A STUDY OF
THE STABILITY OF A SIMPLE
ELECTRICAL POWER SYSTEM
WITH CONNECTIONS TO TEACHING SCIENCE
IN SECONDARY SCHOOLS
by
George Martin Hammond
B.A., University of Northern Colorado, 1972
A thesis submitted to the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Basic Science
A Study of the Stability of
a Simple Electrical Power System with Application
to Teaching Science in Secondary Schools
This thesis for the Master of Basic Science
degree by
George Martin Hammond
has been approved by
Randall P. Tagg
Date
Hammond, George Martin
A Study of a Simple Electrical Power System and Its Application to teaching Science
in the Secondary Schools
Thesis directed by Associate Professor Randall Tagg.
ABSTRACT
This paper develops a nonlinear mathematical model of electric power
generation from basic principles, and examines the equations of the model for multiple
solutions and their stability. A number of experiments are offered in the text that
could be performed by middle or highschool students to involve them in scientific
inquiry and investigations. The aim is to demonstrate the feasibility of connecting
secondaryschool level research with advanced universitylevel research on a problem
that directly impacts students and the public. This work is a prelude to work that will
create scalemodel power systems whose complex behavior will be investigated using
the tools of nonlinear dynamical systems theory. The basic conceptual framework for
this approach is established in this thesis for a single generator attached to a power line
bus whose power demand can be systematically varied.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
iii
DEDICATION
I dedicate this to my mother who ever despaired of its publication and my daughter
who forever keeps me young.
ACKNOWLEDGMENT
My thanks to my committee for their help in keeping me on the straightandnarrow,
and to John Starrett for his computer programs and avid interest. The initial start of
this project was made under the E2020 project at UCD.
Contents
1 Introduction 1
1.1 The Problem......................................................... 1
2 Physical Theory 4
2.1 Introduction................................................ 4
2.2 Assumptions and Basics...................................... 5
2.3 Electric and Magnetic Fields................................ 12
2.4 Electromotive Force and Generator Action.................... 14
2.5 Magnetomotive Force and the Production of a Magnetic Field .... 20
3 Modeling the Synchronous Generator 32
3.1 Required Function Form for B(0).....................................34
3.2 Ideal Rotor Current Distribution Needed to Obtain Desired B(0). . 36
3.3 Realistic Motor Windings and Field Harmonics........................41
3.4 Relation Between Noload emf and Rotor Current......................43
vi
3.5 Armature Connected to a Load: Fields Generated by Armature
Currents......................................................44
3.6 Power Transfer................................................50
3.7 Losses....................................................... 53
3.8 The Swing Equation........................................... 55
3.9 Scaling.......................................................58
4 Analysis of the Model 60
4.1 Introduction................................................. 60
4.2 Fixed Points................................................. 61
4.3 Basins of Attractions........................................ 69
5 Applications to Science Teaching 73
5.1 Equipment.................................................... 73
5.2 Using the Equipment in the Science Classroom..................79
Appendix 83
A Symbols...................................................... 83
B Glossary..................................................... 85
Bibliography 89
vii
Chapter 1
Introduction
1.1 The Problem
Voltage stability on the commercial power grid is one of the most important
problems we have in the world today. Aside from the inconvenience caused
by power outages to the average home owner, industrial consumers suffer
greatly from voltage drops or even a monetary glitch in the delivery system.
For example, one company is reported to have lost $200,000 during a five
cycle (83 millisecond) interruption. Another problem occurs if the power
system cannot damp out transient effects quickly[8]. Also, subtle effects
like subsynchronous resonance can occur where instabilities cause a slow
(1020 Hz) sine wave to be superimposed on the normal 60 Hz wave. This
superimposed wave can literally tear apart power generation equipment by
setting up inertial forces in the rotors of large, coupled machines [8].
The traditional answer to these and other problems by the utility in
dustry is to run the power system well below capacity in order to limit the
instabilities. During peak consumption, standby generators can be brought
up to speed ready to go online to ease the pressure. The fact that this
1
doesnt always work was seen in July, 1996 when two lines in Wyoming
shorted out due to a lightning strike and caused a rolling, 17state power
outage across the western United States. The deregulation of the power in
dustry will bring further problems of stability. Along with cheaper prices
implied in greater competition comes the added problem of synchronizing all
the different power companies on the grid, while ensuring adequate power
quality.
In the manufacturing industry, the answer is to invest in power stabiliza
tion equipment, including capacitorinductor banks, uninterrupted power
systems, and complete backup generation facilities. However, all these so
lutions from the utility and manufacturing industries eventually cost the end
user in increased prices. Over the longer term, the goal is to develop new
strategies of energy generation and transmission so that we continue to have
reliable energy supplies as both producers and consumers proliferate.
This problem also presents a unique way to involve the next generation
of consumers/voters/scientists now in our middle and highschools. This is
the thrust of Energy 2020, a project started by Doctors Gabriela Weaver
and Doris Kimbrough in the Chemistry Department and Dr. Randy Tagg in
the Physics Department of the University of Colorado of Denver. Under this
program, teachers are brought in from local area schools and work closely
with Physics and Chemistry faculty to develop new curricula for use in the
science class room. The aim is to involve students in actual energy research
as a part of their classroom activity. By participating in such research, these
students gain a stake in determining, rather than passively accepting, their
energy future. A key challenge in E2020, then, is to relate significant issues
in electric power generation and transmission to projects that precollege
students can perform.
