ai
L MUR120
40NIO
IuF
242526-
+5v
lIuF
rail
16
_
V+
C2-
-
232-01
232-11
232-02
232-12
IuFf V
cc
I
C
u
C1TTL-Il
TTL-01
TTL-12
L
0
4.)
TTL-02
M15
__ El:5
ir
0
A
Vdd
3
4
C
C
8
9
12u
0
L
Go
U)
2V
-.
11Y
I
$A~
L
1
N ote:
out
bypass capacitors are not shown
MIT: LEES: Environmental Bot
Top Board
Eric
Tung
1
Page
1/1
Figure 5-6: Schematic of top board
Figure 5-6 shows a schematic of the top board; board layout may be found in Appendix K.1. There are connectors for four current sensors, two batteries, two motors, one
82
serial connector, and one inter-card bus connector. One DPDT switch enables the batteries
to be charged in place, while the other DPDT switch chooses between in-circuit programming of the microcontroller and TTL-level serial. Since the wiring to the inter-card bus is
fixed, this dictates which bus lines must be used for certain purposes. Board layout and a
parts list can be found in Appendix K.1
5.3.4
Navigation: LC-resonant wire
There are many possible choices for autonomous navigation including the traditional IR,
sonar, and optical methods. Since the motivation for the course was environmental monitoring, a robust, people-insensitive solution was desired. IR and sonar have the disadvantage
of noise, short range, and susceptibility to false readings due to the movement of nearby
objects such as people. Optical tracks do not have the problem of noise or range, but everyday movement of people across the track was likely to leave the track in an unreadable
state due to dirt, dust, or simple track destruction. Because of this, other options were
considered.
Matthew Mishrikey had previously used a LC-resonant circuit for robot navigation [8].
The magnetic field due to alternating current in a long wire may be detected by nearby
inductors because of the induced current. The wire frequency must be chosen to avoid
interference. Figure 5-7 shows a schematic of the generator circuit, board layout and a list
of parts may be found in Appendix K.2. This approach has the advantage of insensitivity
to passing objects (provided they do not generate magnetic fields at the chosen frequency)
and insensitivity to dirt or dust.
The track is also very simple - just a piece of wire -
and therefore may be quickly deployed or altered. The wire was placed on the ground (as
opposed to on a wall or in the air) so as to minimize the impact on passing people, while
not being limited to wall following behavior.
To detect only one particular wire, an LC circuit is used to filter out undesired frequencies. It would be convenient to eventually have a DC voltage related to distance from
the wire, but the induced voltage is a very small AC signal. After filtering the small AC
signal to isolate it from other potential signals, it is amplified and rectified to get a positive
AC wave. The resulting signal is then put through a low-pass filter to yield a DC value.
Figure 5-8 shows a schematic of the sensor. Board layout and a list of parts can be found
in Appendix K.3.
83
+ 12v
10k
10k
0.22uF
8
LM358
+U
-5v
1
10k
IRF530
4
1k
LM358
-12vIRF9530
-12v
Figure 5-7: Schematic of 2 kHz generator
Ampl i
(
fication
Ful l-Wave Recti
)<
fication
Low-Pass -
) (--
++1.v
IuF
luF
1
9 uF+
U
a.i
-jFLMI
4u
_12d
ndVV
19k--
1
G
N4148
LM358
n
-9.
33-F
Gnd
+5 8
LMU358L
LM358
40
nd
nd
-2
8
Gn
-2
_
Figure 5-8: Schematic of LC sensor
One final note is that two such sensors are needed for navigation. If there was only
one sensor, the robot would not know the correct direction to go when it was off course.
By having two sensors on either side of the track wire, the robot can use a very simple
differential controller to maneuver. Since the wire was placed on the ground, the logical
placement of the LC sensor was on the bottom of the robot, so this led us to choose the
configuration shown in Figure 5-4b because of the larger ground clearance.
5.3.5
Data Collection: Hobo
Environmental data was collected by the use of self-contained units called HOBOs, available through Onset Computer (contact information in Appendix D). These units can be
programmed to periodically record temperature, humidity, light levels, or a variety of other
measurements depending on the particular model. Data can then be transfered back to a
84
PC for analysis.
5.3.6
Data Collection: Current Sensors
Information about the robot itself may be useful for teaching about power consumption.
Figure 5-9 shows a current sensor which connects to the top board. The wire of interest is
looped many times through the sensor in order to increase the current passing through the
sensor. The output voltage is proportional to the current flowing through the sensor.
Connect ot
Figure 5-9: Picture of current sensor
5.4
5.4.1
Future Improvements
Wheel Traction
While the robot worked well on the simple test tracks, in more realistic environments the
robot tends to become stuck on small obstacles because the drive wheels have insufficient
traction and therefore slip. The majority of the mass of the robot is due to the lead-acid
batteries, which are typically located near the front edge of the robot due to the layout
of the robot, and so there is little weight on the drive wheels. The issue of wheel traction
85
was solved by mounting large weights on the back of the robot and covering the wheels
with tape so they coefficient of friction increased. A better long-term solution would be to
redesign the robot so the batteries could take the place of weights.
86
Chapter 6
Assessment and Conclusion
This thesis has described many educational opportunities which were enabled by the development of the 3-phase teaching machine and the introductory robot. This chapter summarizes some possible exercises. Instructors may wish to choose some subset of experiments
depending on available time. If both the permanent magnet and induction machines are to
be used, it is recommended that the induction machine be built first as the controller is a
subset of the permanent magnet controller.
6.1
3-Phase Induction Machine Objectives
The student should be able to assemble the controller from Section 2.4.3 and the tritotem
board from Section 2.4.5 and running open-loop, verify the waveforms look like Figure 211. After doing this they should be able to assemble the tritotem board from Section 2.4.5
and run the induction machine as a motor. The student should understand the difference
between 120-degree conduction and 180-degree conduction, and be able to explain why the
tritotem board does not need shoot-through protection with our setup.
The student should be able to measure the speed resulting from various drive voltages as
in Section 3.5. From the measurements, they should be able to use the scripts in Appendix H
to determine motor parameters.
The student should also be able to obtain the motor
parameters from geometry and explain the general relationship between parameters, as
well as obtaining torque-speed curves. Finally, the student should be able to predict the
acceleration of the induction motor based on the torque-speed curves and geometry, and
compare this to measurements.
87
Figure 6-1 shows an example evaluation form for a TA or LA.
6.2
3-Phase Permanent Magnet Machine Objectives
The student should be able to assemble the controller from Section 2.4.3, the tritotem board
from Section 2.4.5, and the encoder from Section 2.4.2 and run the permanent magnet motor
closed-loop. The student should be able to run the PM machine as a generator and calculate
the motor constant from the speed-voltage curve. The student should then be able to use the
torque arm and a spring scale to verify the motor constant by measuring the torque-current
relationship.
Figure 6-2 shows an example evaluation form for a TA or LA.
6.3
Robot Objectives
The student should understand concepts such as resistance, voltage, current, and power.
The student should understand RC filters, op-amps, and PWM. The student should be
able to interpret circuit diagrams, solder small circuits on a PCB, assemble small circuits
on a breadboard, and write simple programs for a microcontroller.
The student should
understand why a microcontroller needs MOSFETs to drive the motors. The student should
be able to program a microcontroller to drive the robot forwards and to make the robot
turn. The student should be able to understand simple feedback loops and resonant circuits.
The student should be able to combine all these skills to build a robot that navigates by
detecting an AC current in a wire.
Figure 6-3 shows an example evaluation form for a TA or LA.
6.4
Checkoff Sheets
Sample checkoff sheets are shown on the following pages.
