The Analysis and Design of a High-Power, High-Efficiency Generator
By
Andre D. Brown
Submitted to the Department of Electrical Engineering and Computer Science
in Partial Fulfillment of the Requirements for the Degrees of
Bachelor of Science in Electrical Science and Engineering
And Master of Engineering in Electrical Engineering and Computer Science
at the Massachusetts Institute of Technology
May 21, 1999
C
Copyright 1999 MCMXCIX
Andre D. Brown. All rights reserved.
The author hereby grants to M.I.T. permission to reproduce and
distribute publicly paper and electronic copies of this thesis
and to grant othe
ht to do so.
Author
Department of Electrical Engineering and Computer Science
May 21, 1999
Certified by_____________________________
6
Ce e
V
Jeffrey H . Lang
Thesis Co-Supervisor
Certified by
(
Dr. Thomas A. Keim
Thesis Co-Supervisor
Accepted by
Chairman,
Table of Contents
Heading
Page
Chapter 1. Introduction
1.1 Background
6
6
1.2 Tour of Thesis
8
Chapter 2. The Generator Model
2.1 Dimensions
9
10
2.2 Winding Structure
10
2.3 Differences Between a Real Generator
and the Modeled Generator
13
2.4 Summary
13
14
14
Chapter 3. The Design Process
3.1 Design Program
16
3.2 Summary
Chapter 4. Generator Model
4.1 Air Gap Inductances
4.1.1 Stator and Rotor Self Inductances
4.1.2 Stator-Stator Mutual Inductance
4.1.3 Rotor-Stator Mutual Inductance
4.2 Leakage Inductance
18
18
18
20
20
21
4.3 Total Inductances
22
4.4 Resistance
23
4.5 Performance Analysis
4.5.1 Maximum Number of Armature Turns
4.5.2 Back-Electromotive Force
4.5.3 Air Gap Magnetic Flux Density & Tooth Saturation
4.5.4 Back-Iron Thickness
24
24
25
27
31
4.6 Summary
32
2
Chapter 5. Results
5.1 Performance Specifications
33
33
5.2 Optimum Generator
34
5.3 Summary
38
Chapter 6. Summary, Conclusions & Suggestions for Future Work
41
References
43
Appendices
Appenix
Appenix
Appenix
Appenix
Appenix
Appenix
Appenix
Appenix
Appenix
Appenix
Appenix
44
44
49
50
51
52
54
55
60
62
63
66
A. gendesign.m
B. init.m
C. rng.m
D. sdim.m
E. cdim.m
F. lumpparam.m
G. perfm
H. cost.m
I. check.m
J. Mean Path Length
K. Characteristics of Optimum Generators
3
Acknowledgments
First, and foremost, I would like to give praise and honor to the Almighty GOD for giving me
life, health, strength, and knowledge. Without HIM, I would be nothing and my stay and MIT
would not have been possible. I thank HIM for HIS mercy and saving grace, and for never
leaving my side even though I have strayed time and time again.
My parents, Harvey and Gloria Sanders, for providing the daily encouragement and support,
financially and emotionally. I love them very dearly.
My family and friends in Michigan, most notably my sister, niece and special friend who were
always there for me.
I especially would like to express my supreme gratitude the MIT/Industry Advanced Automotive
Electronics Consortium for funding my graduate program. I would like to thank all of the
member companies for always providing the greatly appreciated advice and guidance.
I would like to thank my friends in Chi Alpha Christian Fellowship for their prayers and support:
Andres Tellez, Dedric Carter, David Estrada, Aaron Maldonado, Ted Weatherly, and Mike
Olejarz.
Last, but certainly not the least, my advisors: Jeffrey Lang, Tom Jahns, and Tom Keim. Their
guidance and support throughout have been outstanding. Their knowledge is unmatched. I
learned so much from them in such a little time. I feel very appreciative for being allowed
complete my thesis under their supervision. I am indebted to them. I sincerely thank all of you.
4
The Analysis and Design of a High-Power, High-Efficiency Generator
By
Andre D. Brown
Submitted to the
Department of Electrical Engineering and Computer Science
May 21, 1999
In Partial Fulfillment of the Requirements for the Degree of
Bachelor of Science in Electrical Science and Engineering
And Master of Engineering in Electrical Engineering and Computer Science
ABSTRACT
The purpose of this thesis is to design and optimize a new automotive generator that meets the
increased power requirements set by the automotive industry. Specifically, this thesis develops
the wound field synchronous generator. The design engine employs iterative Monte Carlo
synthesis followed by analysis and evaluation. To do so, the generator is modeled for
electromechanical performance and materials cost, and these models are used to develop a
computer-based design engine. Finally, the design engine is run to develop a 6 kW generator that
appears to be quite inexpensive. In particular, an optimal direct-drive generator is found to have a
diameter and length of 290 mm and 127.5 mm, a mass of 34.9 kg, a cost of 64 dollars, and an
efficiency of 86% at 1500 rpm and 3.25 kW. An optimal 2x-geared-drive generator is found to
have a diameter and length of 196 mm and 141.7 mm, a mass of 20 kg, a cost of 38 dollars, and
an efficiency of 88.8% at 1500 rpm and 3.25 kW.
Thesis Co-Supervisor: Jeffrey H. Lang
Title: Professor, Associate Director, Laboratory for Electromagnetic and Electronic Systems
Thesis Co-Supervisor: Dr. Thomas A. Keim
Title: Research Engineer
5
1. Introduction
1.1 Background
Over the years, the Lundell generator, the current generator in today's automobiles, has
constantly been optimized to reduce cost. These optimizations have resulted in a generator that is
sufficient for today, but because power density and efficiency have been sacrificed, it will not be
sufficient for the future.
The number of loads in an automobile is increasing, and also the power required to drive
these loads is growing. Different features that are being offered in cars today will require much
greater power, totaling approximately six kilowatts. Examples include air-conditioning, power
windows, locks, seats and steering, compact disc players, as well as numerous features that are
essential to the safety of drivers and passengers. Because of this increase in power, efficiency is
now much more important than before. The Lundell generator has an efficiency of approximately
fifty percent [1], thus if it were to supply power to these loads, six kilowatts will also be lost,
which is unacceptable. With this efficiency, other problems will be created, most importantly, a
decrease in fuel efficiency and the need to remove six kilowatts of heat. The automotive industry
has determined that this level of efficiency is unacceptable.
The possibility of the Lundell generator being the generator for future cars is not a farfetched idea. It is believed that the output capability of the Lundell generator is not fully utilized
in the present system [5], due to the high number of cost optimizations. There are several reasons
to believe that the development of an acceptable Lundell generator is possible. Because of the
shape of the alternator, it is fairly simple to change the geometry of the machine to increase the
power output [4]. Also, the Lundell generator performs well over a large speed and temperature
range; it is relatively inexpensive to manufacture; and it comes in a small package and is desirable
because of its low weight [6]. However, there are certain aspects of the Lundell generator that
6
cannot be overlooked. It has high rotor inertia [7] that leads to belt slippage. The Lundell
generator also appears to be too vulnerable to load dump. In addition, it is susceptible to
centrifugal forces at high speeds if more claws are added [3]. Given the current geometry, the
losses due to conduction in the rotor and armature, windage, and pole face are extremely high.
To achieve the desired power rating, current could be increased, but because there is limited
space within the structure this will not be an easy accomplishment. There is limited space for the
field coil [7] and the gauge distribution of the wire cannot continue to grow [2] because the size
of the generator cannot grow. Lastly, since the Lundell generator is belt-driven, it is possible to
increase the belt ratio to increase the power output, but the mechanical limit to that ratio will not
allow for the achievement of the desired six kilowatt goal [2]. Upon examining these issues, it
appears possible for the Lundell generator to be revamped to meet the needs of the automotive
industry and so it still serves as the baseline generator, even at six kilowatts of output power.
However, because of its disadvantages, new designs must be considered. The focus of this thesis
is the study of a wound-field synchronous generator (WFSG). This class of generator was chosen
based on its minimal use of power electronics, absence of commutating brushes, and its
electromechanical performance.
The specifications for a future generator are to deliver 6 kW at an engine speed of 6000
rpm and 4 k.W at 600 rpm. The latter specification is more important because it corresponds to
the highest torque. The power specifications stated above are much larger than the current
estimated output of 1.4 kW at 6000 rpm [3]. In addition to increased power, increased efficiency
is required. As a design specification the generator and its electronics must be at least 75%
efficient at 3.25 kW at 1500 rpm. Last, but not least, is cost. This new generator must be cost
effective, and cost is dependent upon the size and weight of the machine. The generator must be
designed and optimized so that all specifications, power, efficiency and others, are met, but it also
must be reasonable in terms of cost. What cost is reasonable will be determined by industry upon
7
completion of the design. From these specifications, a generator that is adequate for future
vehicles will be designed.
1.2 Tour of Thesis
The remainder of this thesis focuses on the design and evaluation of theWFSG, in both directdrive and geared-drive configurations. The direct drive generator operates at a speed that is
identical to that of the engine, while the geared-drive generator operates at two times the engine
speed. The best generator designs and a comparison between the two configurations are
presented in Chapter 5. To evaluate these designs, an understanding of the method used to find
the optimum generators is necessary. Chapter 2 begins by defining several important mechanical
characteristics as well as the winding structure of the WFSG. Chapter 3 discusses how the design
engine operates. A flow chart is provided highlighting the operations at each stage. Chapter 4
presents the models and analyses used in the performance analysis. A detailed description is
provided. Chapter 5 discusses the results found for the two generator configurations. Chapter 6
presents closing remarks about the optimal generators and gives suggestions for improving the
design yet further. A listing of the software implementation of the design engine is given in the
Appendices.
8
2
The Generator Model
This chapter describes the physical characteristics of the WFSG as modeled here. Included in
the description are definitions of its geometric dimensions, and a description of its winding
structure. In addition, this chapter presents a brief discussion of the differences between a real
generator and the generator modeled.
A 12-pole generator is shown in Figure 2-1. The outer lamination is a stator lamination
and has 72 slots, or six for each pole. Note that the teeth between slots have parallel side-walls.
The inner lamination is a rotor lamination and it has 72 slots to match that of the stator, and teeth
with parallel side-walls. Finally note that this machine is not the optimal generator designed
later. Rather it is presented here as characteristic of the type of generator modeled in this thesis.