2
The main purpose of this paper is to examine the mathematical under
pinnings of power generation to discover points of instability and describe
them in a way that can be understood by a highschool physics teacher. The
method of investigation I will use is the swing equation. I will develop this
equation from a physics standpoint, derive the mathematical model used in
several papers on the subject and analyze it for instabilities. Throughout the
text I will present suggested experiments the teacher can use to help explain
the concepts being presented, and in the final section, make suggestions on
how teachers can use this information in the classroom.
3
Chapter 2
Physical Theory
This section will deal with the physical model of an electric generator and
load, building from basic principles of electricity and magnetism, chapter 3
looks at the detailed physics of synchronous generators, arriving at a math
ematical model used in several papers in the power system literature. This
model is then analyzed in chapter 4 from the point of view of dynamical sys
tems theory. A list of symbols and a short glossary is presented for reference
in the appendices.
2.1 Introduction
A generator1 is a machine that converts mechanical energy into electrical
energy for delivery to a load to do useful work. The mechanical energy is
supplied by a prime mover such as a steam turbine, a hydraulic turbine or
a diesel engine. The prime mover is mechanically coupled to the generator
which creates an alternating (AC) current voltage for delivery across a grid
of transmission lines to the end user (see figure 21). Here the energy is used
1 Italicized words are defined in the glossary.
4
to build houses, make gadgets, cook dinner, watch television, run machines
or do the thousands of things we call on electricity to do. The energy is
consumed as electric power (work per unit time).
Steam turbine Transformer Sybstation
Figure 21: Power generation and distribution system.
In the above paragraph, AC voltage refers to an electromotive force
(emf) whose value or magnitude varies over time in a precise sinusoidal
manner, energy is the capacity to do work and a load is the thing that
does the work. Loads are resistive and/or reactive which refer to the
way the energy is used. Resistive loads (light bulbs, ovens, etc.) dissipate
the energy as Joule heat, while reactive loads (motors, capacitance banks,
etc.) can store energy for delivery back into the system albeit at the cost of
phase differences between the voltage and current (see below).
2.2 Assumptions and Basics
To develop the physical model I will make use of the following assumptions.
These concepts will be clarified through the remainder of the thesis.
A generator can be represented by a voltage source in series with an
inductive reactance (figure 22).
5
/WY>
E O
l
X
Load
Figure 22: Generator with reactance and load
The internal resistance of the generator is negligible compared to the
reactance and the current is 90 out of phase with the voltage.
Loads will be a combination of resistance, inductance, and capacitance,
indicated by a complex impedence Z.
All circuits are drawn as oneline diagrams. The internal symme
try of the generator, based on three identical armature coils, is com
bined with the assumtipn of equal loads attached to the separate coils
(phases). Thus all electrical phases are considered balanced and only
one circuit needs to be represented which results in less confusion.
Also in a balanced load, one phase can represent all three by proper
mathematical treatment (figure 22).
All vectors are denoted by boldface, e.g. v is the velocity vector.
The first step in explaining these ideas is a review of periodic wave char
acteristics. The amplitude of a sinusoidal wave is the amount the function
rises above the x axis while the wavelength is the distance between two points
at the same position. We can also define the frequency of a wave as the num
ber of wavelengths that pass a point in a given time (or the reciprocal of the
wavelength) and the period as the time for one complete oscillation. Figure
6
Figure 23: A cosine curve showing wavelength and amplitude
23 shows the amplitude and wavelength for a cosine wave and the question
arrises: If this wave represents the voltage in a circuit, how do we measure
the average value? The average of a cosine function (when we take the in
tegral) over one period is zero, and yet in real life we observe an average
transmission of power. The answer to the dilemma is to find the root mean
square (rms) of the function where rms is defined for any function g as
1 fb 12
9rms = r / g (ijdt
a b Ja
(2.1)
Figure 24 shows a cos2(x) plot and if we find the rms of this new function
from zero to 27r we get:
which means if eo is the value of the original voltage, the rms voltage is
e Â§C
e_ vr
Pursuing fundamentals a little further, we recall from elementary electric
ity and magnetism that an alternating voltage or current may be represented
by vectors in the imaginary plane called phasors. The magnitude of the pha
sor represents the amplitude of the wave, while the direction (measured as
7
1 2 3 4 5 6 7
* Wavelength >
l................................
Figure 24: A cos2 wave. The wavelength is 7r, one half the wavelength of a
cos wave (27r).
an angle from the horizontal or real axis) represents the phase. Figure 25
illustrates the concept with 25 (a) showing the voltage and current in phase
while going through a resister and 25 (b) voltage and current out of phase
through an inductor. It is important to note here that in 25(a), I lies along
E; the two vectors are not added together in the illustration. In a circuit
consisting only of resistors, voltage and current phasors are related by a real
number R:
V = IR (Ohms Law). (2.3)
In a circuit containing reactive components, such as inductors or capacitors2,
the voltage and current phasors are now related by a complex number, Z,
called the impedance. Impedance can be expanded into real and imaginary
parts
2 Capacitors act very much like inductors in the circuit, that is, they are capable of
storing electrical energy and releasing it back into the circuit. The both act to oppose the
changes in the circuit imposed by the alternating current, but the phasor for a capacitor
leads the voltage phasor while the inductor phasor lags the voltage phasor.