88
Evaluation Guidelines
Excellent (1.5 pts): Student is easily able to demonstrate understanding of key concepts
Good (1 pt): Student has minor errors or confusion which, upon review with LA/TA, do
not impede understanding
Adequate (0.5 pt): Student evidences some effort, familiar with core ideas although with
major errors or confusion; upon review with LA/TA, understands correct concepts
Needs Improvement (0 pts): Student is unprepared or is unfamiliar with core ideas
Induction Machine
Evaluation
Task
Comments
Demonstrate working open-loop controller
by running motor
Determine
motor
parameters
from
voltage-speed measurements
Determine motor parameters from geometry
Determine torque-speed curves from parameters
Compare startup rotational time constant
to predicted
1
Figure 6-1: Sample evaluation sheet for the 3-phase induction machine
89
Evaluation Guidelines
Excellent (1.5 pts): Student is easily able to demonstrate understanding of key concepts
Good (1 pt): Student has minor errors or confusion which, upon review with LA/TA, do
not impede understanding
Adequate (0.5 pt): Student evidences some effort, familiar with core ideas although with
major errors or confusion; upon review with LA/TA, understands correct concepts
Needs Improvement (0 pts): Student is unprepared or is unfamiliar with core ideas
PM Machine
Task
Evaluation
Comments
Demonstrate working closed-loop controller by running as brushless DC motor
Determine motor parameters running as
generator
Determine motor parameters from stall
torque
Figure 6-2: Sample evaluation sheet for the 3-phase permanent magnet machine
90
Evaluation Guidelines
Excellent (1.5 pts): Student is easily able to demonstrate understanding of key concepts
Good (1 pt): Student has minor errors or confusion which, upon review with LA/TA, do
not impede understanding
Adequate (0.5 pt): Student evidences some effort, familiar with core ideas although with
major errors or confusion; upon review with LA/TA, understands correct concepts
Needs Improvement (0 pts): Student is unprepared or is unfamiliar with core ideas
Robot
Evaluation
Task
Comments
Demonstrate understanding of simple programs by using an Atom to drive simple
circuits
Demonstrate understanding of the difference between logic-level signals and power
signals by an open-loop drive dance
Demonstrate soldering skills, understanding of circuit diagrams via a working LC
sensor board
Demonstrate understanding of feedback
and PWM via closed-loop wire following
Demonstrate understanding of op-amps
by explaining how the amplification block
of the LC sensor works
Demonstrate understanding of RC filters
by explaining how the low pass block of
the LC sensor works
Figure 6-3: Sample evaluation sheet for the robot
91
92
Appendix A
3-Phase Machine Mechanical
Drawings
This appendix contains all mechanical drawings sent out to be manufactured, as well as a
list of additional parts for assembly of a single teaching machine.
93
REVSIONS
ZONE
REV-
D
DESCRIPTION
DATE
APPROVED
I-r
EDF-Illoa
-------- L
p
ELI
Jil
L - - - - - - - -
-- - - - - - -
-i
4------------
________________--------------_____
---- - ---
II
I
LI -------
MIT - LEES
Teaching Machine
(A ssemby)
SIZE
A
DWG. NO.
SCALE:12.5
JWEFGHT:
REV
ISHEET I OF I
REVISIONS
ZONE
APPROVED
DATE
DESCRIPTION
REV.
24.00
12.00
( 2.50
2.40
N-.
c~J
4=-
I
1
___0
8D
0
- 1C!%L =q_-
C0
C).
0-
0
-0CDI
04
F
OC'4
-__0
0
C_
I-- :
0
-1
-0
I___4
1.13 )
(1.10)
0
LO
Chamfer for countersunk 10-32
MIT -LEES
Bottom Panel
SIZE DWG. NO.
A
SCALE:1n I WEIGHT:
REV
=
!SHEET I OF I
REVISIONS
ZONE
REV.
DESCRIPTION
DATE
APPROVED
7.00
00.625
L-
- -- J
C0
(D
C15
I
___________
Is
0
U()
PN
'I
Z1.125
0
(y)
6
MIT - LEES
Radial Wall:
Ball Bearing
SIZE
IDWG. NO.
A
SCALE:1:
IWEIGHT:
REV.
ISHEET I OF 5
REVISIONS
ZONE
O
0
poe
CLL
0
-
-
-
-
-
-
-
-
-
10.00
-
APPROVED
0.50 C--
0 .21 8
L_
DATE
DESCRIPTKON
REV.
6
--- - - -
- _-
-
-
__
-
-
-
-
-
-
--
0
~
Chamfer for countersunk 10-326
MIT - LEES
Axial Wall:
With Mounting Hole
S
E DWG.
SCALE:1:2
REV.
NO.
WEIGHT:
SHEET 2OF3
...
ZONE
........
......
.. REVISIO
NS
DESCRIPTION
REV.
DATE
12.00
-!Ea
1.50
1.25
T
LI 1 I
________
______
-I
Q
L
L
00
r)
C~N
*
..
67
L
- -
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
j
-L
(D
LO
6
MIT - LEES
Top Panel
S7ZE DWG.
SCALE:-2
NO.
1 WEIGHT:
REV.
SEET
I
OF
APPRO
~
II
ZONE
00.77 (0.020" deep)
REV.
[
REVISIONS
DESCRIPTION
DATE
APPROVED
0.125
+.002/-.000
Evenly spaced,
dimensions are typical.
U
CY)
0O
0
\,
0(
00.500
to
+.001/-.000
CNN~
MIT - LEES
Magnet Disk
05.00
SIZE
Material: mild steel
I.
NO.
A
SAEA1WEIGHT:
I HEET I
OF 3
ZONE
I
I
I
I
REV.
REVISIONS
I
DESCRIPTION
I
DATE
I
I I
APPROVED
I
Light chamfer,
a convenient edges
R2.50
(DO
MIT - LEES
Induction Disk
SIZE
DWG. NO.
REV.
Material: Copper IA
SCALE:: I
WEGHT:
ISHEET I
OF I
(6.00)
co
"I
101
-m
REVISIONS
Z0ONE
REV.
DESCRIPTION
DATE
APPROVED
Weld edge
0
o?6f
Weld edge
SA
(1.5 : 1)
Large hole is
centered vertically
on piece
1.75
Mou nting holes
will not quite be
cen tered vertically
0.510
(
H
K
0
0D
A
\0
0.542
from center of
mounting holes
CD
UN
2.50
C)
3.50
Material: Sheet aluminum, 0.08 thick
MIT -LEES
0.80
to center of
Motor Mount
mountnia holes
SIZE JUNG. NO.
REV.
A
SCALE:1:1.5 I WEIGHT:
102
ISHEET
1OF I
REVLSIONS
ZONE
Intended to be a light
interference fit wth a 1/2" shaft
APPROVED
DATE
DESCRIPTION
REV.
-- - - - - - - -
-
0
L.-
69
-
- - -
- -
--
-
-- -
0
0
CA~
i i
/
Light chamfer
T
S
-
II
1
Co
_
- --
C;)
MIT - LEES
CD Encoder Mount
Light chamfer
REV.
Material: Plastic or delrin
DR G. NO .
SCALE:2:1
WEIGHT:
SHEET
1OF 1
REVISIONS
ZONE
REV.
DESCRIPTION
DATE
APPROVED
Light chamfer, all convenient edges
0.40
0
0
0
LO)
0
CN
0
\n
--
00
ON
NO
ON
C)
r-1
CN-
Cn)
LO5
0
.O
0.10
S0
0.38-&
0.75
Material: Aluminum
MIT - LEES
Torque Arm
SIZE IDWG. NO.
REV.
A
SCALE:11 I
104
WEIGHT:
1SHEET IOF I
Part Number
Vendor
Quantity
Description
1
Machine Base
Van & Co
1
Winding Guide
Van & Co
2
Steel Disk
MIT CMS
2
Copper Disk
MIT CMS
1
Torque Arm
MIT CMS
1
Ferrite powder core
2
Thrust Bearing Cage, 1/2" ID, 15/16"
T107/65/25-3F3
ELNA Magnetics
5909K31, 6655K17
McMaster
5909K44
McMaster
6383K34
McMaster
1497K3[12345]
McMaster
92530A100
McMaster
3357K12
McMaster
6435K14
McMaster
90267A693
McMaster
OD
4
Thrust Bearing Washer, 1/2" ID, 15/16"
OD
2
Ball Bearing, 1/2" ID, 1-1/8" OD, 3/8"
thick
partial
Keyed Shaft, 1/2"
OD, 1/8"xl/16" key-
way
partial
2
Key, 1/8"x1/8"x12"
Shaft
Collar,
1/2" ID
1-1/8"
OD,
1/8"xl/16" keyway
OD
2
Shaft Collar, 1/2" ID 1-1/8"
4
Knock-in insert, 8-32 Int. Thread, .394"
long
4
8-32 Thumbscrew, 1/2" long
94320A195
McMaster
4
Knife-edge insert, 4-40 Int. Thread, 3/8"
90016A005
McMaster
long
4
4-40 Thumbscrew, 1/4" long
94320A131
McMaster
2
10-24 Flat screw, 3/4" long
90275A245
McMaster
1
10-24 Flat screw, 1" long
90275A247
McMaster
3088A{38,46}2
McMaster
A 6Z 3-36DF03716
SDP-SI
A 6Z 3-12DF03706
SDP-SI
A 6G 3-050037
SDP-SI
as needed
1
Shimming washers, 1/2" ID
Timing pulley, 1/2" ID, 0.2" pitch, 36
grooves
1
Timing pulley, 0.188" ID, 0.2" pitch, 12
grooves
1
Timing belt, 0.2" pitch, 12 grooves
24
NdFeB magnet, 3/4" dia, 1/8" thick
0029
WonderMagnet
24
Ceramic magnet, 0.7" dia, 0.2" thick
0584
WonderMagnet
105
For vendor contact information, see Appendix D.
106
Appendix B
Datasheets
This appendix contains selected pages from datasheets for various components of the teaching machine and the robot.
107
Ferroxcube
Ferrite toroids
T1 07/65/25
RING CORES (TOROIDS)
Effective core parameters
107 ± 2
PARAMETER
SYMBOL
core factor (Cl)
effective volume
effective length
effective area
mass of core
7(l/A)
V.