Figure 2-1: Lamination of a 12-pole generator
9
2.1 Dimensions
Figure 2-2 shows half of a 12-pole generator, and defines the radial dimensions that characterize
the design of the generator. There are six radii; the stator and rotor each have three. The radii on
the rotor are as follows: Rin, inner generator radius;
Rbr,
inner radius to the bottom of a rotor slot;
and Rr,, outer radius of the rotor. The radii on the stator are as follows: Rsin, inner radius of the
stator; Rb,, radius to the bottom of a stator slot; Rout, outer generator radius.
Rbr,
Rout, and Rbs are
designed as random variables chosen by the simulation tool. Rsin depends upon the outer radius of
the rotor and the length of the air-gap,g, which is designed as a random variable in place of Rsin.
The inner stator radius is defined here for completeness. Note that Rin and Rout are selected after
the magnetic flux has been determined to minimize iron mass while avoiding saturation.
Figure 2-2 also defines the angular dimensions that characterize the design of the
generator. The stator tooth pitch, 0,, is an angle that spans the width of one stator tooth at
rotor tooth pitch, Or, is an angle that spans the width of one rotor tooth at
Rbr.
is not shown, but one that is an important parameter, is the machine length,
Rbs.
The
One dimension that
Lnaci.
It is measured
axially and is referred to as the stack length.
2.2 Winding Structure
The generator modeled here is a three-phase machine. It has two slots per phase belt, and a
double layer winding that is wound with full pitch. These are characteristics similar to machines
that are manufactured for automotive purposes. The main difference here is that each pole pair is
modeled as having a separate winding so that all pole pairs can be wound in parallel to reduce the
phase inductance and back-electromotive force (BEMF). Figures 2-3 and 2-4 show how one
phase of the generator is wound. This winding pattern can be extended to the remaining phases
by shifting the pattern by two slots per phase belt. In Figure 2-3, the terminal lead is shown as
10
0t
Or
Rbr
Rrout
Rbs
Fout
Figure 2-2: Cross section showing dimensions of the modeled machine.
11
Key:
-. - .
a23
+al,
+a3,
Figure 2-3: Generator poles and phase winding as viewed from the air
gap with two turns per slot. The labeling is for the 12-pole generator
shown in Figure 2-1. The current is positive for a down arrow, and
negative for an up arrow.
a1
a2
a2 3
a 24
-c 3 -C 4
-Ci
-C 2
bi
b2
b2
b4
-a3 -a 4 C5 c6
-ai
-a 2
c3
c4
-b5
-bi
-b6
-b 2
Figure 2-4: Layout of winding structure for one pole pair as
viewed from the end of the generator.
The labeling is for the 12-pole generator shown in Figure 2-1.
splitting because the pole pairs are wound in parallel. Also note that Figure 2-3 shows only two
turns per slot, but the generator can be wound with multiples of two turns per slot.
12
a24
a2
a4
2.3 Differences Between a Real Generator and the Modeled Generator
There are several differences between the generator shown in Figure 2-1 and a real generator that
are worthy of mention here. First, the cross section shown in Figure 2-1 exhibits cogging. To
eliminate cogging, a real generator would have either skewed rotor slots or a non-commensurate
number of rotor slots. These characteristics are not modeled here in order to maintain simplicity.
Second the generator in Figure 2-1 shows open slots. Most real generators have covers on the
teeth that result in partially closed slots to reduce air-gap flux harmonics and the associated losses
neither of these changes would materially impact the generator cost or any other attributes
considered here. Again, this characteristic is not modeled here.
2.4 Summary
This chapter has laid the modeling foundation for following chapters, in particular Chapter 4
which discusses the magnetics of the generator. The variable definitions that are shown in
Figures 2-1, 2-2, 2-3 and 2-4 are definitions that should be remembered for they will be used
throughout Chapter 4.
13
3
The Design Process
This chapter presents an overview of the generator design process, and the MATLAB program
used to implement that process. It also shows a simplified flow chart of the program. Details of
the program are given in Appendices A-F.
The generator design process employs iterative Monte Carlo synthesis followed by analysis
and evaluation. During synthesis, the physical construct of a generator is randomly created
within a specified limited design space. During analysis, each generator is analyzed for its
electromechanical performance. Finally, during evaluation, a list is created which contains a
description of the least expensive generators that have acceptable electromechanical performance.
This thesis seeks a generator designed for automotive application. Both geared and direct
drive generators are considered. To this end, there are several specifications that a generator must
meet if it is to be considered as acceptable. First, a generator must be sized within specified
mechanical limits in terms of its outer diameter, inner diameter, air-gap length and stack-length.
Additionally, for the geared-drive generator, the ratio of its diameter to length, commonly
referred to as its aspect ratio, must be acceptable. Second, a generator must also meet several
electromechanical performance requirements. The magnetic flux density within its core must be
below a saturation limit; the current density in its windings must be below a specified limit; it
must meet efficiency standards at specified torque-speed points; and it must be able to deliver a
specified power envelope to the load. Beyond meeting these requirements, the best generator is
determined strictly on the basis of cost.
3.1 Design Program
Figure 3-1 shows the logical flow of the generator design program which is implemented as a
collection of MATLAB scripts. An outer shell, in the form of the MATLAB script, gendesign.m,
guides program execution through this flow chart; a listing of this script is given in Appendix A.
14
The second block in the flow chart is the initialization block. This script initializes all
constants, establishes the required electromechanical performance, and clears the list of least
expensive generators. A listing of the initialization script, init.m, is given in Appendix B.
The synthesis block is third. Its job is to design the physical characteristics of each
generator. The synthesizer contains several scripts: a random number generator, a script that
applies these numbers to the design variables discussed in Chapter 2, and a script that continues
generator design by determining other important variables from the synthesized variables.
Listings for all three scripts are given in Appendices C-E.
In general, a random variable X is synthesized according to
X = Xlow + (Xhigh - Xlow ) * 8
(3-1)
where X,10 and Xsg, are the lower and upper limits of X, and 5 is a random number uniformly
distributed between 0 and 1. For some variables Xl,0 ,, and Xgh are specified in the initialization
script, while for other variables these limits depend on variables that have already been
synthesized. The limits of the radii in particular are structured so that each radius is greater than
those internal to it and less than those external to it.
The fourth block evaluates electrical parameters. This script determines the generator
inductances and resistances. A detailed discussion of it is given in Chapter 4 and its script,
lumpparam.m, is given in Appendix F. Also discussed in Chapter 4 is the performance analysis.
This script evaluates power, the number of stator turns, internal flux densities and efficiencies as
well as other optimization quantities. A listing of this script, perf m, is given in Appendix G.
The fifth block evaluates the total material cost of the generator. The cost depends
solely upon the amount of steel and copper used by the generator; and is evaluated according to
Cost=y *m
s
s
+y
c
*m
(3-2)
c
15
where Ys is the cost density of steel, m, is the mass of steel, y is the cost density of
copper, and me is the mass of copper.
After determining the cost of the generator, the design program decides whether it should
save the generator based on its electromechanical performance. It then compares the cost of
generator to that of previously saved generators that meet the electromechanical specifications.
Only then ten generators the lowest cost are saved during a single run. After evaluation, the
design program returns to the synthesis block and iterates through the loop until the number of
designs synthesized equals the number of specified iterations.
To effectively cover the design space, one million iterations are performed for each
design effort. Next, once the lowest cost generator design is found, the design space is narrowed
around its design and another million iterations are performed to optimize the design further.
3.2
Summary
An overall view of the design program is presented in this chapter. The Appendices A through I
provide a full, annotated version of the MATLAB code used to implement the design program. A
basic knowledge of how the program operates is necessary to understand the following chapters,
primarily Chapter 5.
16
Figure 3-1: The design process
showing the basic steps. It is
labeled with descriptions.
No
Yes
17
4
Generator Model
This chapter presents a discussion of the electromechanical analysis of a generator once it is
designed. It discusses in detail the method used to calculate inductances and resistances,
followed by an explanation on the performance analysis. The performance analysis involves
determining output power, stator turns, magnetic flux density, and efficiency. Both sections are
presented for only one phase, but by symmetry the characteristics of the remaining two phases
can be found. A brief summary of the important analysis points concludes this chapter.
Note that for reasons that will become apparent, the inductances and resistances will be
calculated assuming one turn per pole per slot. Later, in equations where resistances and
inductances are used, the actual resistances and inductances will be expressed by the one turn
values multiplied by the appropriate number(s) of turns to appropriate power. This permits
explicit selection of the optimum number of turns.
4.1 Air-Gap Inductances
This section describes models of both air-gap and leakage inductances. It follows Appendix B of
Electric Machinery, 4 1h Edition by Fitzgerald, Kingsley and Umans, which was published by
McGraw-Hill in 1983.
4.1.1 Stator and Rotor Self Inductances
The air-gap inductances modeled here are the stator and rotor self-inductances. The stator selfinductance calculations assume a 3-phase, full pitch generator with two turns per slot. The pole
pairs are wound in parallel with the winding structure shown in Figure 2-4.
The magnetizing inductances are derived from a basic magnetic circuit model using the
magnetomotive force (MMF) and the air-gap flux density,
Bag.
It depends on the number of
turns, area, and the permittivity of free space. The MMF waveform is initially assumed to be
square. Since only the space fundamental component is of concern, the MMF, 3, is
18
=
Ni COS 0
n 2
(4.1)
where N is the number of turns per pole pair, i is the current and 0 is the electrical angle
around the air-gap. The air-gap flux density follows as
Bag -2
i 0 Ni cos 0
7C g
(4.2)
where g is the air gap length. The air-gap flux per pole, ct, can be determined by integrating the
air-gap flux density over the area of one pole according to
j
Bag rd
toNi L mach r
ng
2
where
Lnach
(4.3)
is the machine length and r is the average radius to the air-gap.
The air-gap self inductance of each stator phase per pole, assuming angularly
concentrated windings, can now be found as
L=-
=
oN
7t
L mach r
g
(4.4)
Because the generator actually has a distributed winding, a correction factor must be added to
(4.4). This correction factor is termed the winding factor, kw, the winding factor for the winding
pattern in Figure 2-4. Also, because we wish to write an expression for inductance with only one
turn per slot, the number of turns, N, must be replaced by the minimum series turns per phase per
pole pair, Nap, which is also the number of slots per pole per phase (in this case, 2). The air-gap
inductance is now
4p
L
=
ss
71
)2
(kN
w ap
0
P
where P is the number of poles.