8
Figure 25: Voltage and current through a) a resistor and b) an inductor
showing the wave forms and phasors.
9
Z = R + jX.
(2.4)
Here the real part R represents resistance and the imaginary part, X, rep
resents what is called the reactance of the circuit. The corresponding gen
eralization of Ohms Law is
V = IZ (2.5)
The last point to make about AC circuits is how to represent an instan
taneous value for the voltage and current at any given time. Since the emf
is sinusoidal, we will define the instantaneous voltage, v, as
v = Vq cos(27t ft) (2.6)
where Vo is the absolute maximum value of the voltage. Likewise, we will
define the instantaneous current, i, as
i = Iq cos(27t ft + ip) (2.7)
as shown in figure 26. The instantaneous power in the system, then, is
p = iv (2.8)
= IqVq cos(27tft) cos(27rft + ip)
= \iqVq cos^(1 + cos(27r2 ft))
+xfoVbsinVsin(27r2/t). (2.9)
10
Figure 26: Voltage and current plotted as cosine waves.
Over one period, the time dependant terms average to zero, which leaves us
with the following average power p:
p = ^ Jo Vo cos V (2.10)
Noting that j^Io and are the rootmeansquare current and voltage,
respectively, we define the real power P (measured in watts, W) as
P = IV cosV. (2.11)
The reactive power (imaginary, measured in voltamps reactive, Vars):
Q = IVsmip (2.12)
where I and V are RMS values in both equations 2.11 and 2.12. The final
power used by the industry is the combination of real and imaginary power
and is called the total power (measured in voltamperes, VA) and is defined
as
11
S = P +jQ.
(2.13)
This is usually quoted in terms of its magnitude
S = y/P2+Q2. (2.14)
2.3 Electric and Magnetic Fields
We will start by developing the two basic concepts of the electric and mag
netic fields. The ancient Greeks were the first to notice that a rubbed piece
of amber would attract pieces of straw but it was Faraday that described
this as a vector field surrounding the rod and called it an electric field E.
E can be defined by placing a test charge, qo, in the field and noting
what, if any, force F acts on it. This gives a field strength E at the location
of the test charge according to:
E=Â£ (2.15)
9o
where E is a vector because F is in the same direction as E and qo is a scalar.
This is directly analogous to the definition of the gravitational field g where
a test mass (mo) is used and
g= (2.16)
mo
Note in both cases, the defined field is a force divided by a scalar. Gravity
is newtonskilograms1 and E is newtonscoulomb1.
Magnetism was also described by the ancients when they noticed that
certain rocks attracted iron and that those rocks, when suspended, pointed
12
north. It wasnt until the last century that the concept was quantified and
electricity and magnetism were found to be related. The field, called the
magnetic flux density, B, is represented by lines of induction (as E is repre
sented by lines of force).
B is also defined by a test charge but this time qo is fired through the
field of B with a velocity v and a sideways deflection of qo is observed.
By varying the direction of v but not its magnitude, the maximum force
of deflection on go is found when v lies in a direction normal to B. These
physical observations lead B to be defined as the vector that satisfies the
relation
F =gov x B. (2.17)
This means the force F is at right angles to the plane of v and B and will
always be a sideways force.
The last concept to develop in this section is flux. If we place an arbitrary
surface into an electric or magnetic field, we could measure the integral of
the normal component of the field across the surface. This quantity is called
the flux, $*, of the field and is represented graphically by the total number of
vectors through the surface. The more vectors through the area, the stronger
the flux (see figure 27).
Mathematically, the magnetic flux is defined by the surface integral
$ = J B hdA (2.18)
where n is the normal vector to the surface.
13
vector field
/wy
surface
Figure 27: Flux through a surface showing the field and the normal to the
surface h.
2.4 Electromotive Force and Generator Action
Before the 19th century, electricity and magnetism were thought to be similar
phenomena but not related. The connection between the two was established
by Oersted, in 1820, when he performed the classic experiment that caused
a compass needle to deflect when a current flowed in a nearby wire, proving
the link between electricity and magnetism. Ampere, the same year, showed
a magnetic field was generated around a current carrying wire (see below).
Then, in 1832, Faraday proved a magnetic field could produce a current in a
wire and with that discovery, electricity and magnetism were proven to be,
in a sense, different sides of the same coin.
We will start our study with a definition of the electromotive force and
show how this will lead to Maxwells equation relating the electric field E to
the flux density B.
A circuit is just a convenient way to confine an electric or magnetic field
to where we want it to go. In the case of an electric field, we use a wire with
a high conductivity, a. The wire will contain the field and the charge carriers
that are moving under the influence of E.
The force moving the charge carriers is the emf and is defined as the
14
line integral of E over the length of the current loop, or:
Â£ = emf = N ~Edl (2.19)
for any length of wire in the circuit and where N is the number of turns in
the circuit.
Next, we define flux linkage. If we take the surface in figure 27 and
wrap it with an Ntum coil of wire as in figure 28 the linkage of the coil is
Figure 28: A surface wound with an Nturn coil.
the flux through a single turn of the coil multiplied by the number of turns
or
i/> = N$. (2.20)
Faradays great insight (after nearly nine years of work) was that an emf is
generated by a changing magnetic field. This is known as Faradays law and
is expressed:
15
e = *t
dt
(2.21)
where the minus sign indicates that the emf will drive a current that will
induce its own B field (see below) that opposes the change in flux.