1,
A
m
VALUE
UNIT
0.504
133000
259
514
-680
1
mm-
3
mm
mm
2
mm
g
Coating
Coated cores are available on request.
65 ± 1.3
-
25 Z0.75
Dimensions in mm.
Fig.1 T107/65/25 ring core.
Ring core data
GRADE
3F4
3F3
E
E
2004 Sep 01
TYPE NUMBER
AL
(nH)
- 750
-1800
1870 ±25%
4485 ±25%
2
108
T107/65/25-3F4
T107/65/25-3F3
Ferroxcube
Material specification
3F3
3F3 SPECIFICATIONS
A medium frequency power material for use in power
and general purpose transformers at frequencies of
0.2 - 0.5 MHz.
SYMBOL
9a
B
Pv
p
Tc
density
CONDITIONS
VALUE
3F3
UNIT
3'
25 *C; :510 kHz;
0.25 mT
2000 ±20%
10
100 *C; 25 kHz;
200 mT
25 *C; 10 kHz;
1200 Nm
100 *C; 10 kHz;
1200 Nm
100 *C; 100 kHz;
100 mT
100 *C; 400 kHz;
50 mT
-4000
104
DC;25*C
-440
mT
10
-370
580
kW/m
3
10
10-1
5150
-2
Om
>200
*C
- 4750
kg/m
5000---w----
3
3000-
300
2000-
200
1000-
100
50
150
T (Bc)
0
-25
250
Complex permeability as
a function of frequency.
-
3F3
-10C
0
25
50
150
Fig.3 Typical B-H loops.
160
109
102
------
H (A/m)
Fig.2 Initial permeability as a
function of temperature.
2004 Sep 01
f (MHz)
Fig.1
B
(MT)
400
10
~~25~,
-
500
3F3
4000-
0-50
1
500
250
Ferroxcube
Material specification
3F3
Material specification
3F3
8000
-
-
-
CBW44
-
00
- 1C3F3
9a
10
6000
3F3
Mrev
4000
10 3
-
2000
10 2
0
0
100
200
300
400
B (mT)
10
1
Fig.4
10
Amplitude permeability as
function of peak flux density.
4
Fig.5
BW4
400
3F3
T - 100 "C
10
102
-
w___
3F3
(kW/m )
ikot
103
Reversible permeability as a
function of magnetic field strength.
3
(kW/m3)
H (A/m)
f
B
(kHz) (mT)
300
A
103A
200
100
400
50
25
100
200
100
102
100
10
10
1
102
6 (mT)
010
10
Fig.7
Fig.6
Specific power loss as a function of peak
flux density with frequency as a parameter.
2004 Sep 01
161
110
40
80
T (,C) 120
Specific power loss for several
frequency/flux density combinations
as a function of temperature.
Data Sheet
Atom 28 Pin Module
Description
The Basic Atom is a tiny computer, or better known as a microcontroller.
The Basic Atom was designed for use in a wide array of applications. The
Basic Atom is built around the 16F876 PlCmicro MCU@, which contains
internal memory (384 Bytes of RAM and 8K of FLASH / program space).
Each ATOM has a built-in 5-volt regulator, a number of general-purpose
1/O pins (TTL-level, 0-5 volts), commands for math and 1/O pin
VIN
SOUT
lion
SIN
a0
vss
operations
Eg
*
P1
*'s
Hardware
P3
P4
can be used for digital
o
P10
0
P9
P8
P7
AX2
AXO
AXI
It)
P11
P6
accessed using their associated pin labels as shown. All 4 pins can be analog.
The Basic Atom 28 pin module can DEBUG and use analog at the same time.
This was a limitation with the 24 pin Atom.
It)
P12
00
P5
1/O or Analog inputs. These additional 1/O pins can be
S
P14
P13
P2
The Basic Atom 28 pin was designed to be pin compatible to the Basic
Atom 24 pin module. The Atom 28 pin version has an additional 4 pins that
VDD
P15
PO
and a RS232 level converter, for in circuit programming.
vss
RES
ATN
m
o
m
AX3
Atom 28
Pin
IC
.62" (16mm)
Programming Connection Method I
Without the development board; the ATOM must be connected to a
free PC serial port as shown. DTR or RTS must be connected to the
ATN pin. Power must be supplied with a common ground to the Atom
and serial cable ground. A maximum of 9 volts can be used to power
the Atom from the VIN pin. Otherwise a regulated 5 volts can be supplied
to VDD. If you have trouble connecting to the Atom double check your
wiring. Most programming problems arise from incorrect wiring or faulty
31
0--0-=-43D
0-
0 000-5
Serial Port
RX
TX
DTR or RTS
VSS/GND
ATOM
SOUT
SIN
ATN
vSS
[AN
serial cables.
Programming Connection Method II
With the ATOM development board; first insert the Basic Atom module with power off. Pin one on the Development is clearly
marked. Pin one on the BasicAtom, is the pin labeled SOUT, shown above. Connect the PC serial port to the DB-9 connector.
Then plug the power adapter in and power up the development board. If you are having problems programming the Atom
double check your connections. Make sure power is applied and the LED is lite on the development board. Most programming
problems are usually related to bad serial cables.
Software Setup
To begin software installation, follow the instruction in the Basic Atom programming manual. Once you have the software
installed make sure to select the correct COM port, the Basic Atom is attached to. Go to the Tools menu, select System
Setup. The setup screen will only display the COM ports available. If a certain COM port is not displayed, windows is not
reporting its status to the Basic Atom software. Check your device manager to ensure the COM port is installed correctly.
( 1999-2002 Basic Micro.com a All Rights Reserved No portion of this work may be
reproduced without prior written consent from Basic Micro Inc.
111
A.i M ino inc.
Atom 28 Pin Module
Data Sheet
Pin Descriptions
S_OUT
Serial Out 115K, connects to PC serial port RX pin (DB9 pin 2 / DB25 pin 3) for programming the ATOM.
S_IN
Serial in 115K, connects to PC serial port TX pin (DB9 pin 3 / DB25 pin 2) for programming the ATOM.
ATN
Driven Reset, connects to PC serial port DTR (or RTS) pin (DB9 pin 4 / DB25 pin 20) for programming
The ATOM
VSS
Power / Serial Ground.
VDD
5-volt DC input/output. Unregulated voltage applied to the VIN pin will output 5 volts on VDD. Regulated
voltage between 4.5V and 5.5V should be applied to VDD if no voltage is applied to VIN.
RES
Driven low to force a reset. This pin is internally pulled high and can be left disconnected. Do not drive
high.
VIN
Power Input 5-12 VDC. Internally regulated to 5 Volts. Can be left disconnected if 5 volts is applied to
VDD.
P0-P1 5
AX0-AX3
General-purpose 1/O pins. Max for each pin is; sink 25 mA and source 20mA. Total for all pins should not
exceed 50 mA (sink) and 40 mA (source).
General-purpose
1/O pins,
Analog or Digital pins. Max for each pin is; sink 25 mA and source 20mA.
Total for all pins should not exceed 50 mA (sink) and 40 mA (source).
Power Consumption
Normal operation, no loads
Sleep Mode
Nap Mode
5ma
200ua
600ua
Warranty
Basic Micro warranties its products against defects in material and workmanship for a period of 90 days. If a defect is discovered, Basic Micro will
at our discretion repair, replace, or refund the purchase price of the product in question. Contact us at support@basicmicro.com No returns will be
accepted without the proper authorization.
Copyrights and Trademarks
CopyrightV 1999-2001 by Basic Micro, Inc. All rights reserved. PlCmicro@ is a trademark of Microchip Technology, Inc. The Basic Atom and Basic
Micro are registered trademarks of Basic Micro Inc. Other trademarks mentioned are registered trademarks of their respective holders.
Disclaimer
Basic Micro cannot be held responsible for any incidental, or consequential damages resulting from use of products manufactured or sold by Basic
Micro or its distributors. No products from Basic Micro should be used in any medical devices and/or medical situations. No product should be used
in a life support situation.
Contacts
Email: sales@basicmicro.com, Tech support: support@basicmicro.com, Web: http://www.basicmicro.com
Discussion List
A web based discussion board is maintained at http://www.basicmicro.com
Technical Support
Technical support is made available by sending an email to support@basicmicro.com. All email will be answered within 48 hours. All general syntax
and programming question, unless deemed to be a software issue, will be referred to the on-line discussion forums.
C
1999-2002 Basic Micro.com V All Rights Reserved No portion of this work may be
reproduced without prior written consent from Basic Micro
Inc.
112
3A MC
-
MiD i c.
Appendix C
Controller Parts and Schematics
This appendix contains schematics, board layout, encoder disk pattern, and parts list for the
position encoder (Section 2.4.2) and schematics, board layout, and parts list motor controller
(Section 2.4.3). Part lists are a guideline only, many other parts may be substituted. For
vendor contact information, see Appendix D.