19
r
mach
g
(4.5)
The rotor self-inductance, Lif, is directly related to (4.5). The assumptions made for the
stator are also made here. Therefore,
L
=
ff
4pt0
L mahr
0o(k
N )2 mach P
n
wf f
g
(4.6)
where kwr is the winding factor on the field and Nf is the number of series turns per phase
per pole pair, which is effectively the series turns per pole pair because the rotor has only one
phase and also the number of slots per pole, because this is a one-turn inductance. The number of
pole pairs is not included because the rotor is wound in series.
4.1.2 Stator-Stator Mutual Inductance
The mutual inductance, M, between stator phases is assumed to be given by
M
=
1
- L
2 ss
(4.7)
4.1.3 Rotor-Stator Inductance due to the Field
The mutual inductance between the rotor field winding and a stator phase arises from flux linked
by both the stator and rotor windings. The analysis is similar to that presented in (4.1 - 4.3), but
both windings must be simultaneously considered.
From a basic magnetic circuit model with two windings, the flux linking both is a
function of the current in both windings. As a result the mutual inductance is also a function of
the geometry of both windings. Equation (4.3) applies here, except that it is a function of both
the number of stator and the field turns. Since the stator is wound in parallel, an adjustment
similar to that made in (4.4) is also made here. The mutual inductance between a stator phase and
the rotor field winding, is defined as the variable, Lsr, is then
20
wkaN
4
L
=pkfNfw
sr
o
f
L mahr
apjmach
P
g
(4.8)
The value given in (4.8) applies when the rotor winding is fully aligned with a stator
phase; the mutual inductance is assumed to vary as a cosine function with the electrical
angle measured from the fully aligned position.
4.2
Leakage Inductance
All the flux that links the stator and/or rotor windings does not cross the air-gap. Because of this
phenomenon, leakage inductance components are necessary. The leakage component of
inductance is generally difficult to find exactly. An approach related to the energy stored in a slot
is taken here.
The magnetic intensity, H, is assumed to contain only the one component, that points
directly across the slot as shown in Figure 4-1. By making this assumption, Ampere's Law is
easily applied to find the magnetic intensity. The H field is given by
fH-dl = Total Current -> H W
= NI a
aD
y slot
(4.9)
slot
where Wio, is the average slot width, Ia is the current through phase a, N is the number of
armature turns per slot, D3i10 is the height of a slot and the value of x is defined as the coordinate
of height in the slot above its bottom.
The energy stored in a slot is directly determined from the H field. The energy is given
by
D2
E=L
mach
W
slot
slot
H2d
f
0 2 o
0
N2D
l_
6 o
21
L
slot mach 12
W
slot
(4.10)
H~
Figure 4.1: A simple drawing of
Dsio,
_
stator slot is shown. Notice how the
H field only has one component.
Also shown are the dimensions of
the slot.
_
Wsiot
By equating E to
-
L i 2 , the leakage inductance, L1, may be found. This inductance also must
2 1
be adjusted to comply with the winding structure. This adjustment involves using the series turns
per phase per pole pair. The inductance is not divided by the square of the pole pairs because,
initially, the number of pole pairs multiplies the leakage inductance. This initial multiplication
provides the inductance for the entire machine in one phase. Thus, leakage component is
L
=
1
D
p N L
o ap mach slot
3PW
slot
(4.11)
4.3 Total Inductances
Since both the stator phase inductance and leakage inductance are now known, several
quantities can be found. The total self-inductance, Ls, for a stator phase is
4
L =L +L =
s
ss
1
0
(kN
w
r
)2L
mach
g
ap
7E
D
N L
+ o ap mach slot
3P W
slot
(4.12)
The synchronous inductance is
6
L
syn
=L
ss
+M+L
1
=
x
r
)2
(kN
( w ap
mach
p
g
22
D
pNL
+ o ap mach slot
3PW
slot
(4.13)
4.4 Resistance
The resistances of the stator and rotor windings both depend upon the number of turns; mean path
length, Lmp, which are different for both laminations; the conductivity of copper, cy,; and the area
of a slot, As.
The area of a slot can be calculated from Figure 4-1, by multiplying the slot depth by the
average slot width. The stator and rotor slot areas are both calculated in the script, cdim.m, which
is presented in Appendix E. The mean path length is more complex. A discussion of this
calculation is found in Appendix J.
The stator phase resistance is
2N
R=
s
L
ap mps
P
c f ss
(4.14)
aGpA
where the packing factor, pf, is defined as the percentage of the slot area devoted to the winding,
Nap
is the number of series turns per phase per pole pair, Lmps is the mean length of travel of
copper wire on the stator for one turn, As is the area of a stator slot, and P is the number poles.
This resistance accounts for the parallel winding structure by dividing by the square of the
number of pole pairs. The resistance in (4.14) is for the entire generator. The square of the poles
is not found because it is cancelled by the total number of turns per phase. The rotor resistance is
similar in form to that of (4.12). It is
L
R
rotor
=
N P
mpr
(4.15)
f
2ac p A
c f sr
where P is the number of poles, Lmpt is the mean length of copper wire on the rotor, Asr is the area
of a rotor slot and Nf is the number of field turns per pole pair.
23
4.5 Performance Analysis
The loads in an automobile require DC power, but a generator produces AC power. A 3-phase
rectifier is therefore used to perform the required conversion. To maintain simplicity, it is
assumed that the 3-phase rectifier has a constant voltage at its load with an assumed sinusoidal
phase current. The analysis here then follows from the paper titled Analysis of Three-Phase
Rectifiers with Constant Voltage Loads by Caliskan, Perrault, Jahns, and Kassakian. The goal of
this section is to establish the number of armature turns, the back-electromotive force (BEMF),
the number of field ampere-turns, calculate flux density, both air-gap and the flux across the
tooth, and the optimum stator and rotor back-iron thickness.
There are several known quantities that are used throughout this section. In Chapter 5,
they are discussed in detail because they are directly related to the performance requirements.
The known quantities are the output power, Pa,, the current density, J, the rotor speed, n, the load
voltage, VL, and the saturation flux density limit of the iron core,
Bsat.
4.5.1 Maximum Number of Armature Turns
In the previous sections of this chapter, the analysis is based upon the assumption of one turn per
stator slot for convenience. However, the actual generator design could have any number of turns
per slot. To bound the number of turns, note that the stator amperes per unit area must not exceed
a pre-specified limit, J, due to thermal concerns. Thus, a maximum number of stator turns can be
found as
N
max
=
Jp A
J f s
I
(4.16)
where I1 is rms value of the maximum sinusoidal. The DC load current is simply
24
P
I I=out4
ou
L
(4.17)
V
L
We may estimate Nmax by assuming that the rms phase current is equal to the DC load current.
N
max
=
Jp A V
f s L
(4.18)
P
out
The approximations used here are crude, but in the course of this study it became evident that the
favored designs do not have number of turns near Ninax. Nnax then serves as a boundary to limit
the range of search. Since the optimum is being found far from the boundary, inaccuracy in the
estimation of Nm.ax is relatively unimportant. The use of (4.18) is discussed in Chapter 5.
4.5.2 Back-Electromotive Force
The assumed constant-voltage rectifier model allows us to use the analysis found in Analysis of
Three-PhaseRectifiers with Constant-Voltage Loads to determine the average load current. This
average is
2V
I =3
L/
7
E
af
Eaf
d
(4.19)
7
N2
a
where
+4V
L
2
X
c
is the BEMF, and Vd is the voltage drop across a diode and
Na
is the number of series
turns per slot on the armature. The commutating reactance, Xc, is given by
(k
6pt
X
=
0
D
r p N2 L
N )2 L
w Pap
mach + o ap mach slot
g
3P 2 W
slot
Q
(4.20)
Equation (4.20) is obtained by adding (4.12) and (4.7) and then multiplying by Q, the electrical
angular frequency.
25
Preliminary designs were performed by requiring the stator output power at each of three
operating points to be equal to the corresponding specified value. It was noted in these
preliminary designs that the low speed high power design point (600 rpm, 4000 watts) required a
very substantial excitation power. As a result, the requirement at this design point has been
interpreted as being 4000 watts net of excitation power. The following derivation shows how it is
possible in closed form to calculate the BEMF which will result in the desired net power. The
input power generated in one phase of the stator is found by multiplying (4.19) by the load
voltage
2V
KL3V
P
V
in
I
L(\ L
E
L
L
2(
af
+4V
Na
4.)
(4.21)
2X
N
a
where
d)
c
is the number of series turns per slot on the armature. The generated power can be
equated to the sum of output power and rotor copper loss
P
in
=-3
P
out
+ N fI f 2R
f( fjrotor
(4.22)
where the NfIf term is the field ampere-turn product per pole pair. Nf appears in (4.22) because
Rrotor is the value for Nr equal to 1. Note that a factor of 3 is present because there are 3 phases.
The ampere turn product can next be written as a function of the BEMF
E
N I
af
=
ff
(4.23)
QN L
a sr
Combining (4.21), (4.22), and (4.23) yields a single equation for the output power
P
=
out
L
nN 2 X
a
E
2
2
+ 4V
'2V
9V
L
d
af
2
E
af
!Q N L
R
I
af
ca
26
(4.24)
rotor
By manipulating (4.24), a quadratic equation in the BEMF can be found
2
2P
E4 +E2
af
af
QN L
a af
out
R
otorNX
2
_V
3V4
L
1QLfIL
R
a c rotor
rotor
4
af )
(4.25)
+
Na Laf)
R2
rotor
9
p2 +
VL
7r2 N 2X
out
a c.)
2V
+4V
L
j
=0
d)
Equation (4.25) has two roots. Either both of the roots will be positive or both will be imaginary.
If both are imaginary, then the generator cannot meet all the constraints. However, if the roots
are positive real, the lowest root that is at least the line-to-neutral value of the internal voltage, is
the optimal value for the Back EMF.
4.5.3 Air Gap Magnetic Flux Density & Tooth Saturation
The air-gap magnetic flux density is computed on the basis of the winding structure shown in
Figure 2.4. The flux density in the air-gap,
Bag,
can be analyzed by determining the stator and
rotor MMF. The flux density has a maximum value when the MMFs are aligned and a minimum
value when they are misaligned. The flux density will therefore be a function of both the MMF
magnitudes and the angle between stator and rotor magnetic axes,
6
s,.