If we use Stokes Theorem, equation 2.19 can be changed from a line
integral to the surface integral:
N
j) Edl = N J
V x E dA.
(2.22)
where S is an arbitrary surface bounded by or capping the coil. But from
Faradays results and the definition of flux
(2.23)
where S is the same capping surface in equation 2.22. In circumstances
where the area of the capping surface does not change with time we can
move the derivative inside the integral and equation 2.23 becomes
dip
dt
r (IB
= N /
Js dt
n dA.
(2.24)
This means the right hand sides of equations 2.22 and 2.23 are equal to each
other and we can write
N f V x EcL4= N [ ^
Js Js dt
n dA
(2.25)
or
is
(V x EdA + ^) = 0.
s dt
(2.26)
Since the surface is arbitrary (as long as it is bounded by the coil), the only
16
way this integral can equal zero is for the integrand to be zero. This leaves
us with Maxwells equation:
V x E = 
SB
dt "
(2.27)
Experiment A simple demonstration of this phenomenon is illustrated
in Figure 29. Two wire rails are set up perpendicular to a magnetic field B
and a length of wire is moved with velocity u through the field. If the wire
travels a distance x, the emf generated in the wire is
B
Figure 29: Generating an emf with a wire passing perpendicular to a flux
field B. Arrows in the circuit show direction of current flow. This current
produces a magnetic field in a direction (see next section) adding to B and
opposing the decrease in flux through the coil.
T J
emf = (B lx) = B l = BZu.
cit at
(2.28)
Experiment We could use this device to make a serviceable alternating
current by just moving the wire up and down in the field as shown in figure
17
Figure 210: A simple AC machine
210 where we use a wheel to convert rotational into linear motion and the
field B is into the plane of the paper between the rails. 11
Fortunately, there is an easier way as shown in figure 2
11. Almost every school I have been in has one of these classic old hand
generators or something very similar to it. The machine is simple to set
up and use but care should be taken in connecting a power source to the
electromagnet; the magnet coil has a very low resistance and therefore draws
a large current (somewhere around three amps) and needs a large power
supply. Since the strength of the emf is proportional to the length of the
wire as well as the field and velocity, we can increase the length by forming
it into a loop (windings in figure 211) and spinning it in the magnetic field
with an angular velocity uj (figure 212). In this case, however, we must be
careful to find only the portion of the velocity vector that is perpendicular
to the magnetic field B. Figure 213 shows a side view of the coil (the coil is
into the page) rotating in the field. If the coil rotates with a frequency of u>t,
the portion of the velocity vector u, whose magnitude is urr has a component
perpendicular to the flux given by usin0 and the emf becomes:
Experiment
18
Figure 211: An electromagnetpowered handcranked generator.
=  N^ (BZ2r sin 0(t)) 
= I AT3Z2r(sinwt)
at
= 2Blru cos ut.
(2.29)
Using this formula, the student can calculate the generated emf and
compare this to an actual reading. The sinusoidal output is best viewed on
an oscilloscope connected to the takeoff lugs (figure 211).
A synchronous machine essentially turns this model inside out and puts
the magnet (called the rotor) in the center of a hollow cylinder (the stator)
around which the coils are wound. We will develop the synchronous machine
19
Figure 212: A coil of length l rotating in a magnetic field,
in considerable detail in the next chapter.
2.5 Magnetomotive Force and the Production of a
Magnetic Field
We have not yet specified how the magnetic field is produced. As stated
earlier, Amp6re proved in 1820 that electric currents produce magnetic fields.
The field is formed around the wire using the righthand rule: with the
right thumb pointing in the direction of the current, the magnetic field will
curl around the wire like the fingers. Ampere also proved that the field is
proportional to the current, I, in the wire. This field, produced by externally
imposed currents, will be denoted by the symbol, H. In free space (vacuum),
the relationship between the magnetic field H (measured in ampereturns
per meter) and magnetic flux density B (measured in Webersmeter2 or in
Tesla) is via a simple constant of proportionality:
B =/x0H (free space). (2.30)
20
Figure 213: Velocity of a rotating coil element
The constant fi0 is called the permeability of free space and has the value:
/z0 = 47r x 10 7 Newtonsamps 2 (Henriesmeter 1)
(2.31)
In the simple configuration of figure 214, the magnitude H of the field at
radial distance r from a long straight wire carrying current /i is given by
H
_ h
2tvt '
[Experiment: Current Balance) Suppose another long straight wire carry
ing current, I2, is brought to within a distance d from and parallel with the
first wire carrying current I\. Each wire produces a magnetic field according
to 2.30 above which then exerts a force on the charges moving in the other
wire.
The force per unit length F/L on wire two due to the flux density, Bi,
produced by wire one is (by adaptation of equation 2.17):
21
wire
r
Figure 214: Magnetic field around a wire.
f=B^
(2.32)
Since
Bi
h
2trd
(2.33)
we have
F Ijh
L fi27zd
= 2^ x 107 Newtonsm1. (234)
a
A common experiment available in physics labs is a current balance that
measures this force. If we take L = 30 cm, d = 1 mm and /i = h = 10 amp:
22
F
2 103
X 107
x 0.3
= 6. x 103 N
(2.35)
This is the weight of a mass, m = 6 x 10 ^kg = 0.6(7. Students
can learn that this force between current carrying wires is the basis of most
electric motors, but might winder how high torque motors can be made when
the above calculation yields such a smalt force. The answer lies in both
increasing the length of wires that interact and in intensifying the magnetic
flux density through the use of ferromagnetic materials. Such materials are
also vital in generator design, as discussed below.