C.1
Position Encoder
Board design for the position encoder by Warit Wichakool.
+5v
180
+5vhoobypass
47k
,
1 5 0
+
a
+5o
I
4.
68 3.3k
3k
+
+5v
1k
U
3.3k
150k
100k
680kC
Photodiode/transistor
4 C
5U
lk
3.3k
113
7
Out
74HC14
-
aCbypass
C.1.1
Base
Quantity
Description
1
Encoder base
2
DIP-8 socket
ED3108-ND
Digikey
4
10 kQ potentiometer
3386F-103-ND
Digikey
2
LM358
296-1395-5-ND
Digikey
2
2-position terminal block
ED1609-ND
Digikey
1
6-pin 0.156" header w/friction lock
A1973-ND, WM4624-ND
Digikey
1
Hex Schmitt-trigger inverter
MM74HC14N-ND
Digikey
1
DIP-14 socket
ED3114-ND
Digikey
2
180 Q resistor
180QBK-ND
Digikey
6
3.3 kQ resistor
3.3KQBK-ND
Digikey
2
47 kQ resistor
47KQBK-ND
Digikey
2
100 kQ resistor
100KQBK-ND
Digikey
4
150 kQ resistor
150KQBK-ND
Digikey
4
680 Q resistor
680KQBK-ND
1
22 pF bypass capacitor
3
0.047 pF bypass capacitor
C.1.2
Part Number
Vendor
ExpressPCB
1107PHCT-ND
Digikey
Sensor Board
The sensor board has been carefully sized so that when attached to the encoder base and
when the base is sitting on the wood base of the 3-phase machine, the sensors are aligned
with the pattern on the encoder wheel. Changing headers or the sensor board layout may
require changes to the encoder wheel.
Part Number
Vendor
Quantity
Description
1
Sensor Board
1
6-pin 0.156" header
WM3303-ND, WM3513-ND
Digikey
2
Long focal length photointerrupter
GP2S28
Digikey
ExpressPCB
114
D2 G 02
DI G01
0
6800
r-
+5V
rl
C
10kon
C.1.3
ND
I
wO
0
10k
10
C
on
Shaft Encoder v3.0
0
i
Encoder Wheel
This pattern is meant to be affixed to a CD and mounted on the encoder bushing (see
Appendix A).
115
-- 4-A
/
i
-. 09A
lmwmlww
/
116
Motor Controller
C.2
The motor controller takes input from an onboard clock or from the position encoder described in Section 2.4.2. It outputs control signals suitable for the tri-totem board described
in Section 2.4.5. Although we include a board layout, it is not tested; experiments were
done using the circuit built on a breadboard.
+5v
A6ec
-J
0.6n8U2-J-
3. 3k
10k
+12V
Y2
A3
r-
.33uf
3n
1Y6
2
In765out
1
N
+v22
f~N
0.u
1k
4. 3k
A5
2.2k,
03
CHI)
B2
o-V1
SW
..
2a
GN
V2
B43A3
Y3
High A
Low A
- -- - - - - -
Y
3C4-
+5-
CP
0
CB~Y
C
--
CE
Y4
YND
Y5
+5v
117
B
Low B
High C
C
-+- Low
--
Quantity
Description
Part Number
Vendor
1
4-bit binary counter (74ALS163)
296-1488-5-ND
Digikey
1
3 -> 8 line demux (74ALS138)
296-1485-5-ND
Digikey
1
Hex Schmitt-trigger inverter (7414)
MM74HC14N-ND
Digikey
2
Quad 2-input NAND (74ALSOO)
296-1477-5-ND
Digikey
1
Quad 2-input AND (74ALS08)
296-1123-5-ND
Digikey
1
Differential Comparator (LM311)
LM311NFS-ND
Digikey
1
5V regulator
LM7805CT-ND
Digikey
1
DIP-8 socket
ED3108-ND
Digikey
4
DIP-14 socket
ED3114-ND
Digikey
2
DIP-16 socket
ED3116-ND
Digikey
1
510 Q resistor
510QBK-ND
Digikey
2
2.2 kQ resistor
2.2KQBK-ND
Digikey
1
3.3 kQ resistor
3.3KQBK-ND
Digikey
1
4.3 kQ resistor
4.3KQBK-ND
Digikey
1
1 kQ potentiometer
1
10 kQ potentiometer
1
DPDT switch
2
10 nF bypass capacitor
1
33 nF bypass capacitor
1
0.33 piF bypass capacitor
1
0.68 pF bypass capacitor
1
0.1 pF bypass capacitor
118
C
C)
3-phase Motor Co ntroiie
Eric Tung, 2005
1N
4. k
0
74HC 1 +
a
74LSO*
0)
a
3.3k
+
a,
a,
LL
A
7
C
1
0
0
V
a
LB
HA
0~
0
To Enc
119
120
Appendix D
Vendor Contact Information
This appendix contains contact information for vendors used in this paper.
All Electronics
http://allelectronics.com/
All Electronics Corp.
14928 Oxnard St.
Van Nuys, CA 91411-2610
T: 888-826-5432
F: 818-781-2653
Digikey
http://www.digikey.com/
701 Brooks Avenue South
Thief River Falls, MN 56701
T: 800-344-4539
F: 218-681-3380
ExpressPCB
http://www.expresspcb.com/
support@expresspcb. com
121
Ferroxcube (via ELNA Magnetics)
http://www.ferroxcube.com/
http://www.elnamagnetics.com/
PO Box 395, 234 Tinker Street
Woodstock, NY 12498
T: 800-553-2870
F: 845-679-7010
J&J
http://www.j-jfab.com/
22A Thayer Rd
Waltham, MA 02454
T: 781-899-2373
F: 781-899-2393
McMaster-Carr
http://www.mcmaster.com/
P.O. Box 440
New Brunswick, NJ 08903-0440
T: 732-392-6200
F: 732-329-3772
MIT Central Machine Shop
http://web.mit.edu/cmshop/
cmshopamit.edu
Massachusetts Institute of Technology
Central Machine Shop, Building 38-001
Cambridge, MA 02139
T: 617-253-2392
122
F: 617-258-6158
Newark InOne
http://www.newark.com
4801 N. Ravenswood
Chicago, IL 60640-4496
T: 800-463-9275
Onset Computer
http://www.onsetcomp.com/
sales@onsetcomp.com
470 MacArthur Blvd.
Bourne, MA 02532
T: 800-564-4377
F: 508-759-9100
Proxy
http://www.proxyinc.com/
info@proxyinc.com
55 Chase Street
Methuen, MA 01844
T: 978-687-3138
F: 978-794-8635
SDP-SI
http://www.sdp-si.com/
support@sdp-si.com
2101 Jericho Tpke
Box 5416
123
New Hyde Park, NY 11042-5416
T: 516-328-3300
F: 516-326-8827
Surplus Center
http://http://www.surpluscenter.com/
TechHelpOsurpluscenter.com
Surplus Center
PO Box 82209
Lincoln, NE 68501
T: 800-488-3407
F: 402-474-5198
Van & Company, Inc.
http://www,vanandcompany.com/
ContactUs~vanandcompany.com
Van & Company, Inc.
547 Weeden St.
Pawtucket, RI 02860
T: 401-722-9829
F: 401-728-5210
WonderMagnet
http://www.wondermagnet.com/
ffawondermagnet.com, forcefldgverinet.com
2606 W Vine Dr
Fort Collins, CO 80521
T: 877-944-6247
124
Appendix E
Armature Construction
Instructions
E.1
Preparation
Materials needed:
T107/65/25 core
winding guide
42m AWG18, enamel insulated
soldering station
heat shrink
wire cutter
We construct the armature by first winding segments, then connecting segments into
phases. The phases will then be connected into the "wye" configuration to create the threephase armature. From the back-of-the-envelope calculation in Section 3.4.1, we know a
turn around the armature will take about 10cm. From the power requirements, we know
we want approximately 20 turns per winding segment. We need to include a little extra
length per segment in order to account for slightly sloppy winding and connections to other
segments. We suggest 2.3m of wire per winding segment; excess can be removed later. If
you have already verified that this length is appropriate, you may wish to cut the other 17
wires at this time. Mark one side of the winding guide with tape. Since we want to always
wind the wire in the same direction, this makes it easy to remember which side we start on.
125
Figure E-1: Winding guide; one side marked for winding direction consistency
E.2
Wind Segments
Keep winding, keeping the turns close together. One easy way to count is to examine the
outer circumference - each strand is one turn. Figure E-2, there are 10.5 turns, or half of
the segment. The other half segment will come from winding back over the existing loops.
Figure E-2: Half-wound segment showing direction of winding
126
Once the winding is complete, tape the wire down on both sides to keep the it from
shifting while winding the next segment.
Figure E-3: Tape keeps the segment from shifting as you wind other segments
To spread out distortions, we will not wind segments sequentially. Instead, skip one
segment and wind the next.