The armature and rotor MMF can be written as a function of the magnetic field intensity,
H, by using Ampere's Law. By imagining a contour that circles one pole of the generator
including both the stator and rotor, the total current can be determined. Again it should be
remembered that the number of pole pairs divides the current through each phase because the
pole pairs have separate windings. Also, at the instant where the current in one phase is
maximum, the current through the other two phases is half that of the peak current. The choice to
have the peak current through phase a is done arbitrarily, the analysis can readily be extended to
any of the phases.
27
The current contribution from each phase is doubled because there are 2 slots per phase
per pole. Using Ampere's Law yields the peak magnetic intensities on the stator and rotor, Hs
and H,
2(
2H g= - 4N N I
S
P
sa a g
2H g=N
r
where
Na
sf\
(4.26)
(N I
f f
is the number of armature turns per slot, Nsa and Nst, are the number of slots per pole
on the armature and rotor, and NfIf is the series ampere-turn product for one rotor slot. Also, the
gross phase current is defined as, Ig. The current is multiplied by the number of pole pairs which
represents the parallel winding structure. To be consistent with other values that are being taken
from the AC side of the rectifier, such as the generated power which is used to calculate the Back
EMF, the gross phase current is used. Its relation to the load current, IL,is
C
I
g
=1
L
+
2
f f)
rotor
VL
(4.27)
The armature and rotor MMF over one pole can be extracted from (4.26), and taking the
fundamental component and using algebraic techniques
2
S
k N
w
sa
N I
a g
(4.28)
P
E
2
3
2
r
=-k
n
wf
N
sf
(4.29)
N I
f f
where a winding factor has been added because the winding is distributed and has two layers.
The two MMFs add vectorially, so it is necessary to determine 6 sr, the angle between
them, in order to determine the total air-gap MMF. Figure 4.2 is essential in determining 5sr. The
28
figure shows a 4- by-4 matrix that relates the phase and rotor flux linkages to the inductance and
phase and field currents.
L
X a
a
b_
-M
c
- M
-M
-M
L
-M
L
L
b
L cos
_af
bf
L
- M
f
f)
L
bf
cos 0
(f
2-j
3 )
L
cf
L
c
af
cosr f
kf)J
-
cos 0f
27
3)
cosc 0cf
f
3
cos 0 +
(f
3 )
a
b
L
f
ff_
Figure 4-2: Shows a 4x4 matrix that relates flux linkage,
inductance. and current
By taking the derivative of the field flux linkage, kr, the torque, T, can be found
T = ak[i L cososj+i L cos
86O
a af
f)
b bf
C
f
2-n)
3 )
L
f
coss
f
2x
3
i
Jf]
(4.30)
where the electrical angle between the rotor magnetic axis and the phase a magnetic axis is
defined by Of. Torque is not a given quantity, therefore an equation that relates torque to a given
quantity is desired. This is found by relating it to the input power
P. = -- o T
in
m
(4.31)
where om is the mechanical angular velocity. The negative sign in (4.31) is because the analysis
is performed for a generator.
Substituting (4.30) into (4.31) yields an equation for the generated power
Po
P =
in
L
m
2
Isr f
i sin
a
o+i
f
sin(Of _ 27 + i sin (
+ 27C
be
3)
e
f
3
The phase currents are assumed to be sinusoidal and have the form
29
(4.32)
i
a
;
=ICos 0
L
Is)
i
b
=I
L
Cos 0
s
2
3)
;i=ICos
c
L
0
s
+ 2T)
3)
(4.33)
where Os is the electrical angle between phase a and the stator magnetic axes. By substituting
(4.33) into (4.32) and applying trigonometric relations, a simplified equation can be found
relating the input power to the angle between the magnetic axes,6r. Note that the angle found by
subtracting
0
, and 0,, is not the desired angle. If two vectors are imagined as representing the
magnetic axes, the angle found is the one measured from tail-to-head as shown in Figure 4-2a.
The correct angle is the one measured when the magnetic axes are positioned tail to tail as shown
in Figure 4-2b.
Rotor
magnetic
axis
Stator
So
Stator .magnetic
magnetic
axis
axis
E-6s,
6
sr
Rotor
ymagnetic
axis
Figure 4.2b: The desired
angle.
Figure 4.2a: The angle
calculated using (4.31)
and (4.32).
The angle between the magnetic axes can now be written as
2P
sr
-
= sin-
(4.34)
in
Po N N I L I
m a f f af g
The magnetic flux density in the air-gap can now be found by using Figure 4-2b and the law of
cosines to obtain the resultant MMF, 3
B
ag
=
_sr
o g
o
g
sr
cos
2 +2 -20
sr)
s r
r
s
30
(4.35)
Although the air-gap flux density is of concern, it is not the quantity that will be compared to the
saturation flux density. The flux density that flows across the teeth of each lamination is the
value that is desired. Both can be found by properly scaling (4.35). The flux is continuous across
the boundary, therefore
B rO6L
= B rOL
st av t mach
ag av st mach
(.6
(4.36)
where B,, is the tooth flux density on the stator, 0, is the span of a stator tooth, and Os, is the stator
slot pitch. Solving for the tooth flux density yields
B
B
0
ag t
0
=
St
(4.37)
St
The rotor tooth flux density, B, is similar. It follows as
B
B
=
rt
0
ag r
(4.38)
0
rt
where 0, is the span of a rotor tooth and 0s, is the rotor slot pitch. The two quantities in (4.37) and
(4.38) are monitored to determine whether the generator saturates.
4.5.4 Back-Iron Thickness
It is desirable to have the optimum thickness in the back-iron of both laminations. That is, it is
desirable that the magnetic flux density is near saturation on the return path of the flux in the rotor
and stator back iron. To find the optimum thickness, the flux in the back-iron is found first as the
air gap flux integrated over one half pole. With the assumption of a sinusoidal air-gap flux
density, the expression
31
7t
2
B
sin 6 rdO = B
0 ag
T
(4.39)
sat
can be written to equate the air gap flux collected by a half pole to that which is present in the
back iron. The back iron flux density is assumed to be at the saturation level of the core. Here, T
is the desired core thickness for both the rotor and stator. Integrating and solving for T gives the
optimum thickness
2B
T
r
agav
PB
(4.40)
sat
This thickness is designed into the generator by adjusting Rin and Ro, after they have randomly
designed.
4.6
Summary
This chapter presents assumptions and methods, by which the electrical parameters are calculated
and by which the generator is analyzed for its electromechanical performance. The methodology
will be recalled in Chapter 5, which discusses the results of several generator designs. This
chapter only discusses the important parameters and does not model all of the quantities used. A
full listing of the code used to design the generator is given in Appendices F and G.
32
5
Results
This chapter presents the generators designed by applying the methods discussed in Chapters 2, 3,
and 4. The results are given for direct-drive and geared-drive generator configurations in which
the geared-drive generator operates at twice the engine speed. The specifications that drive the
design process are given first. This is followed by a table which presents the important
parameters for the optimum generators and a comparison of the two generator configurations.
This chapter concludes with design and performance details.
5.1
Performance Specifications
The generators are analyzed at three different speeds: 600, 1500, and 6000 rpm. At each speed
the generator is required to meet a specific output power requirement: 4000 W at 600 rpm, 3250
W at 1500 rpm, and 6000 W at 6000 rpm. In addition, the generator has to be efficient when
generating the desired output power. The requirements only specify that the design be at least 75
percent efficient at 3250 W and 1500 rpm. However, in this thesis the efficiency at all speed
points is computed for completeness. Since a thermal model is not used, a current density
requirement is used as a replacement to ensure that the generator does not overheat.
The stator
and rotor current densities cannot exceed 2000 amperes per square centimeter.
Each generator is designed with the assumption of using an M-19 steel core. The flux
through the teeth of both laminations must be less than the saturation flux density which is
approximately 1.8 T. The air-gap spacing must be at least 0.635 mm, and it is allowed to range
from this value to 10 times this value. The direct-drive and geared-drive generators must have a
machine diameter no larger than 300 mm. The length of the direct-drive generator is allowed to
vary from 0 to the value of the machine diameter. This restriction allows "pancake" generators,
but for the geared-drive generator, pancakes are not allowed; it has a requirement on the aspect
ratio, the ratio of the machine diameter to length may be no more than 2.
33
Knowing now the specifications for each generator design, a word must be added on how
the design engine determines several parameters, most notably the number of armature turns. As
mentioned in Chapter 4, the maximum number of armature turns can be found from the current
density limit. The design engine then uses this value and, iteratively, counts down to the
minimum value, which is one turn; fractional turns are not allowed. Once a value is found that
keeps the magnetic flux density below its saturation value, the design engine then uses this
number of turns to analyze the efficiency criteria. If the number of stator turns passes the
efficiency requirement, then the generator can be considered an acceptable design, if not, the
machine generator is discarded and the design engine returns to the beginning of the loop to begin
the process again.
The determination of the total cost of the machine in this thesis depends only on the mass
of steel and copper. The cost given here is simply a lower bound on the expected total cost of the
generator. The specifications require that the cost of steel and copper be $0.45 per pound and
$2.27 per pound, respectively. Other necessary parameters are mass densities; they are 7462
kilograms per cubic meter for steel and 8960 kilograms per cubic meter for copper.
The mechanical and electrical specifications are displayed in Table 5-1 and Table 5-2,
respectively. A generator must meet these requirements if it is to be considered an acceptable
design. Beyond this, the optimal generator is the one with the lowest cost.
5.2 Optimum Generators
The two types of generators, direct-drive and geared-drive are optimized here, meaning that they
meet the mechanical and electrical specifications shown in Tables 5-1 and 5-2 and they are the
lowest cost generators. The optimum generators are found by first running 3 million iterations for
generators with varying numbers of poles. Here, generators with 8, 10, and 12 poles were all
designed for direct-drive generators, as were 4, 6, 8 and 10 pole generators for the geared-drive
configuration. This choice was made based on experience. The design space was then narrowed
34
and run for another million iterations for the number of poles that appeared to be optimal in that it
provided the lowest cost. This exhaustive search method is done for both types of generators the
detailed listings of the initialization scripts for all wide and narrow design runs are given in
Appendix B.
Figures 5-1 and 5-2 compare the lowest cost generators for each number of poles after
running the first million iterations for both the direct-drive and geared-drive generators. Also
displayed is the lowest cost generator of each type after the refinement process. The figures
follow a parabolic shape for each type as expected.