When certain materials, called ferromagnets (such as iron, nickel and
cobalt), are placed in a magnetic field H, the flux density, B, inside the
material is greatly intensified according to the relation:
B =7zrJu0H =/j, H (2.36)
where we define fi = fiTfiQ and [iT, the relative permeability, is a dimension
less number ranging between 1,000 and 10,000 for common ferromagnets.
Anyone who has played with an ironcore electromagnet knows that this
intensified flux density extends beyond the material, at least at smalt dis
tances. Thus, in designing generators, we want to make the rotor out of a
layered steelcore, electromagnet (fiT ~ 5,000 6', 000) and then minimize
the air gap between the rotor surface and the stator coils where the rotating
magnetic field induces an electromotive force.
We must note here that the permeability is not constant and we should
23
write fir = /ir(H). As the exciting current is increases (and H increases
proportionally), the resulting magnetic flux density saturates. This, and a
related phenomenon called hysteresis, are discussed more in a later section.
Usually, though, we begin the modeling of generators by treating fiT as a
constant.
We need to be able to calculate the emfihduci'ng fields from the knowl
edge of the current applied to the rotor coils (called field coils). In doing so,
the ideas of magnetic circuits and magnetomotive force become useful. The
simplest magnetic circuit is a loop of high permeability magnetic material
called a core, with a current carrying wire wrapped around all or part of it
(fig 215). The purpose of the magnetic material is to provide a path for the
induction field so that B is confined to the core much the same way a wire
provides a path for E.
Figure 215: A simple magnetic circuit.
To determine the magnetic flux within a generator, we will follow this
chain of reasoning: (1) Identify the distribution of currents that act as a
source for the magnetic field H; (2) find the induced flux density B using
24
the geometry of the ferromagnetic materials used to confine and guide the
field; and (3) account for fringing effects that occur when B crosses air gaps
between the rotor and stator.
The first step, calculating H, begins with Maxwells equation:
7xff=j+ (2.37)
where j is the current density and D the displacement field, related to the
electric field, E, by the expression D = eB, where e is the dielectric constant
of the material. At the small frequencies 60 Hz) used in our power
systems, the timederivative of D can be neglected5 and we obtain, to a
very good approximation, the simplified equation
V x H = j (2.38)
Consider a currentcarrying coil in the geometry shown in figure 216 and
mark a fictitious closed path, F, as shown. If S' is the surface enclosed by
this path, we can integrate equation 2.38 over the surface to get:
f (V x H)dA= / ydA.
Js Js
The righthand side is just the total current flowing through the surface
J ydA = Nt (2.40)
where N' is the number of times the coil of wire penetrates the circuit (the
number of windings) and i is the current through the coil.
The lefthand side is converted to a line integral around the dotted path 3
3We can safely ignore D because it really describes effects happening more than a wave
length away. Since a wave length at 60 Hz is close to 300 miles, D can be neglected.
25
Figure 216: A simple coil circuit,
using Stokers Theorem:
P ETdT
(2.41)
By analogy to equation 2.19 (the definition of the emf), we now define mag
netomotive force (mmf) by
$ = mmf' = (p Hdl
r
and established the simple relation:
mmf = Ni. (2.43)
The advantage of this point of view is that all the complexities of mag
netic field distributions in the intricate geometry of' a generator are related
to the quantities N~ and i that are easily specified. The complexities are
unraveled as we make simplifying assumptions about the geometry of the
field thanks to the guiding and near confinement of R by the ferromagnetic
26
core materials.
Figure 217: Volume V bounded by a surface S.
This geometric simplification of the distribution of magnetic flux density
B uses another of Maxwells equations:
V B =0. (2.44)
Consider a volume V containing flux lines B (figure 217) and integrate
equation 2.44 over V:
I V B =0.
Jv
We can convert the left hand side to a surface integral:
(2.45)
/ V B=/
BdA
(2.46)
and so
l
BdA = 0.
(2.47)
Thus, the flux density entering an enclosed volume must equal the flux den
sity leaving the volume.4
If we make our volume coincide with a section of the magnetic core, then
to a good approximation, the flux lines B, are parallel to the surface except
where we cut the core to let flux in or out.
BdA=0 over this part of
Figure 218: The path of B through a ferromagnetic core.
Referring to figure 218 we see that equation 2.47 implies we can write
BA = constant(v) (248)
where A is the crosssection at any point along the magnetic core. We now
show how these ideas can be used to calculate the magnetic flux in practical
situations.
If we introduce an air gap in the magnetic circuit, which is small com
pared to the length of the core, the flux is constrained to the core and is
continuous through the magnetic circuit[12] (figure 21S).
Thus, we can treat figure 219 as a circuit with series components. For
4 This is analogous to current flowing in a wire: The current density entering a volume
of wire equals the current density out unless there is some means to store charge within
the volume. In the case of magnetic flux density, there is rigorously no storage of magnetic
"charge'.
28
Figure 219: A simple magnetic circuit with an air gap.
the core we have
ft = x (2)
and for the gap:
Bg = Â£ (2.50)
Ag
Note that if the gap is small, we can ignore fringing so that Ag ~ Ac. Then
Bs = Bc=j (2.51)
This is why we said earlier that the magnetic flux density, intensified within
the ferromagnetic core, remains nearly constant (and hence still intensified)
across small gaps between portions of the core.