Figure E-4: Two finished, taped segments
Note the wire tends to expand into a block rather than the desired arc; there's not much
127
space on the inside between the windings. If one were to wrap sequentially, the distortion
would build up as you progress. By skipping segments, every other segment will be regular,
so any distortions will be spread out evenly.
Keep winding segments, skipping every other one. After this is done, go back and wind
the skipped ones, making sure to move tape as necessary so it doesn't get stuck under the
wire. Leave the ends sticking out, the segments will be connected at the same time.
E.3
Connect Segments
It is easier to envision what is happening on a flat bar rather than a toroid. Figure E-5 shows
the correct connections between segments. The small arrows show an example current in the
highlighted phase; the large arrows show the resulting flux. In a phase, adjacent segments
produce magnetic fields in opposite directions; when the fields meet, they push out and will
interact with the rotors. In Figure E-5, note that the distance between the connecting wires
and the core is exaggerated for clarity.
Figure E-5: Connections between phase segments; distance is exaggerated for clarity
To actually connect the segments, trim the connecting wires to length, strip the ends of
insulation, and add a piece of heat shrink. Solder the segments together and heat the heat
shrink.
E.4
3-phase Connection
The partially-connected stators should have six loose ends; we want to turn this into three
phase connections and a neutral, or star, point. Suppose the loose ends are, in order, A,
128
B, C, A', B', C'. A, C, and B' (or alternately B, A', C' - this reverses rotation direction)
should be connected together to make the star point. The unconnected ends are the phase
connections.
E.5
Flux Check
Hook the star point to ground and B to a high DC value, putting about 2A through the
windings. The flux probe should show 3 positive areas interleaved with 3 negative areas
(i.e. 0, since the meter only shows positive flux. You can flip the direction switch on the
to verify there is a negative flux). The magnitude is not particularly important, but the
sign is. Then disconnect B and connect A', repeat measurements, then change to C' and
measure one more time. You should see the positive flux area move smoothly around the
stator (see graphs and multicolored stator for example).
129
130
Appendix F
Permanent Magnet Machine
Assembly Instructions
Materials needed:
9/64" allen wrench
2 PM rotors
2 keyed shaft collars
2 shaft collars
2 thrust bearing sets
2 ball bearings
shaft key
keyed shaft
spacer
spacing washers as necessary
temporary standoffs
131
Start with a PM rotor on the end of the keyed shaft with the keyways roughly aligned.
Add in the shaft key and spacer.
132
0
0
Keeping the rotors far away from the stator so they do not attract, add the second PM
rotor and the keyed shaft collars.
133
Insert temporary standoffs between the rotors and stator before allowing the rotors near
the stator. Make sure the shaft spacer is large enough to keep the rotors from touching the
stator. If not, remove a shaft collar and rotor to add shimming washers, then replace the
rotor and shaft collar. Once the spacing is correct, add the ball bearings to the shaft and
position them in the retaining wells.
134
00
000
Center the shaft as desired, then snug up interior shaft collars and lock. Make sure the
shaft key passes through the PM rotors and the keyed shaft collars. Add thrust bearings
and exterior shaft collars.
135
Close the case, then snug up the needle-roller thrust bearings and shaft collars. Lock
the shaft collars, making sure they do not compress the case - this will cause the needle
roller bearings to not rotate freely. Once the external shaft collars are locked, open the case
and remove the temporary standoffs.
If the rotors are not centered, you may wish to center them. Only one external shaft
collar (the one on the side of the rotor which is closer to the stator) should be opposing the
attractive force of the permanent magnets. Loosen the other shaft collar, shift by about half
the positioning error, and lock. Shift the shaft by pressing on the side you just adjusted;
this should allow you to reposition the shaft collar on the other side.
136
Appendix G
Induction Machine Assembly
Instructions
Materials needed:
9/64" allen wrench
2 induction rotors
2 keyed shaft collars
2 shaft collars
2 thrust bearing sets
2 ball bearings
shaft key
keyed shaft
spacer
shimming washers as necessary
137
Start with an induction rotor on the end of the keyed shaft with the keyways roughly
aligned. Add in the shaft key and spacer.
138
0
ID
Add the second induction rotor and the keyed shaft collars.
139
-F--
Make sure the shaft spacer is as small as possible while still keeping the rotors from
touching any of the windings. If not, remove a shaft collar and rotor to add or remove
shimming washers, then replace the rotor and shaft collar. Once the spacing is correct, add
the ball bearings to the shaft and position them in the retaining wells.
140
000
000
Center the shaft as desired, then snug up interior shaft collars and lock. Make sure
the shaft key passes through the induction rotors and the keyed shaft collars. Add thrust
bearings and exterior shaft collars.
141
Close the case, then snug up the thrust bearings and shaft collars. Lock the shaft collars,
making sure they do not compress the case - this will cause the needle roller bearings to
not rotate freely. If the rotors are not centered, you may wish to center them.
142
Appendix H
Induction Machine Analysis Scripts
H.1
Amplitude and Phase Extraction: fa3.m
This script reads in a single dataset and applies the built-in fminsearch to find the best-fit
curve according to the error function err. m (Appendix H.2). It must be run manually for
each dataset.
% simple analysis of a near sinusoidal waveform - extracts amplitude and phase
%
global td vd
used to pass data to fminsearch
% initial guess
fO
=
fO is frequency guess in Hz
[50,50);
AO = [4,2];
AO is amplitude guess
phO = [0,0);
phO is phase guess
f = zeros(2);
f is actual frequency
A = zeros(2);
A is actual amplitude
ph = zeros(2);
ph is actual phase
load TEK00006.CSV;
load data (current)
load TEK00007.CSV;
load data (voltage)
time-dat = [TEK00006(:,1), TEK00007(:,1)];
% first column in files is time
vi-dat
%
=
[TEK00006(:,2), TEK00007(:,2)];
second column in files is measurement
Numpoints = [length(time-dat(: ,)) ,length(timedatC(:,2))];
number of datapoints
dn = Numnpoints./10;
approximately 1/10 of the # of datapoints
XO
= [f0.',
XX
AO.', phO.');
for count = 1:1:2
is state vector
% for each dataset
% Following loop is to avoid getting stuck in local extrema
for N = dn(:,count):dn(:,count):Num-points(:,count)
vidat(1:N,count);
XO(count,:)
= fminsearch('err',
consider 1/10th of the set, then 2/10...
% N/10 of the time set
td = time-dat(i:N,count);
vd =
% first
%
XO(count,:));
143
N/10 of the measurement set
% fit a sinusoid to the partial set
end
X = fminsearch('err', XO(count,:));
% finally fit a sinusoid to the entire set
f(count) = X(;
% extract frequency from the state vector
A(count) = X(2);
% extract amplitude from the state vector
ph(count) = X(3);
% extract phase from the state vector
end
% now display the data
fprintf('I: Frequency = %8.3f Hz,
Amplitude = %8.3f A (Peak),
Phase = .8.3f\n',
f(1),
A(l),
ph(l))
fprintf('V: Frequency = %8.3f Hz,
Amplitude = %8.3f V (Peak),
Phase = 78. 3f\n', f(2),
A(2),
ph(2))
fprintf('Phase shift = %8.3f rad (%8.3f deg)\n\n',
(ph(2)-ph(1)),
(180*(ph(2)-ph(1))/pi))
f igure (1)
plot(timedat(:,1),
vidat(:,1),'y.',
time-dat(:,1),
A(1)
.*
cos(2*pi*f(i)
time-dat(:,2),
vidat(:,2),'b.',
time-dat(:,2),
A(2)
.*
cos(2*pi*f(2)
time.dat(:,1)
.*
.*
+ ph(i)),
time.dat(:,2) + ph(2)),
ylabel('Yellow: Current [A], Blue: Voltage EV]');
xlabel('Seconds');
H.2
Error From Sinusoid: err.m
This function computes the error from the dataset described by [td, vdl and the sinusoid
A cos (27r. f . td + ph).
function e
= err(X)
global td vd
f
=X(1);
A =X(2);
ph = X(3);
t
=
td;
v
= A
e
=
.* cos(2*pi*f
sum((v
H.3
-
.* t + ph);
vd) .- 2);
Deriving Parameters From Data: fanal.m
This script collects the data produced by fa3.m (Appendix H.1) and calculates the parameters for the model described
in Section 3.3 by applying the built-in fminsearch with the error function derr.m (Appendix H.4).