Parameter
Direct Drive
Geared
Outer Diameter
(mm)
Inner Diameter
(mm)
<300
<300
>165
>82.5
Air-Gap
0.635
0.635
(mm)
Length
-
> outer radius
Packing Factor
35%
35%
Cost of Steel
dollars
0.45
0.45
2.27
2.27
lbs
Cost of Copper
dollars
lbs
Table 5-1: The mechanical specifications that
must be met by each generator design.
35
600
1500
6000
Output Power (W)
4000
3250
6000
Efficiency (%)
-
75
-
Terminal Voltage (V)
42
42
42
Magnetic Flux
Density(M-19 core)
<1.8
<1.8
<1.8
<2x10 7
<2x10 7
<2x10 7
(T)
Rotor and Stator
Current Density
(Am
2 )
I
_
_
_
_
_
I
_
_
_
_
_
I
_
Table 5-2: The electrical specifications that must be met by
each generator. The specifications are the same for the direct
drive and geared generators, except the geared uses a multiple
of the engine speed.
Total Cost vs Number of Poles
90
80
70
-0. -'
60
-. -
--
direct-drive
---
geared-
-
drec
refined
0
o-50
240
0
-search
-30
20
refined
-~-
10 -search
0
2 4
6 8 1012 14
Number of Poles
Figure 5-1: Total Cost versus Number of Poles for
direct drive and geared type generators.
36
_
_
_
_
Several important parameters from the optimum generators of both types are shown in
Table 5-3. The direct-drive generator is a 10-pole generator with 9 armature turns per slot. A 2dimensional drawing is shown in figure 5-2a. A complete listing of generator characteristics is
given in Appendices K, L, and M.
Speed
Number of Poles
lx
10
2x
6
Cost (dollars)
Mass (kg)
64.26
34.9
38.18
20
Outer Diameter (mm)
-290
-196
Length (mm)
127.5
141.7
Aspect Ratio (Diameter to Length)
2.27
1.38
Efficiency (at 1500 rpm)
86%
88.8%
Stator Current Density (worst)
1.99 X 10
1.99 x 107
Rotor Current Density (worst)
1.65 x 107
1.41 x 107
Peak Stator Tooth Flux (T)
1.66
1.60
Peak Rotor Tooth Flux (T)
1.78
1.76
Peak Shear Stress
1.547
1.34
( Am__2
)
Table 5-3: Comparison of results
found for direct drive and geared
type WFSM generators
The optimum direct-drive generator has 10 poles and 9 stator turns per slot. The physical
appearance of the machine is reasonable. The generator was not fully optimized after the search
was refined. All parameters are against the upper bound of their specified limits, except that of
efficiency and the diameter. The efficiency could be made closer to the lower limit of 75 percent
by decreasing the area of the rotor slots, for example, thereby increasing the rotor copper losses
and decreasing the efficiency. However, the rotor current density will then increase and thus can
37
be tolerated until it violates the thermal constraints. Thus, the cost optimal generator will more
nearly meet the efficiency specification.
The optimum geared-drive generators have 6 poles and 8 stator turns per slot. The
generators were fully optimized after the search was refined. The physical appearance of the
machine also reasonable, except that of the stator slots. In order to achieve the required output
power, the design engine has chosen deep stator slots to allow the stator current density to float
towards its upper bound. All other parameters are against the upper bound of their specified limit,
except that of efficiency. The efficiency could be closer to the lower limit of 75 percent. Butjust
like the direct drive, a manual optimization can be done. The rotor current density is not near its
maximum, therefore further decreases in slot area can be tolerated until other specifications are
violated.
5.3
Summary
Both types of generators are reasonable in terms of air-gap length, and stator and rotor
slot make-up, more so the direct-drive generator. However, the direct-drive generator is a more
expensive machine than the geared-drive generator. Since cost is the ultimate deciding factor, the
direct drive generator probably is not the best type. A cost associated with gears is not included,
but that cost is not likely more than 50% of the direct drive cost.
The geared generators have an advantage because they are much smaller machines,
approximately 196 mm in diameter compared to 290 mm for the direct drive thus the geared
generator would be much easier to fit inside an automobile. The smaller generator should result
in a higher shear stress, but in this case it does not. The direct-drive generator has a shorter stack
length, but not by enough to make it superior.
In all aspects except that of the mechanical dimensions and cost, the direct and geared
drive generators are similar. From the automotive viewpoint, a small, inexpensive machine
would be desirable. The geared generator meets this criterion.
38
Figure 5-2a: Direct Drive
Generator. The characteristics are
shown in Table 5-3. column 2.
39
Figure 5-2b: Geared-drive generator.
The characteristics are shown in Table
5.3, column 3.
40
6
Summary, Conclusions & Suggestions for Future Work
The research presented in this thesis discusses a method to find the optimum WFSM generator for
automotive applications. In Chapter 2, the characteristic design of the generator is discussed. It
provides definitions for the design variables and the winding structure, and it discusses how the
generator as designed here is different from a real model. In Chapter 3, the basics of the design
engine are covered. It gives an overview of how an acceptable generator is found. Chapter 4
presents a detailed description of the derivation of the electrical parameters. It also discusses the
analysis method. Results from running the design engine are given in Chapter 5. The
specifications for each generator are covered in addition to the optimum designs for both direct
drive and geared type generators are covered and a comparison between the two types is given.
6.1
Conclusion
Both configurations of optimum generators, direct and geared drive, are highly efficient and
relatively inexpensive. In fact, the geared-drive generators are significantly more efficient than
the design specification because the thermal design limit imposed on the stator and rotor current
densities demands enough copper to force high efficiency. Both generator configurations seem to
favor deep stator slots, extra stator heating and lower field ampere-turns. But these are not
particularly troublesome except possibly for a solution for stator heat dissipation. Except for the
problems of gearing, the geared type is more desirable for the automotive industry because of its
cost and small package compared to that of the direct drive, approximately 38 dollars and 20 kg
to 64 dollars and 35 kg. Given the results, a low-cost, highly-efficient and reasonably sized
generator certainly appears feasible.
6.2
Suggestions For Future Work
Because of the high current densities in the optimal generators, a thermal model should be
derived to provide a more accurate assessment of thermal performance. Also, the Lundell
41
generator, and perhaps another candidate generator, should be examined in a comparison the
WFSM discussed here. The specifications should be the same and neither generator should use
extensive power electronics, but a simple 3-phase rectifier as is used here. Lastly, the slots
should be examined, more specifically, partially closed slots and skewing of the rotor slots, all of
which prevents cogging should be designed. This might change a few dimensions but the overall
performance and cost of the optimum generators should not change. Load-dump is another factor
that should be considered given the high back emf voltage at high speed.
42
References
1. Afridi, K. A Methodology for the Design and Evaluation of Advanced Automotive
Electrical Power Systems. MIT Thesis, 1996.
2.
Correspondence with Dr. John Miller of Ford Motor Co. July 1998.
3.
Gutt, H. and Muller, J. "New Aspects for Developing and Optimizing Modern
Motorcar Generators. IEEE IAS, October 1994, Denver, CO.
4.
Kuppers, Henneberger, and Ramesohl. "The Influence of the Number of Poles on the
Output Performance of a Claw-Pole Generator." ICEM, September 1996.
5. Liang, F., Miller, J., and Zarei, S. "A Control Scheme to Maximize Output Power of
a Synchronous Alternator in a Vehicle Electrical Power Generation System."
6.
Mohan, B and Macke, E. "ElectroMagnetic Components in Dual Voltage Systems."
7.
Naidu, M., Boules, N., and Henry, R. "A High Efficiency High Power Generation
System for Automobiles." IEEE Transactions on Industry Applications, vol. 33, No.
6, November/December 1997.
43
Appendix A.
%script
init;
Start-up, gendesign.m
runs the alternator
design program.
%makes call to init
script
while desno <= n,
%establishes loop management
fprintf('\nBeginning Design Number %d\n',desno)
rng;
%generates random numbers
sdim;
assiqns random numbers to synthesizedvr
cdim;
%caicuiates remaining dimensions
lumpparam;
ealculas
lumped pa rametea
f:1ro di mensio
perf;
cost;
tevaluate cost
cst
mach nes
saves bs.a
nd lest:
check;
desno=desno+l;
fprintf('\nCost
= %g,Best Cost=%g\npoles=%g\n',tcost,min(savetc),
ri
f ( ' \ncount e
g
, couter
saveef-=2,
if
[rout;rbs;rsin;rrout;rbr;tthang;rtthang;L]
e imen
eyn
end
end
44
2
*pp);
Appendix B. 1 Direct-Drive Initialization Script, init.m
The following is the script, init.m. It's in its MATLAB form. The percent signs means that the
code is commented out. This is the version of the initialization before the refinement
initialize
cnstants to be used trsSIogout
n=le6;
%nuber of design iterations
emaximur alternator
radius in m for dr
1.5!nimum
.altrnator
radius C inm
for de
driv
..rinmnin=8.25e-2;
:sminf
inner radius for direct drive
desno=1;
:tses des igr no
s
60
1500 6000 ;
speed in rpm for direct
P=[4000,3000,6000];
pp=4;
spb=2;
esltspe
phaso
bel
sp=6;
mniumr[
ar -g ap length (m)
gmin=0 . 33e-3;
1J
pf=0.35;
or
spakin fact
cs=0.45;
dollars
per nound--cost for steel
cc=2.27;
%doll
pe rs kg -- tee1(
etric
csm=cs*2.205;
-t-e
0'r''
.
ccm=cc*2.205;
sigmac=4.45e7;
sd=7462;
per m''3--c-ppe
nk
akg per m'- -see
cd=8960;
Nap=2;
nUmr
z
f slot
ton pe phSe.rmtu.