Putting all this together, and noting equation 2.38 reduces to Hclc over
a closed loop, we get:
29
mmf = Ni = Hclc + Hgig
= lc + ^9 (2.52)
**
where fi0 is the permeabihty of air. Rewriting equation 2.52 in terms of the
flux gives:
mmf = <2?(H~p)
(2.53)
'Me
Note how this is analogous to Ohms law: emf = IR. Here flux, plays
the role of current, and we define magnetic reluctance, (the analogue of
electrical resistance) of a portion of a circuit by
fiA
(2.54)
where I is the length of the circuit portion, A is its crosssectional area, and
H is its permeability (the analogue of electrical conductivity). This gives us,
in equation 2.53, a series circuit
mm/ = $(5ic+!R9). (2.55)
where is the reluctance of the core and 3^ is the reluctance of the
gap. Note that the reluctance is inversely proportional to the permeability
and we can make n fi0 with a suitable choice of material. Thus, the
reluctance of the core will be small with respect to the air gap and 2.53
becomes: 9
9 9
30
Next we apply this to a simple rotating magnetic circuit and develop the
principles of the electric generator in the next chapter.
31
Chapter 3
Modeling the Synchronous
Generator
We can now apply the ideas of the preceding chapter to obtain a physical un
derstanding of threephase synchronous generators. The machine has three
separate coils spaced 120 apart that are connected either endtoend (delta
configuration) or have one end connected in common with the other end free
(Y connection). Figure 31 is an illustration of a threephase machine with
a salient twopole rotor in the center.
Let us suppose that we have designed the machine so a purely sinusoidal
emf is induced in each coil. We choose the origin of time so that the emf of
coil of coil a is maximum at t = 0. Then, since the coils are identical and
symmetrically spaced about the perimeter of the stator, the three emfs are
given by:
EaJO = Em sin(utf) (3.1)
EbO = _ . 3tt, Em sm (wi ) (3.2)
32
a coil
c coil b coil
Figure 31: Schematic of a delta connected synchronous machine showing
the balanced coils and the rotor. Eqo is the noload emf developed by coil a.
Eco = Emsm(ujt + ) (3.3)
where the amplitude Em will be derived below and where 6 = u>t such that all
degrees are measured in mechanical radians. We initially assume no current
flows out of the coil, hence the subscript 0 in Eao, Ef,o, Eco If the rotor
has only two poles, it is easy to see that the machine will have one electrical
revolution per mechanical revolution. Machines with multiple poles on the
rotor will have more electrical degrees per revolution by the relation:
0e = \0m (3.4)
where p is the number of poles on the rotor. For analysis, it is convenient to
work with a single pair of poles and realize that by symmetry the conditions
on the other poles are identical to the ones under consideration.
The following discussion will be restricted to the geometrically simpler
case of cylindrical rotors. We want to understand how sinusoidal voltages
33
are produced in the stator coils of this geometry. We first establish that
a rotating sinusoidal field is required to do this and then we establish the
ideal pattern of field currents in the rotor that will produce this magnetic
field. Real machines approximate this pattern through various field coil
winding designs. After understanding how to produce a sinusoidal emf in
the armature (stator) coils, we attach a load to the stator terminals so that
current flows through the armature. We will show that the consequence of
this armature current is the production of a superimposed rotating magnetic
field that lags the field generated by the rotor, resulting in a countertorque
on the rotor. Thus, we see that the prime mover must do work to keep the
rotor spinning against this countertorque, and it is this mechanical work
that is converted into the electrical energy of current flowing out into the
load.
Finally, we arrive at a circuit model for the generator that allows us
to construct an equation modeling the dynamics of the generator under
variations in load. This is the socalled swing equation whose solutions we
will explore in the next chapter.
3.1 Required Function Form for B(0)
Let us consider an IVtum coil aa wound on the inside of the stator, as
shown in figure 32, subtending an angle p about the center C of the gen
erator. From Faradays Law, the emf Ea generated across the terminals at
tached to coil aa due to a timedependent magnetic flux $ passing through
a surface bounded by the coil is
Eao = Ns
dt
(3.5)
34
+
Figure 32: The flux through a stator coil aa imposed by a cylindrical
rotor.
Let us choose the surface to be the cylindrical section defined by the arc
 < 0 <  and extending length l into the paper. Consider the radial
component Br of magnetic flux density crossing this arc. We write
Br = Sr(r2,7,t) (3.6)
where r2 is the radius of the stator inner wall (ignoring variations due to
slots cut into the wall to house the coils) and 7 is the angle from the axis
that bisects the arc oa. Then
Ea0{t)
~Ns^t jBr(rh'ht)dA
~Nsli / Â£ Br(r2t)lr2d'y
(3.7)
35
We would like Eao(t) to have the form of equation 3.1
Eao (t) = Em cos Ldt (3.8)
and this will be obtained if Br(ro, 7, t) has the form of a rotating sinusoidal
wave
BT(r2,7, t) = Brm(r2) sin(7 u>t).