% attempt to fit parameters to measured frequency response
DS = [.3 .325 .001 .0001];
% initial guess at paramters
D = fminsearch('derr', DS);
X note
derr only fits amplitude - it ignores phase
Ra = D(;
. fitted armature resistance
R_2 = D(2);
% fitted rotor resistance
L-m = D(3);
% fitted magnetizing inductance
L-a = D(4);
% fitted magnitizing leakage
144
'k--',
'k--')
fprintf ('Frequency Response Analysis of Induction Motor\n');
fprintf('Armature Resistance = %g ohms \n', Ra);
fprintf('Rotor Resistance
= %g ohms \n', R_2);
fprintf('Magnetizing Inductance = %g Hy\n', L-m);
fprintf('Leakage Inductance
= %g Hy\n', L-a);
,
fa = logspace(l, 3, 100);
om
= 2*pi .*
zm = (j*L-m
fa;
space of interest in Hz
% convert to radians
.* om)
.* R-2 ./(j*L-m
zt = Ra + j*L-a .*
.* om + R_2);
% impedence of (magnetizing inductance
om + zm;
II
rotor resistance)
% total impedence of motor
Za = abs(zt);
Zp = angle(zt);
% data - this should be identical to the data in derr.m
f
=
[45 48 75 100 200 300 500);
za = [1.725/4.892
1.910/5.312 1.855/4.618 1.944/4.436 2.160/3.802 2.282/3.291
2.385/2.486);
zp = [.355 .402 .478 .527 .718 .846 1.067);
figure(i)
subplot 211
loglog(fa, Za, f, za, 'o')
title('Frequency Response Analysis of Induction Motor: Data Points and Best-Fit Line')
ylabel('Ohms')
subplot 212
semilogx(fa, Zp, f, zp, 'o')
ylabel('Radians')
xlabel('Hz')
H.4
Error From Data: derr.m
This function computes the error between the collected datasets (hardcoded in as f, za, zp)
and the model described in Section 3.3. The error function is based only on the difference
in magnitude; phase is ignored.
function error = derr(D)
% extraction of elements of induction motor parameters based on measurements
% experimental data
f = [45 48 75 100 200 300 500);
za = [1.725/4.892
zp = [.355 .402
Z = za .*
1.910/5.312 1.855/4.618 1.944/4.436 2.160/3.802 2.282/3.291 2.385/2.486);
.478 .527
.718 .846 1.067);
exp(j .* zp);
% now analytical prediction: D has the parameters
R1 = D(1);
R2 = D(2);
Lm = D(3);
145
Li = D(4);
om
= 2*pi
.*
zm = (j*Lm
f;
.*
om)
zt = R1 + j*Ll .*
.*
R2 ./(j*Lm .*
om
+ zm;
om
+ R2);
error = sum(abs(Z-zt));
H.5
Deriving Parameters From Dimensions: tmot.m
This script calculates the parameters for the model described in Section 3.3 based on dimensions of the machine.
% This file is adapted from one provided by Prof.
J.L. Kirtley Jr.
in 2005
% This is approximately the 6.131 motor, an axial-flux induction machine
% This file takes in physical measurements of the motor and attempts to extrapolate
%
electrical parameters and performance
C = .0254;
X Conversion
Ro = C*4.25/2;
% outer radius of stator
Ri =
C*2.5/2;
between m and in
% inner radius of stator
w = C*1.0;
X axial
p = 3;
% this is a six pole machine
N-c = 20;
% turns per coil
g = ((C*1.68)-w)/2;
% SWAG at effective air-gap
t = C*.0625;
% Thickness of copper sheet
dw = C*.0403;
% AWG 18 on the stator
sig = 5.9e7;
% copper conductivity
muzero = pi*4e-7;
7.constant
I_1 = 5.0;
% if we run at a current limit (peak number)
V_1 = 12;
% or a voltage limit (peak number)
f = 60;
% electrical frequency
s = logspace(-3, 0, 100);
% slip space we're interested in
length of stator
for equations
Rwo = Ro + 2*dw;
Rwi = Ri - 2*dw;
Rs =
C*.25;
% radius of the shaft
% from measurement
mR-arm
= .312;
mR-rot
= .152;
mL-mag = .000211;
mL-leak = .000252;
X housekeeping
N
kw
= 2*p*N-c;
=
sin(pi/6)/(pi/6);
sigs = sig*t;
% total number of turns on armature
. winding factor for distributed winding
. rotor
surface conductivity
146
% first, compute winding resistance
Aw = (pi/4)*dw~2;
% area of wire
1 = N-a*2*(w+Ro-Ri+pi*dw);
% guess at wire length
Ra =
'
1/(sig*Aw);
% this is stator resistance
magnetizing inductance
L-m = (3/2)*(4/pi) * (muzero*N-a^2*kw-2*(Ro-2-Ri-2)/(2*g*p-2));
% guess at leakage inductance
XL-a = muzero*(3/2)*(4/pi)*(N-a-2
* kw-2 *w)/(2*p);
% Aw* constants are divided by \mu_0 J-y
Awon = ((Ro^(2*p))/(2*p*(4-p-2)*Rwo^(p-2)))
* (2+p-(4*Rwo^(p-2))/(Ro^(p-2)));
Awop = (3+p/2-2*Rwo~(p-2)/Ro^(p-2))/(p*(4-p^2)*Ro^(p-2));
% in 1/m
Awin = (Ri-(p+2))/(2*p*(4-p^2))*(((2-p)*Rwi^(p+2)*Ri^(p-2))/(Rs-(2*p)-Ri^(2*p))
+ ((2+p)*Rs-(2*p)*Ri^(p-2))/((Rs^(2*p)-Ri^(2*p))*Rwi-(p-2))
...
- (4*Rs-(2*p))/(Rs^(2*p)-Ri^(2*p)));
Awip = (1)/(Ri~(p-2)*2*p*(4-p^2))*(((2-p)*Rwi-(p+2)*Ri~(p-2))/(Rs^(2*p)-Ri~(2*p))
+ ((2+p)*Rs^(2*p)*Ri^(p-2))/((Rs^(2*p)-Ri~(2*p))*Rwi-(p-2))
% in m^5
...
- (4*Rs-(2*p))/(Rs^(2*p)-Ri^(2*p))
L-ao = (4/pi)*(muzero*4*w*kw^2*N-a^2)/((Rwo^2-Ro^2)^2)*(Awop*(Rwo^(p+2)-Ro^(p+2))/(p+2)
+ Awon*(Rwo~(-p+2)-Ro^(-p+2))/(-p+2)
- (Rwo^4-Ro^4)/(4*(4-p^2)));
L-ai = (4/pi)*(muzero*4*w*kw^2*N-a^2)/((Ri~2-Rwi^2)^2)*(Awip*(Ri^(p+2)-Rwi^(p+2))/(p+2)
+ Awin*(Ri^(-p+2)-Rwi^(-p+2))/(-p+2)
- (Ri^4-Rwi^4)/(4*(4-p~2)));
L-a = L-ao + L-ai;
% Rotor resistance
R_2 = (3/2)*(4/pi)*((N-a~2 * kw^2)/sigs)*(2*(Ro-Ri)/(Ro+Ri));
fprintf('Toy Induction Motor Analysis\n');
fprintf('Outer Radius = %g in \n', Ro/C);
fprintf('Inner Radius = %g in \n', Ri/C);
fprintf('Axial Length = %g in \n', w/C);
fprintf('Magnetic Gap = %g in \n', g/C);
%gin
fprintf('Rotor Cond
=
fprintf('Stator Cond
= %g in \n', dw/C);
fprintf('Turns/Coil
=
fprintf('Pole Number
=
%5.Of
%5.Of
\n', t/C);
\n', N-c);
\n',
2*p);
fprintf('Armature Resistance = %g ohms \n', Ra);
fprintf('Rotor Resistance
= %g ohms \n', R_2);
fprintf('Magnetizing Inductance = %g Hy\n',
fprintf('Leakage Inductance
'
L-m);
= %g Hy\n', L-a);
Now we are going to try to run it
Nm = (60*f/p) .*
(1-s);
...
% mechanical speed in RPM
147
...
+ 4);
% synchronous speed in radians/second
oms = 2*pi*f/p;
Rr = R_2 ./
.
s;
Xm
=
2*pi*f *L-m;
X1
=
2*pi*f *L-a;
rotor resistance
% magnetizing reactance
% leakage reactance
Za = j*X1 + Ra;
% impedence of (mag leak + stator)
Zr = (j*Xm .* Rr)
./
(j*Xm + Rr);
X
impedence of (rotor 11
mag
branch)
% terminal impedance
Zt = Za + Zr;
mXm = 2*pi*f*mL-mag;
mRr = mR-rot ./
s;
mZt = ((j*2*pi*f*mL-leak)
% first,
+
(mR-arm))
+ (j*mXm
.* mRr)./(j*mXm+mRr);
current limited
% Solid lines are predicted by measured,
Ir = I-l*j*Xm ./
Vc = abs(Zt)
(j*Xm + Rr);
dashed is predicted by analytical
% rotor current
Il;
.*
% this will be terminal voltage
Tc = (p/oms)*(3/2)
.*
mIr
= I-l*j*mXm
(j*mXm + mRr);
mVc
= abs(mZt)
./
.*
(abs(Ir) .^2)
.