Nf=6;
muzero=4*pi*10^ (-7);
V1=42;
Bgsat=1.8;
Jmax=2e7;
Jrmax=2e7;
Vd=1;
%counts the number of machines found
counter=0;
rouLtmax=.15;
ro ut1i-r
45
Appendix B.2 Geared-Drive Initialization script, init.m
The following is the script, init.m. It's in its MATLAB form. The percent signs means that the
code is commented out. This is the version of the initialization before the refinement.
in. it
iize
cons tantCs to b~~eused th roug~hout
n=le6;
number of desi n.
routmax=.0825;
maximm ateriato
troutin=.22e-2;
rinmin=4.125e-2;
desno=1;
teratio s
r radius i (m) for geared
rive
eminimum a.te~~rnator radius inm m fo~r gearea drLv
innier radius (m) for geare drive
design
4sets no
speed=2.*[600,1500,6000];
rm
o
e
P= [4000, 3000,6000];
pp=3;
spl
pairsF-is
spb=2;
sp=6;
mi iu
i - a
e g h m
gmin=0.33e-3;
slOts pr
pf=0.35;
pol
spakin.fato
cs=0.45;
cc=2.27;
csm=cs*2. 205;
ccm=cc*2. 205;
c
g-opr(e
per
sigmac=4.45e7;
ity
in m
fr
copper
sd=7462;
JkJ per
cd=8960;
I):1 pe.
Nap=2;
o turrns
lt
per phase (armaturt
Nf=6;
muzero=4*pi*10^(-7);
spereabiity f fee sp~ace n
V1=42;
Bgsat=1.8;
Jmax=2e7;
urr
t ds
l
in A/m'2
Jrmax=2e7;
A/'I
Vd=1;
counter=0;
%counts the number of machines found
46
Appendix B.3. Direct Drive Initialization Script After Refinement, init.m
The following is the script, init.m. It's in its MATLAB form. The percent signs means that the
code is commented out. This is the version of the initialization after the refinement.
Note that in the stack length, and stator and rotor tooth angles, have both minimum and maximum
values which are not shown on this sheet. They are as follows: the length ranges from 0.068 m
to 0.092 m, the stator tooth angle ranges from 0.053 rad to 0.0718 rad, and the rotor tooth angle
ranges from 0.0437 rad to 0.06 rad. It also worth mentioning that the air gap has a maximum
value of 0.448 m.
nitializes
n=le6;
routma:=.142;.
ruti=.109;
rni=.5-
constants
to be used throughout
number of d4esign iterations
raiu
s in
radiuas in
m~ for
mn f or
m
iumaleratr
minimu
aternatr
;rmin
desno=1;
%st
speed= 600, 15;0i
P=[4000,3000,6000];
pp=4;
spb=2;
sp=6;
gmin=0.33e-3;
pf=0. 35;
eslo
cs=0.45;
cc=2.27;
csm=cs*2.205;
ccm=cc*2.205;
sigmac=4. 45e7;
pcngC
1-d:llars
sd=7462;
cd=8960;
Nap=2;
Nf=6;
muzero=4*p i*10^ (-7);
Vl=42;
Bgsat=1. 8;
Jmax=2e7;
Jrmax=2e7;
Vd=1;
counter=0;
inne
ras
di
direct..
for
directa
directdrv
dri
drive
no
rpm for direct
S
0
pe-
pole
faco
per
:
E!":
pud-os
fo
dollars
pound-~coft
per
fo
e~~1
aa
r..i>C
copper,
see
I
/a5.
voltage
i r
sla,
maYximtum flux density
in steel
%.urrent
density limi
in A/m^2
urr
d
limit on rotor A/^ 2
(fowarddio
drop
%counts the number of machines found
47
cor.
Appendix B.4 Geared Drive Initialization Script After Refinement, init.m
The following is the script, init.m. It's in its MATLAB form. The percent signs means that the
code is commented out. This is the version of the initialization after the refinement.
Note that in the stack length, and stator and rotor tooth angles, have both minimum and maximum
values which are not shown on this sheet. They are as follows: the length ranges from 0.07505
m to 0.10 15 m, the stator tooth angle ranges from 0.0584 rad to 0.079 rad, and the rotor tooth
angle ranges from 0.04964 rad to 0.06716 rad. It also worth mentioning that the air gap has a
maximum value of 0.401 m.
i
al
zes constants
n=le6;
routmax= .0911;
6.73-2;
to b
use d
throughout
m
mniu
ae
or
alentr
raiu
in
mL
or
(
gear:3
rinmin=4 .125e-2;
a
od
desno=1; 8.25e-2;
1
min inner
speed=2. *[600,1500,6000);
gIr Ered
see CIn r. m
P=[4000, 3000,6000];
-poepuairs
pp=3;
%slos
per phase e
spb=2;
,slots
sp=6;
per (pole
(mnimum air-gap lentha
m
gmin=0.3 3e-3;
pf=0.35;
cs=0.45;
1dollars per pound.- - CoStL for st eel
cc=2.27;
p
kg
do
csm=cs*2 .205;
ccm=cc*2 205;
ed lar
per kg-cpe
)
(mtrc
sigmac=4 .45e7;
in1 mosmfrcper Tcnutvt
sd=7462;
cd=8960;
Nap=2;
Ynubeof
slo
turns"
pe
phs
(amtD
unmer o
Nf=6;
il
un
e
oepi
eris/
muzero=4 *pi*1 0^ (-7);
spaceinspermeeabJ iity -ffe
V1=42;
Bgsat=1. 8;
% ren
density
limit
inl A/m^
Jmax=2e7
%nty l
Jrmax=2e 7;
rotor A/m^2
Vd=1;
%fourad dthe derop of
counter= 0;
%counts the number of machines found
48
ri
driv
Appendix C. Random Number Generator, rng.m
This script sets up a vector that includes 8 random numbers between 0 and 1. Each entry in a
vector is assigned to a mechanical dimension found in Appendix D.
%Raridomr
number generator
A=rand (8, 1);
nol=A (1)
no2=A (2)
no3=A(3)
no4=A (4)
no5=A(5)
no6=A (6)
no7=A (7)
no8=A (8)
49
Appendix D. Random Variables, sdim.m
%script calculates synthesized
dimensions
souter radius of alternator
+ no i* ( routmax
.rout roumi
-routmin);
-ar-apradius
g
Xgi
+ no8
*
(10*gcmin-gmiln);
of rotor dimensions
%Erlradiu
rrout= rinmin + no2*(routg - rinmin);
h r',-i.
r.n radius
of
dimensos
rotor
rbr= rinmin + no3*(rrout
-
rinmin);
%in rner rad:i us orf stat or
rsin=rrout+g;
stator
rbs=
-
back-iron radius
rsin + no4*(rout
nu4.
r*.ou -O ;
n*
Out(7i
r
i2
haau
c.,k
i..
-
rsin);
dire
t
diI
ve:
P*Y)>
-Ao)
ttooth
50
It
to
c
Appendix E. Synthesizer, cdim.m
script uses
syn tensized
va
iables
to
remain
cal culate
uimesion
%onersot.s
& rotor slots
%Number of stator
slts=sp*pp*2;
%stator
slot-tooth angle--from middle of slot
to middle of slot
ica l radians
%in mhan
slthang=pi/ (sp*pp) ;
ooth
s pan
in
cal
mh
radians
tthang=2*tthangl;
slsp=slthang-tthang;
trot
inians
wtsslt=rsin*slsp;
stthw=rsin*tthang;
wbsslt=rbs*slthang-stthw
sot-o
w
n
dt
avwsslt=(wbsslt+wtsslt)/2;
rslthang=pi/(sp*pp);
rtthang=2*rtthangl;
rslsp=rslthang-rtthang;
rotor
top and bottom
slot
in
mechanical
Q
radians
widths
wtrslt=rbr*rslsp; iroto
top sl
rotor t
rtthw=rbr*rtthang;
wbrslt=rrout*rslthang-rtthw;
wdth
Voth
ao
bottom sl
ar rotor slt width
avwrslt=((wtrslt+wbrslt)/2);
-stator
and rotor
sslht=rbs-rsin;
rslht=rrout-rbr;
slot heights,
reospectively
%stator
rot or
%regi nni
Estim :g on
)natiof mean oath ergth
angle using average slot width and stator
tooth width
theta=asin(avwsslt/(avwsslt+stthw));
meanpath
legth
around armature, single turn per slot
mpL=2*L+(pi*avwsslt + 4*(3*stthw + 2.5*avwsslt)/cos(theta))*1.2;
51
%calculation for stator slot area
taslt=avwsslt*sslht;
ffor one slot
uaslt=pf*taslt;
usable area for ne s
tuaslt=uaslt*slts;
%usable area for al s ot s combined
calculation of rotor
slot
area
tarslt=avwrslt*rslht;
uarslt=pf*tarslt;
tuarsit=uarslt*slts;
%Length of wires--for rotor
he following is the Iength of semicircle
Ll=2*pi* (avwrslt + rtthw) /2;
%average rotor slot width plus
%width
L2=2*pi*((avwrslt + rtthw)/2 +
avwrslt+rtthw);
L3=2*pi*((avwrslt + rtthw)/2 +
2*(avwrslt+rtthw));
Lt=(6*L + Li + L2 + L3) ;
%total ength of conduc ort
%polI1e
%(single turn/slot
52
)
tooth
o
Appendix F. lummparam.m
arid Resi stances
%Calcullation of Inductances
3 phase machine
%full-pitched,
%each pole pair has a separate wouncinn--
Unless otherwise noted egs
.
pDlaced in
1
ur
per
Slot
parallel
appenaix e
at,
at
Fitzqera'o
fr
taken
AssuminPg
Factors
%Breadth factor, elct rical
Kb= sin (spb*pi/sp/2) /(spb*sin (pi/sp/2));
Kbf=sin(Nf*pi/sp/2) /(Nf*sin(pi/sp/2)
Kw= (Kb);
avrag=rrout+g/2;
savera
ai
cr
Lff=(4/pi)*muzero*Kbf^2*Nf^2*L*avrag/g;
j.
hs
au to sl..ot-laagelu
notes
-Jf's
9
meotho
K singiEnergy
comone
Lal=pp*muzero*Nap^2*L* (sslht/ (3*avwsslt)) ;C
Lal)/
L=Lsl;
/pp^2;
(-0.5*Lss)
M=
S~atolr
+eponen;
Mt. 1aL i
f'or
rLcae(d
f2ora
uctace -sat
ce
elf Mnucta nc
'Tntat
T(tal
;b*
/r(pp^2)
Lal)
La=(ssi)
%Tota sf-nuanc
%L=b~b + Lbl;
Ls=(1
5*Lss
'
duaoargpnlxo
sefinutac
sao
LoLS; e
g
taor'slf-nucane
r-i
C'
i.AZA
d
s
%
(4/pi)*muzero*Kw^2*Nap^2*L*avrag/g;
Lss=
sL
"b><'C
phase-
ph.uases--
a pgap&
avran(
coprn
e p or phal-
for phso
+ Lal)/ (pp2)
%Statf-or-Rotor Mutual induc-tance
Laf= (4 /pi) *muzero* (Kbf*Nf ) *(Kw*Nap) *L*avrag/ (g*pp);
ewd
induct.ance beween p has-9e7 a and fielI-F
P:bf= Laf %Pae b to f:iel1.d
jClclIon
t
.,,4of ReistanceA
Rrotor=Lt*(Nf)*pp/(sigmac*uarslt);
.l ati on of armatiure resistanCe
Rarm=Nap*mpL/(sigmac*uaslt*pp);
53
SMut1-ual
ps
Appendix G. perf m
The performance analysis takes place here. The first several lines initialize the variables, most
will be a 1x3 vector.