(3.9)
To see this, substitute the above expression into equation 3.7
d fi
Eao = NS I ^ BT(r2,7,t)lrQd'y
d
= Afe / BTm(r2) sin(7 u>t)lr2d,y
d Â£
= Nslr2BTTn(r2) [ cos(7 art)] Â£
ut 2
= Nslr2BTm(r2)^ cos( art) + cos(^ art)]
= Nslr2Brm(r2) [ cos ^ cos art sin  sin art
p p . ,
+ cos cos art sin sm art
2 2 J
= Nslr2Brm(r2)2sin.^(smart)
P
= JVs2Zr2sinSrm(r2)wcosa;t.
(3.10)
3.2 Ideal Rotor Current Distribution Needed to
Obtain Desired B(0)
Let us first assume that no load is attached to the terminals of coil aa\
so that no current flows in this coil. In this case the flux density B(r, 7,
36
Figure 33: A simple rotating magnetic circuit passing radially through a
rotor of radius 77, across an air gap of thickness g, and then azimuthally
through a stator whose inside wall has radius r^. Both rotor and stator are
assumed to be made of a high permeability material fj,.
t) is due entirely to the magnetic field H(r, 7, t) produced by currents in
windings of the rotor. Since the fields linking the stator coils are in the air
gap,
B(r, 7, t) = HqH. (r, 7, t). (3.11)
Suppose current flows along the walls of the rotor in a direction parallel to
the rotor axis and let this current have a linear density k(7, t). Applying
Amperes Law to the path T shown in figure 33 gives
L
rSurface
dA
37
rO+ir
= / (7.*)
Je
r\drf
f ^r  dl = f k(7, i) 77 dy = mmf{6) (3.12)
Jr M Je
Here the current is positive if it comes out of the page. Because of the high
permeability of the rotor and stator cores, we expect that the lefthand side
is dominated by the air gap
L
dl
rr.

Jri
' dr + r Br(r,e+^,t)Jr
Jt
Mo Jt
and we anticipate a symmetry BT(r, 0 + 7r, t) = Br(r, 6, t) so
(3.13)
/SfcaUisjfMifl* (314)
Jr M Jr i Mo
Because of the round rotor we also expect that the flux density is oriented,
to a very good approximation, in the radial direction in the air gap. When
ever fields emerge from a high permeability medium to air, the boundary
conditions on the fields show that1
Bt (air gap) ^ Mr
5 (air gap) V1 + Mr
The divergence free condition on B is
(3.15)
V B = 0 (3.16)
and in the air gap, V B ~ ^1. This means ~ 0 and we take Br to be
essentially constant across the gap. Thus,
1 Strictly speaking, this argument applies in the absense of surface currents.
38
(3.17)
f Br(r,0)dr ~ Br(r2,9,t)g.
Jri
Finally, equation 3.12 becomes (after flipping left and righthand sides)
mmf{9, t) = 2
From the preceding subsection
Br(r2,9,t)g.
Mo
(3.18)
... . 2Brm(r2) sm(0 ut)
mmf{B, t) =g
Mo
(3.19)
or
P (7, t) rlli7 =
Mo
(3.20)
which will be satisfied if /c(7, t) = Km cos(7 art). To see this and to find Km
/0+7T
Jo
K(7i tyidry
r6+n
= Jo
Km cos(7 ujtyid'y
= Km [sin(7 ort)]g+7r 77
: 2KmT\ sin(0 art)
(3.21)
so
= =^i. (3.22)
Mo
(We have suppressed the radial dependance of Brm since we have assumed
BTm(r 1) Brmi e. the field is nearly constant across the gap.) In
conclusion, then, we want a current distribution around the rotor that is
given by:
39
kOj, t) = \Brmg gQg/ _
n Vo
(3.23)
Figure 3 4 shows a radial plot of this current density distribution as a
function of 7 at different values of u>t.
Figure 34: Current density indicated by radial displacement of curve from
rotor surface, a)ut = 0 b)wt = j c)u>t = indicates current into the
page and indicates current out of the page.
The question now arrises: How shall we generate such a rotating current
distribution? One way is to wrap windings, each carrying a constant current
Ir, around the rotor with linear density
n(~f') = ^Tm^ cos 7' (324)
77 fj.0Ir
windings per unit length along the rotors circumference. Here V represents
the angle relative to an arbitrary reference point. Then, we simply rotate
the rotor at uniform angular velocity so that
7' = 7 bit
(3.25)
40
Figure 35: A rotor with k(j) = o (into the paper) for <7 Lot < 
and /c(7) = kq (out of the paper) for the other half of the circle.
describes the relation between the rotor frame of reference and the stator
frame. Finally, as food for thought, note that we might consider other ways
to produce a traveling wave of current density around the rotor even without
having the rotor actually rotate.
3.3 Realistic Motor Windings and Field Harmon
ics
In practice, it is much easier to put windings on the rotor with a uniform
density as depicted in figure 35. We will show that this gives a triangular
dependence of B on 9 by considering first the case uit = 0.
Here (at t = 0)
nBr(r0,9)g
Vo
pit/l r0(7T
/ (Ko)n^7+ / (o)n^7
Je A/2
2kqt\9
(3.26)
(3.27)
41
for 0 < 0 < For f < 0 < ^ and ^ < 0 < 2n respectively we get
2Br(ro,0)g = 2Kori(7r ^ (3.28)
Mo
and
2Br(r0,6)g = 2Kor^_2jr + (3.29)
Mo
which is a sawtooth plot that can be expanded as a Fourier sine series
Br(r0,6,0)
9
E
4 .