.*Rr;
. rotor
this should be torque
current
I-l;
mTc = (p/oms)*(3/2) .*
(abs(mIr) .-2) .*mRr;
figure(1)
subplot 211
plot(Nm, Tc, 'r--', Nm, mTc, 'b-')
title('Toy Motor:
Current Limited Predictions (Solid = from measured,
Dashed = from analytical)')
ylabel('Torque, N-m')
subplot 212
plot(Nm, Vc, 'r--', Nm, mVc, 'b-')
ylabel('Terminal Voltage');
xlabel('Speed, RPM')
% Now voltage limited
% Solid lines are predicted by measured, dashed is predicted by analytical
I =
V_1 ./
% terminal current
Zt;
Ir = I .* j*Xm ./(j*Xm + Rr);
Tc = (p/oms)*(3/2) .*
mI
mIr
V_1 ./
=
=
mI
.*
(abs(Ir) .^2)
% rotor
.*Rr;
branch current
% this should be torque
mZt;
j*mXm ./
(j*mXm + mRr);
mTc = (p/oms)*(3/2) .*
(abs(mIr) .-2) .*mRr;
figure(2)
subplot 211
plot(Nm, Tc, 'r--', Nm, mTc, 'b-')
title('Toy Motor: Voltage Limited Predictions
(Solid = from measured, Dashed = from analytical)')
ylabel('Torque, N-m')
subplot 212
plot(Nm, abs(I),
'r--', Nm, abs(mI), 'b-')
148
ylabel('Current')
xlabel('Speed, RPM')
figure(3)
plot(Rr, abs(Zt));
figure(4)
plot(real(Zt), imag(Zt))
H.6
Comparing: fresp.m
This function plots the frequency response derived from collected data (f a3.m, Appendix H.1)
and calculated behavior (tmot .m, Appendix H.5).
% Frequency Response Analysis of Induction Motor
% This file plots measured frequency response (from fa3) and extrapolated response (from tmot)
% measured response from fa3.m
f-meas = [45 48.4 75 100 200 300 500);
z-meas = [1.725/4.892 1.910/5.312
ph-meas = [.355 .402
.478
1.855/4.618 1.944/4.436 2.160/3.802 2.282/3.291
2.385/2.486];
.527 .718 .846 1.067];
% extrapolated response from tmot.m
f
om
% in Hz
logspace(1,3,100);
=
=
2*pi .* f; % in rad
R1 = .251;
R2 = .139;
Lm
=
.000386;
Ll
=
.000307;
om)
zm = (j*Lm .*
zt = Ri + j*Ll .*
.*
R2 ./(j*Lm
om
+ zm;
.*
om
+ R2);
figure(i)
subplot 211
loglog(f, abs(zt),
f-meas, z-meas, 'x')
'-',
title('Induction Motor Frequency Response: Measured data and analytical prediction')
ylabel('Ohms')
grid on
subplot 212
semilogx(f,
(180/pi)
.*
angle(zt),
'-',
f-meas, (180/pi)
grid on
ylabel(' degrees')
xlabel('Hz')
149
.*
ph-meas,
'x')
% fixed points
% f
=
om
=
zm
=
zt
[48.335 61.744 45 75 100 200 300 500 1000);
2*pi
.*
(j*Lm
.*
fmeas;
om) .*
R1 + j*Ll .*
om
R2 ./(j*Lm .*
om
+ R2);
+ zm;
fprintf('Expected data based on extrapolated response\n');
for kk =
1:length(f.meas);
fprintf('f = %g
IzI
=
Xg
angle(z) = %g deg\n',
end
150
f-meas(kk),
abs(zt(kk)),
(180/pi)*angle zt(kk)));
Appendix I
Permanent Magnet Machine
Analysis Scripts
1.1
Motor Constant Identification Via Speed-Voltage:
speed-voltage.m
This script takes data hardcoded into the script and applies the built-in fminsearch to find
the best-fit line according to the error function speed-voltage-err.m (Appendix 1.2).
global vdd freq
vdd = [2 3 4 5 6 7 8 9 10];
% in V
v2f = [6.6 9.3 12.2 15.5 18.6 22.1 25.6 28.9 31.9];
% V2
freq = v2f .*
(= electrical frequency) in Hz
(2*pi/3); % convert to mechanical frequency in rad/sec
X_0 = [1 0];
X = fminsearch('speed-voltage-err', X_0);
fprintf('Best-fit
line: freq [rad/sec] = %g v + %g\n', X(1), X(2));
fprintf ('Best-fit line: v = %g freq[rad/sec) +
%g\n',
vline = linspace(0,11,12);
figure(1);
plot(vdd,
freq, 'x', vline, X(1)
.*
vline
+
X(2),
'-');
ylabel('Rotational speed [rad/sec]');
xlabel('Applied DC voltage, 50% PWM');
axis([0 11 0 72));
151
1/X(1), -X(2)/X(1));
1.2
Error From Speed-Voltage Line:
speed-voltage-err.m
This function computes the error from the dataset described by [vdd, freq] and the line
m Vdd + b.
% Finds best-fit line to data
function e = speed-voltage-err(X)
global vdd freq
M=
X(1);
b =
X(2);
m .* vdd
f=
+ b;
e = sum((f - freq).^2);
1.3
Motor Constant Identification Via Torque-Current:
torque current. m
This script takes data hardcoded into the script and applies the built-in fminsearch to find
the best-fit line according to the error function torque-current -err.m (Appendix 1.4).
global i torque
i = [0.5 1 1.5 2 2.5];
force = [2 1.9 1.9 1.85 2.5);
hole =[1
spacing
% in N
2 3 4 4);
=
0.025;
% in
m
torque = force .* hole * spacing;
X_0 = [1 0);
X = fiminsearch('torque-current-err', X-0);
fprintf ('Best-fit line: torque [N-m] = %g i + %g\n', X(i), X(2));
iline = linspace(0,3,4);
figure ();
plot(i,
torque,
'x', iline, X(1)
.* iline + X(2),
'-');
ylabel ('Maximum Torque [N-m');
xlabel('Applied DC current');
axis([o 3 0 0.3));
1.4
Error From Torque-Current Line:
torque current err .m
This function computes the error from the dataset described by [i, torque] and the line
m - i + b.
% Finds best-fit line to data
152
function e = torque-current-err(X)
global i torque
m = X(;
b = X(2);
t =
m
e=
sum((t - torque).~2);
.*
i + b;
Motor Constant Identification Via Voltage-speed:
1.5
gen-voltage-speed.m
This script takes data hardcoded into the script and applies the built-in fminsearch to find
the best-fit line according to the error function gen-voltage -speed-err.m (Appendix 1.6).
global vout omega
vout
=
% in V
[1.6 2.32 3.1 3.92 4.68 5.52 6.2 7];
% V2 (= electrical frequency) in Hz
v2f = [3.73 5.48 7.52 9.32 11.26 13.18 15.03 16.96);
omega = v2f .*
(2*pi/3); % convert to mechanical frequency in rad/sec
X_0 = [1 0];
X = fminsearch('gen-voltage-speed-err', X_0);
fprintf('Best-fit line: Vab = %g omega [rad/sec] + %g\n', X(i), X(2));
vline = linspace(0,40,41);
figure (1);
plot(omega,
vout,
'x', vline, X(1) .* vline + X(2),
'-');
xlabel('Rotational speed [rad/sec]');
ylabel('AB voltage');
axis([0 40 0 8]);
Error From Generator Voltage-Speed Line:
1.6
gen-volt age -speed-err. m
This function computes the error from the dataset described by [omega, vout] and the line
vout = m - omega + b.
% Finds best-fit line to data
function e = gen-voltage-speed-err(X)
global vout omega
m = XC1);
b = X(2);
v =
e
m
.*
omega
= sum((v -
+ b;
vout).~2);
153
154
Appendix J
Robot Mechanical Drawings
This appendix contains all mechanical drawings for the robot, as well as a list of parts for
assembly of a single robot.
155
ZONE
0
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a
5o
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DATE
DESCRIPTION
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30A
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0
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ol
~Bend
C;'
- i +0.015
I(~oo
IT
-I
o o
2.355
-0.00
Bends
1/4" dia holes
on a 1" grid
centered on panel
0
Bend
0.128
11
C
t0
-Weld edge
MIT - LEES
A (1 :1)
-
Mobile Card Rack
SIZE
NOnREV.
hDeG.
SCALE:1:3
!W EtGHT:
Material: Sheet aluminium, 0.09 thick IA2
!$HEET I OF I
o0
(i
400
0
Unless otherwise
0
0
0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0
noted,allholes
arel/4"dia
C
0
0
B
B
ooooooooooo
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0 0o
00
00
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000
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0
0
0
0
0
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--- 0 0 0
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0
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0
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B (1:2)
0
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cK
000000000000000 00000 0 0 0 01
holes on
a I"gridj]00A
V
0
4.00
Weld edge
0.94
-*-Weld
edge
0
Fhis pattern is
centered c n
the grid, ar d
rep eats every 2 "
Weld edge
0
C1J
r)
Weld edge
A (1 :2)
Material: Sheet aluminium, 0.09 thick
0
0
0
-
I
0
0
0
0
0
MIT - LEES
Wheel Base
Y
0.81
REV.