%Evaluates performarnce of machine
Eaf=[];
Na=[];
N=[];
B=[];
Bt=[I];
If=[]
A= [I;
C=[];
D= 0;
F1=[];
F2=[]
F3=[]
Q1=[]
Q2=[];
Q3=[];
Vs=[];
Vs1=[]
Vs2=[];
Vs3=[];
Currl=[];
Curr2=[];
Curr3=[];
saveef=2;
rin=rinmin;
T=0;
j =0;
%each entry [600rps, 1500rpms,
n.net
Iln=P/Vl;
6000rpms]
load current, power
ver load voltage
freq=pp*speed/60;
icycles per see, returns values at 3sp3eeds
omegae=2*pi*freq;
elect ri.cal
-
per sec
%mechaical radians per sec
omegam=omegae/pp;
Xc=omegae.*(La
raians
M);
reactance dependant
rcommutating
%commutation induct ance, Ls-M
%Calculate the range for number of armature turns-Density Limit
Nal= floor((Jmax*uaslt*sqrt(2))./Iln);
'Current
%Taking minimum value of Na
Na=min (Nal);
54
upon
if
Na<1,
fprintf('Design %d No Good, Turns less than 1\n',desno);
else
-Determini
ng the number of turns that. satisfy saturat.i-on
mi
for i=Na:-1:1,
if all(Bt<=Bgsat),
if saveef==2,
%Establishing Constants -a Ck Em
1
twe need
in
order
to
meet net
will be used as substitutes into
output
cower
A= Rrotor./((omegae*i*Laf).^2);
C= (9*Vl./(pi*i^2*Xc));
D= (2*Vl + 4*Vd)/pi;
F1=[A(1).^2
(2*P(1).*A(1)-(C(1) ).^2)
F2=[A(2).^2 (2*P(2).*A(2)-(C(2) .^2)
F3=[A(3).^2 (2*P(3).*A(3)-(C(3) ) 2)
Q1=roots (Fl);
Q2=roots(F2);
Q3=roots(F3);
%WVoltage need ed to
Vsl=sqrt(Q1);
Vs2=sqrt (Q2) ;
%Voltage need ed to
Vs3=sqrt (Q3);
%Voltage need ed to
(P(1)
(P(2)
(P(3)
^2 + (C(1) *D) ^2)];
^2 + (C(2)*D) .^2));
^2 + (C(3)*D) ^2)];
achieve a net of 4kw
achieve a net of 3kw
aieve a net of 6kw
if imag(Vsl)-=O,
Bt=2*Bgsat;
elseif Vsl(l)<Vl/sqrt(3)
Eaf(1)=9e9;
j=2;
save lowroot
elseif Vsl(2)>=Vl/sqrt(3)
Eaf(1)=Vs1(2)
else
Eaf(1)=Vs1(1)
end
if imag(Vs2)-=O,
Bt=2*Bgsat;
elseif
Vs2(1)<Vl/sqrt(3)
Eaf(2)=9e9;
j=2;
save lowroot
elseif Vs2(2)>=Vl/sqrt(3)
Eaf (2) =Vs2 (2)
else
Eaf(2)=Vs2 (1);
end
if imag(Vs3)-=O,
Bt=2*Bgsat;
elseif Vs3(1)<Vl/sqrt(3) ,
Eaf(3)=9e9;
j=2;
save lowroot
elseif Vs3(2)>=Vl/sqrt(3)
Eaf(3)=Vs3(2);
55
else
Eaf (3)=Vs3 (1);
end
"P i ::i
C! ak an
Kas
ro
vol ag
on
:r e ci a
%Wve connected voItage
.2 + (2* (V + 2*Vd) /'pi) 2
. .*i^"2.*Xc/3)
.Eaf~sqrt ( (pi
%So
rc
,
.."t
......
if isempty(Bt),
if all(Eaf<9e9),
in am pere-urns
current
Cuield
If=Eaf./(omegae*i*Laf) ;
%Gross 1lad curren t..
Ilg=(P+If.^2*Rrotor)/Vl;
%loa
Ilf=(3.*(sqrt(Eaf.^2-D^2)))./(pi*Xc) ;
Fr=2*Kbf*Nf*If/pi;
Fs=2*Kw*2*Nap*i*Ilg/pp/pi;
dsr=asin(2*(Vl) ./(3*pp*omegae.*If*i*Laff));
if all(dsr>=O) & all(dsr<=pi),
if rtthang <= tthang,
2
+ Fr.^2 + 2*(Fs.*Fr).*co s (piBt=muzero*sqrt(Fs.^
dsr))/g* (slthang/rtthang)
else
Bt=muzero*sqrt(Fs.^2 + Fr.^2 +2*(Fs.*Fr).*cos(pidsr))/g* (slthang/tthang)
end
c
tenreors
if
all(Bt
s see
<=
is
at all speIeds
if
oftrn
he nume
Bgsat)
sloss=(3/2)*(Ilg(2))^2*Rarm
.*
i^2 + (If (2) ) ^2*Rrotor;
slossl=(3/2)*(Ilg(2))^2*Rarm.*iA2;
sloss2=(If(2))^2*Rrotor;
for Ia
susing rm valus
fiiny -n
%auation
f
eta=P(2)./(sloss + P(2));
osses
l
%Roor Cu rrent
Jr=If/uarslt;
Densi
iciecy--npu
eqaS
ass
oupu
ty
Density using Gross
ratcr
Cur rent
S
Js=i*Ilg/(sqrt(2)*uaslt);
if
saisfied
hat stSfidi
eta >= 0.75 & all(Jr
saveef=eta;
Ns = i;
gap flux denst
ar
B=muzero*sqrt(Fs.^2
lced current
<= Jrmax) & all(Js <= Jmax),
y
+ Fr.^2 + 2*(Fs.*Fr).*cos(pi-dsr))/g
56
plS
fprintf
(' \nEfficiency = %g' , saveef)
fprintf('\nStator Turns = %d\n',i);
%Shear Stress
ss=P. / ( 2 *pi* (avrag) ^2*L*omegam*6. 895e3);
Flux across Rotor tooth
Btr=muzero*sqrt(Fs.^2 + Fr.^2 + 2*(Fs.*Fr).*cos(pidsr))/g*(slthang/rtthang);
%Flux across Stator toot-h
Bts=muzero*sqrt(Fs.^2 + Fr.^2 +2*(Fs.*Fr).*cos(pidsr))/g*(slthang/tthang);
Bs=max(B);
£!inding the optimun back-iron Thickness
T = Bs*avrag/(Bgsat*pp);
%Creating the optimized rout and rin;
rout = rbs + T;
rin = rbr - T;
if
rout > routmax,
saveef=2;
end
rin < rinmin,
saveef=2;
end
if
end
else
fprintf('Design %d No Good\n Cannot Meet Flux Density\n',desno)
end
else
fprintf('Dsr
is out of range\n');
end
end
end
end
end
end
end
if saveef==2 & all(Bt<=Bgsat)& Na>=1 & ~isempty(Bt),
fprintf('Efficiency is lower than 75 percent')
end
if rout > routmax,
fprintf('\n Bad machine, outer Radius Greater than max')
end
if rin < rinmin,
fprintf('\n
Bad machine,
end
inner Radius Less than min')
if saveef~=2 & rout <= routmax & rin >= rinmin,
%Determination of efficiency at remaining operating points
57
sloss=(3/2)*(Ilg).^2*Rarm * Ns^2 + (If).^2*Rrotor;
slossl=2*(Ilg).^2*Rarm.*Ns^2;
sloss2=(If).^2*Rrotor;
Pg=P + sloss2;
KSrhear Stress
ss=Pg. / ( 2 *pi* (avrag) ^2*L*omegam*6. 895e3)
%Calul.ation efiLCiency
eta=P./(sloss + P);
end
58
Appendix H. Cost.m
The cost of the machine is found in this script.
%Evalu2ates cost
of entire
machine
% Calculations for Stator
%back-iron depth--stator
bid=rout-rbs;
%Volume of stator
back-iron strip
Vbs=pi*L*(rout^2-rbs^2);
Stator Back-iron Steel mass
massa=sd*Vbs;
%mass in kg
Sta tor Teeth Volu me
sth=rbs-rsin;
tooth heigth
stthv=stthw*sth*L;
%vol.
massat=sd*stthv;
tmassat=slts*massat;
:ooth
of one :ooth
omas
mass of all teet
Total mass of sta or st
tsmass=tmassat + massa;
SL
one
o
in
emass
kg
volume
tsltv=0.5*(tuaslt*mpL);
toal
tatr
lotL
voIlme
Lstator slot copperI ma s s
cmass=tsltv*cd;
Calcu. at ions
.Rotor
copper mass in st:a.or
For rotor
volume;
Vr=pi*L*(rbr^2-rin^2);
stel
maSs
massr=sd*Vr;
%mass
in:kg
%Caiculat.ion of Rotor teeth volume
rth=rrout-rbr;
rthv=rth*rtthw*L;
%rotor tooth heighth
%Rotor Teeth Steel mass;
massrt=sd*rthv;
%mass of one rotor toot h
tmassrt=slts*massrt;
%mass of all
rotor teeth
%Total mass of rotor steel
trmass=tmassrt + massr;
%rotor slot volume
trsltv=tuarslt*(L+0.5*L2);
>mass in
lbs
%totalrotor slot volume--L2 is
59
average
;r otor
copr peri mass
cmassr=trsltv*cd;
stotal cost of steel in dollars
tscost=(trmass+tsmass)*csm;
%Ltotal. cost of copper in do.la
orn rotor anI stator
tccost=(cmass + cmassr)*ccm;
tcost=tscost + tccost;
60
Appendix I. Check.m
This script decides whether a generator design should be saved.
tk lws c
if exist('savetc')==O
savetc=[9e9 9e9 9e9 9e9 9e9 9e9 9e9 9e9 9e9 9e9];
end
if all(Bt<Bgsat),
if saveef-=2,
counter=counter+1;
if tcost < max(savetc),
if savetc(1)==max(savetc),
eval('save templ');
fprintf('Saved
Design Number
%d\n',desno);
savetc(1)=tcost;
elseif savetc(2)==max(savetc),
eval('save temp2');
fprintf('Saved Design Number
d\n',desno);
savetc(2)=tcost;
elseif savetc(3)==max(savetc),
eval('save temp3');
fprintf('Saved Design Number %d\n',desno);
savetc(3)=tcost;
elseif savetc(4)==max(savetc),
eval('save temp4');
fprintf('Saved Design Number %d\n',desno);
savetc(4)=tcost;
elseif savetc(5)==max(savetc),
eval('save temp5');
fprintf('Saved Design Number %d\n',desno);
savetc(5)=tcost;
elseif savetc(6)==max(savetc),
eval('save temp6');
fprintf('Saved Design Number
savetc
(6) =tcost;
d\n',desno);
elseif savetc(7)==max(savetc),
eval('save temp7');
fprintf('Saved Design Number %d\n' ,desno);
savetc(7)=tcost;
elseif savetc(8)==max(savetc),
eval('save temp8');
fprintf('Saved Design Number %d\n',desno);
savetc(8)=tcost;
elseif savetc(9)==max(savetc),
eval('save temp9');
fprintf('Saved Design Number %d\n',desno);
savetc(9)=tcost;
elseif savetc(10)==max(savetc),
eval('save templ0');
fprintf('Saved Design Number %d \,desno);
savetc(10)=tcost;
61
Appendix J. Mean Path Length
Determining the stator and rotor resistances requires the total length of copper of wire. The
length is measured down one slot and back through another. The difficulty arises in determining
the length of the end turns. A derivation is given here for both the stator and rotor.
On the stator, an assumption is made on the shape of the turns of the winding. Figure J-1
shows the shape of the winding as seen from the back-iron.
T_
Sml
0
Limach
Figure J-1: Defines paramenters
discussed in the following
derivations.
The copper wire covers the stack length,
Lmach,
of the alternator twice. The second is the length
of a segment shown in the figure. Pythagoras' Theorem can be implemented to find this length.
Since the winding is full pitch, the horizontal distance is the width of 3 teeth and 2.5 slot widths
therefore the length of a segment is
62
3T
L
+2.5S
w(1)
w
=
seg
cos0
where T, is the width of one tooth; S, is the slot width; and 0 is the angle shown in Figure J-1.
Similarly, the angle can be found. Looking at the figure allows Pythagoras' Theorem to be
applied again. The angle is
S
0 = sin-1
S
w
+T
(J.2)
The length of the six arcs can be found by assuming the copper wire passes through the middle of
a slot. Because of this assumption, each angle is
0
1
=
7TS
W
(J.3)
6
To find the total contribution of the arcs, (J.3) must be multiplied by 6 because there are six of
them.
Combining (J. 1) and (J.2), and then adding that result to 6 times (J.3) and 2 times the stack
length, the mean path length on the stator is
4 3T +2.5S
L
mps
=2L
mach
+ S
+
wS
,71T
cos sin-
(
+
J.4)
w
S
+T
On the rotor, the mean path length is simpler. The end turn shape is assumed to be a
semicircle. Therefore the circumference of a full circle plus the 2 times the stack length yields
the mean path length on the rotor, Lmpr. Figure J.2 shows the configuration of one pole pair.
63
Lnach
Figure J-2: Shows semicircle
shape of turns on rotor.
64
Appendix K. 1
Characteristics of Optimum Direct Drive Generator
Total Cost= 64.2611
Design Number #27268
Number of Poles =10
Total Machine Mass of Steel =27.6845 kg
Stator Turns= 9
Total Machine Mass of Copper=7.35037 kg
Number of slots = 60
Stator Steel Mass= 18.6419kg
Air Gap = 0.000639326 m
Stator Copper Mass= 6.15706kg
Machine Length=0.127478 m
Rotor Steel Mass= 9.04262 kg
Back-iron Thickness= 0.011789 m
Rotor Copper Mass= 1.19331 kg
Self-Inductance= 2.60879e-005H
Leakage Inductance= 8.21512e-007H
Mutual Inductance= 2.60879e-006H Synchronous Inductance= 7.99068e-006H
Inductance in Stator due to Field= 5.21759e-005H
Rotor Resistance = 0.0588632 Ohms
Stator Resistance = 8.63865e-005 Ohms
Mechanical Dimensions on Stator
Machine Diameter = 0.289649 m
Stator Back-iron radius = 0.133036 m
Stator Inner Radius = 0.109874 m
Stator Tooth angle = 0.0609478 rad
Stator Slot Span = 0.0437719 rad
Stator Bottom Slot = 0.00723492m
Stator Top Slot = 0.00480938 m
Stator Tooth width = 0.00669655 m
Stator Slot Height = 0.0231622 m
Stator Mean Path Length= 0.469186 m
Total Usable Stator
Slot Area= 0.00292921 m^2
Mechanical Dimensions on Rotor
Rotor Outer radius = 0.109234 m
Rotor Back-Iron radius =0.102521 m
Machine inner radius= 0.090732m
Rotor Tooth angle = 0.0568773 rad
Rotor Slot Span = 0.0478424 rad
Rotor Bottom Slot = 0.00560786m
Rotor Top Slot = 0.00490485 m
Rotor Tooth width = 0.00583112 m
Rotor Slot Height= 0.00671319 m
Rotor Mean Path Length = 1.07836 m
Total Usable Rotor
Slot Area = 0.000741025 m^2
Performance Characteristics
Efficiency
Total Loss
Stator Loss
Rotor Loss
Gross Power
At 600rpm
0.597922
2689.84
329.32
2442.85
6442.85
Gross Current
Load Current
Commutating
Reactance(Ohms)
Back-Emf
Field Current
(A turns)
Gap Flux
Density
Rotor Tooth Flux
T
Stator Tooth Flux
T
Stator Current
Density(A/m^2)
Rotor Current
Density (A/m^'2)
Sheer Stress(psi)
153. 401A
95. 2381A
0.00251035
At 1500 rpm
0.860641
526.253
108.317
445.016
3695.02
87.9766
77.381
0.00627587
At 6000 rpm
0.913341
569.291
317.967
330.816
6330.82
150.734
142.857
0.0251035
110.594
30. 053V
32.0677 V
203.716
86.9492
74.9672
0.968484 T
1.78313 T
0.40789 T
0.750988 T
0.318738 T
0.586844
1.66404 T
0.700832 T
0.547651
1.99966e+007
1.14682e+007
1.96489e+00 7
1.64947e+007
1.547
7.04019e+006
0.354887
6.07002e+00 6
0.15201
Appendix K.2 Characteristics of Optimum Geared-Drive Generator
Design Number #683703
To tal Cost= 38.1824
Tota 1 Machine Mass of Steel =15.4705 kg
Number of Poles =6
Tota 1 Machine Mass of Copper=4.56147 kg
Stator Turns= 8
Stator Steel Mass= 10.2087kg
Number of slots = 36
Stator Copper Mass= 3.71394kg
Air Gap = 0.000715871 m
Rotor Steel Mass= 5.26178 kg
Machine Length=0.141694 m
Copper Mass= 0.847534 kg
Rotor
Back-iron Thickness= 0.010434 m
Leakage Inductance= 4.9079e-007H
Self-Inductance= 2.81293e-005H
Sy nchronous Inductance= 1.42283e-005H
Mutual Inductance= 4.68822e-006H
Inductance in Stator due to Field= 5.62586e-005H
Rotor Resistance = 0.0361488 Ohms
Stator Resistance = 0.000183695 Ohms
Mechanical Dimensions on Rotor
Rotor Outer radius = 0.0710419 m
Mechanical Dimensions on Stator
Machine Diameter = 0.196381 m
Stator Back-iron radius = 0.0877567
Rotor Back-Iron radius =0.0654614 m
Machine inner radius= 0.0550274m
Stator Inner Radius = 0.0717578 m
Tooth angle = 0.0780917 rad
Rotor
rad
0.0861383
=
Stator Tooth angle
Slot Span = 0.0964412 rad
Rotor
rad
=
0.0883946
Span
Slot
Stator
Rotor Bottom Slot = 0.00728715m
Stator Bottom Slot = 0.00913534m
Rotor Top Slot = 0.00631318 m
Stator Top Slot = 0.006343 m
Rotor Tooth width - 0.00511199 m
Stator Tooth width = 0.00618109 m
Rotor Slot Height= 0.00558047 m
Stator Slot Height = 0.0159989 m
Mean Path Length = 1.18697 m
Rotor
Stator Mean Path Length= 0.531375 m
Total Usable Rotor
Stator
Total Usable
Slot Area = 0.000478146 m^2
Slot Area= 0.00156011 m^2
Performance Characteristics
At 600rpm
0.721169
1546.55
370.055
1269.01
5269.01
Efficiency
Total Loss
Stator Loss
Rotor Loss
Gross
Power
Gross Current
Load Current
Commutating
Reactance(Ohms)
Back-Emf
Field Current
(A turns)
Gap Flux
Density
125. 453A
95.2381A
0.00536392
31.7904V
187.364
Stator Tooth Flux
0.789128 T
1.76368 T
1.59893 T
Stator Current
Density(A/m^2)
1.63758e+007
Rotor Current
Density(A/m^2)
Sheer Stress(psi)
1.41068e+007
1.33986
Rotor Tooth
Flux
At 1500 rpm
0.887893
410.349
166.604
285.397
3535.4
84.1761
77.381
0.0134098
37.6901 V
88.854
0.364856 T
0.815444 T
0.739269 T
1.09878e+007
6.68989e+006
0.359608
66
At 6000 rpm
0.876357
846.522
551.586
432.832
6432.83
153.163
142.857
0.0536392
185.662
109.424
0.420774 T
0.940419 T
0.85257 T
1.99929e+00
8.23862e+00
0.163581
6
7
67