I
W
SCALE:I:4
.N
G
WEIGHT:
SHEET
I
F1
I
REVISIONS
ZONE
CD
C(4
Co
0
REV.
DATE
DESCRIPTION
APPROVED
)0.400
C-
Er)
(NC;
3.25
0)
-~1
C).
04.00
II
0)
i'
MIT - LEES
Wheel Lock
Material: Steel
SEIDW G. NO.
SCALEI1I
WEIGHT:
REV.
SHEET 1OF
Quantity
Description
Part Number
Vendor
1
Card Rack
J&J
1
Wheel Base
J&J
2
Wheel Lock
2
Plate swivel caster
78155T61, 2390T2
McMaster
16
1/4-20 bolts, 1/2"
92865A537
McMaster
32
1/4-20 nuts
90494A029
McMaster
2
Shaft Collar, 10mm ID
57485K68
McMaster
2
Buehler
sold out
Surplus Center
right
angle
motor
(1.61.031.107.03)
6
washers, 7/16" ID
91081A031
McMaster
6
M6-12mm pan screw
90353A312
McMaster
2
wheel, 7.5" dia, 1.5" width
1-2686
Surplus Center
12
Card Guide (PCG3925)
81N1649
Newark
For vendor contact information, see Appendix D.
168
Appendix K
Robot Controller Parts and
Schematics
This appendix contains schematics, board layout, and parts list for the top board (Section 5.3.3) and 2 kHz generator (Section 5.3.4). For vendor contact information, see Appendix D.
169
K.1
Top Board
..............................
........................
.....
......
.......
..... .........
........
....
...........
....
.....
.........
..........
-.11
.......
..
..
...........
............
......... ..
,,
.0-00
103
.................
CF
.
IL
-
.
.
.
.
.
.
.
.
.
U Fvl' -"
6 6
... . ... .. ...
ry
0-0-0-0
0-0-0.0
(DOODNOWO
0-0.0-0,
0-0.0-0
..
...
0.00.0
0-0.0-0.
0-0.0-0
0.0-0-0
0-0.0-0.
0.0-0-0
0.0-0-0
0.0-0-0
00.0-0
0.0-0-0
0-0-0-0.
0-0-0-0.
0-0.0-0
0.0-0-0.
0-0.0-0
0-0-0-0.
c"00000 .. ..... .
0-0-0-0
0-0-0-0
..... .
0-0.0-0
OMOMC>10
0-0-0.0
000.0
0-0-0-0
0-0.0-0
0-0-0.0
O-ONO-0
0.0-0-0
0.0-0-0
... .. .
oc*000 ..0.0-0.0
O.OX0.0
OMOMOMO
OMC*OMO
010,010
QPONOMO .. .
OWOMOWO
01010,10
00.0-0.
000-0.
000-0
0-0-0-0.
0-0.0-0.
000.0-0.
000.0
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0.0.0.0.
O.ONO-O
0.0.0.0.
000.000
.
4010,010
Omo
0-0.0-0
0-0-0-0
0-0-0-0.
0-0.0-0.
0-0-0-0- 0-0-0-0.
0-0.0-0.
000.0-0.
0-0-0.0 .. .... ...
0.0-0.0 .. ... ....
0-0.0-0
0-000.0
0.0-0=
0-0-0-0.
60 6
e*C)
0-0.0-0.
0.0.0.0.
0-0-0-0.
0.0-0-0 .....
.
.
.
.
k
Ul
. ... .
... .... .
..
..
..
..
..
.. ... ... ... ....
.......
........
.............
.
...
..
............
.
.......
..
.
........
.
.....
....
........
..
........
....
-.
- - 1.111.1
...
...........
........
. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. .
170
I
CI
C
+
1)
+0
a
MD
C
0
m
a)
-0
0*
LC
2
0
tD
0
Ct
0
-q
0 :3
C 0
0 .0 ID
~0
0~
m
cJ)
m
m
I-
+
Mr,
1
Scn r-
-1 --
u +u I
Serial connector
To LC sensor
+
To current sensors (x4)
-
ru
ru
C
flfl
I
I
I
;,0W-J0%cA
Finger conne ector
TLT
C
u
+I
3
wa
C
D
D
I
N
Ft-
C CA
-.
+
Quantity
Description
1
Top board
1
5V regulator
LM7805CT-ND
Digikey
2
40N10 NFET
RFP40N1O-ND
Digikey
2
MUR120 diode
MUR120RLOSCT-ND
Digikey
1
8-pin DIP socket
ED3108-ND
Digikey
1
Dual FET driver (TC4427)
TC4427ACPA-ND
Digikey
1
16-pin DIP socket
ED3116-ND
Digikey
1
TTL-to-RS232 converter (MAX232)
1
DSUB-9 male connector
609-1482-ND
Digikey
4
Male quickfit terminal
1266K-ND
Digikey
2
10kQ resistor
10KQBK-ND
Digikey
1
130Q resistor
130QBK-ND
Digikey
2
DPDT switch
5
luF capacitor
1
0.33uF capacitor
1
0.luF capacitor
1
Red LED
Part Number
Vendor
ExpressPCB
172
Digikey
N
0
+12V
10k
1ok
0.2uF
-5v
+
LM358
-12V
5
0
'*1
IRF530
IRF9530
49
MIT - LEES
2kHz Generator
M.
Mishrikey
Rev 0
B'
825/2005-
Page
Pged
1/1
'
..
. .
In i c
.O
.
GNN
4 7 0uF
-.....
-COD
.
n
Part Number
. . .
Vendor
Quantity
Description
1/2
2kHz Generator Board
1
LM358
296-1395-5-ND
Digikey
1
IRF530 NFET
568-1159-5-ND
Digikey
1
IRF9530 PFET
IRF9530-ND
Digikey
1
15kg resistor
15KQBK-ND
Digikey
3
lOkQ resistor
1OKQBK-ND
Digikey
1
40Q power resistor
1
1kQ potentiometer
2
470uF capacitor
1
0.22uF capacitor
174
. .
ExpressPCB
ImH
> (-
Full-Wave Rectification
)(
Amplification
(
/
Low-Pass -
-12
+2 .IF
IuF
+43
IuF
IuF
-LM358
uF-
0.
4
33F
12
+12v
-j H. + Ul
+\8_
1
U2
I.F
-2
IuF
-P
I
LM358
Gn
++2
43
+U58
LM358
LM358
4I0u
0k
Tn
+12v
+2
1
-
RL
-12v4
20k
k2
-12VK
10k
leek
51k
1NM4148
10+U
Gnd
1
U4
%Nr -1N4148
look
0.33uF
20k
Gnnd
/d
20k
Cnd
Gnd
12v
IMH
0. 1uF CndV
0. IuF
I.33u
IF5
,
10
1
0
+
k
4184nd
d
.
3
2.3
F
+438
Mk1
0kGrT
kk
U
F
Ind
Gnd
20k
\/
20k
Cnd
MIT:LEES:Environmental BOO
LC Board
Note: LRLC and RRLC circuits are identical
Eric Tung
8/24/20_5
Page 1/1
.
....................
.............
................ .
. . . . . .
. . . . . . .
...........
.
.........
...........
...........
...........
...........
. ............
...
...........
...........
..........
I...........
...........
............
............
............
... ...
....
. .. ....
.
.
.
.
.
.
.
.
.
.
.
Pq ....
. . . . . . . . . . . .
.............
. . . .
. .........
.1-. 7
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.............
..
........... .
.
..
.........
..................
..
. .. .. . .. . .
';,A h
6NO
. . . . . . . . . . . . .
. . I
. . . . . . . . . . . .
. . . . . . . . . . . . I
...
.........
--........
.............
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
...
...
......
Part Number
Quantity
Description
I
LC Sensor board
2
ImH inductor
12
IuF nonpolarized capacitor
2
0.33uF nonpolarized capacitor
Vendor
ExpressPCB
sold out
176
AllElectronics
Bibliography
[1] Mariano Alvira. Courseware development for a laboratory class in power electronics.
Master's thesis, MIT, June 2005.
[2] BasicMicro. BasicAtom syntax manual v3.0.
http://www.basicmicro.com/downloads/docs/atom.pdf
(24 May 2006)
[3] A. E. Fitzgerald, Charles Kingsley, Jr., and Stephen D. Umans. Electric Machinery.
McGraw-Hill, 6th edition, 2002.
[4] Herman A. Haus and James R. Melcher.
Electromagnetic fields and energy.
http://web.mit.edu/6.013.book/www/book.html (24 May 2006)
[5] James L. Kirtley.
Class notes for 6.188, brief on permanent magnet motor.
MIT,
October 2004.
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