~ sin
(y)sin(n0)
(3.30)
Moori
5
sin0 sin 30 +
7r 97r
4
257T
sin 50 ..
For other times, we simply replace 0 with Q uit \
Br(ro, 0,0) = ^^ri ( sin(0 cot) ~ sin3(0 u>i) + sin5(0 Lot)....)
g 7T 97T 207T
(3.31)
Compare this to the flux density obtained by the cosine (rather than square
wave) current distribution
Br(r0,0,0) = tLoKri sin(0 urt (3.32)
9
To ensure the same flux density in the fundamental sin(0 u>t) term of
equation 3.31, we require
Mo^on 4 Momn
9 tt g
(3.33)
42
or
Ko ^ Km (3.34)
.79/Cm.
Thus if we uniformly wrap the rotor with coils of linear density equal to 79%
of the maximum linear density of a cosine wrapping, we achieve the same
flux density in the fundamental term of the sine expansion.
In a threephase generator, the third and ninth harmonics will produce
equal voltages at each of the three output terminals, thus contributing zero
to the output current (since this requires difference in voltages between the
terminals). Thus it is necessary only to deal with fifth, seventh, eleventh, etc.
harmonics. Textbooks on motors show that techniques such as fractional
pitch windings (p < 7T in figure 32) can lead to suppression of the fifth
and seventh harmonics without detracting much from the strength of the
fundamental.
3.4 Relation Between Noload emf and Rotor Cur
rent
So far, we have assumed that no load was attached to the generator termi
nals. Let us call the emf generated across the terminals of coil aa under
this condition Eon This was shown to be expressed as
Eao = Ns2lrz sin ~rBTTnu> cos u>t (3.35)
43
where BTTn = /Jp^ri 4 for a rotor wound with uniform linear current density
Ko ampsm1. If Nr windings carry current Ir aroimd the rotor then we can
write
o =
NrIr
7rri
(3.36)
because the N sides of the windings carrying current in one direction are
distributed over a semicircle of length nr. This means
Brm
HqNr Ir 4
7TZ
and substituting the above into equation 3.35 gives
(3.37)
EaO (t) = NslT2 sin ITU COS U)t
7TZ 2 g
= LTau;ITcosujt (3.38)
where
Lra = ^NsNr1^ sin % (3.39)
7TZ g 2
can be interpreted as a mutual inductance that relates the rotorinduced emf
to the current IT in the rotor. Note that Lra is proportional to the product
of the rotor and stator coil windings NsNr.
3.5 Armature Connected to a Load: Fields Gen
erated by Armature Currents
Suppose a balanced load is attached to the armature
44
Ia{t) = lLCOs(u)t Ip) ) 3 (3.40)
h(t) = II cos(uit ip (3.41)
m = II cos (ut ip 47T. (3.42)
Here ip represents an asyetunspecified phase lag between the aa coil ar
mature current and the rotorinduced emf in the aa coil.
Each of these currents acts as a source of magnetomotive force that
creates magnetic flux that also links the stator coils. Therefore, the actual
emf induced in each coil is due to the superposition of flux produced by the
rotor and flux produced by the stator currents. We will see that this situation
can be reduced to a relatively simple circuit model for the generator.
First, what is the angular flux density distribution produced by the cur
rent in coil aa at any given time? We will concentrate on the simplest
situation, where coil aa is concentrated into a single slot that spans a
fullpitch angle (referring bade to figure 33) of p = tt.
By an argument similar to that for the field produced by currents dis
tributed around the rotor
2 BÂ£\r2,e,t)g
Vo
Wafi(7)Wa(*)(7 + f)
d'y
(3.43)
where the deltafunctions represent the concentration of the coil windings at
angles within the stator wall. Integrating, we have
45
(3.44)
B^\r2,9,t)g
Vo
%NSIa for = <*
< ~^NsIa for Â§ < 9 < tt, >
7T < 9 <
V 2 /
which gives a square wave that varies from Nsla when 9<\ or
and the value ^Nsla when  < 9 < We write (r2,9, t) as a cosine
series
B$?\r2,9,t) = ^Nsla (cos 9 i cos39 + \ cos 59 ] . (3.45)
1 2g tt \ 3 5 )
As before we will retain the fundamental mode, noting that higher harmonics
may be suppressed by various methods, including distributing the windings
across a finite spread of angles in the stator and setting the windings at a
fractional pitch (p < tt).
Thus we write
cy
Bj?\r2,9,t) = finNsIl oos(u>t ip) cos 9 (3.46)
ng
and similarly
Br\r2,9,t) = HqNsIlcos{ut xp ^)cos(0 ^) (3.47)
'ng o o
/ \ 2 47T 47T
Bf\r2,9, t) = /iQNsil cos(oit ip) cos (9 ). (3.48)
TT g o o
The total field due to armature currents is
46
B^\r2,0,t)
fj,0TV's/_Â£, [cos (wt ip) cos 6
ng
/ , 2tt i 2tt i
+ COS(wt V ~ g) COS(0 g")
+ cos(a;f V y) cos(0 y)]
3 2
Â£i0iV57x, cos(0 U)t + ^).
2 ng
(3.49)
Note that this is a rotating wave with the same angular frequency as the
rotor.
The induced emf in coil aa is then (taking p = 7r)
Eas(t)
i /*
~dt lNsB{S\ro,e,t)lrd0
^~2
4;Nslr2(NslL) [2 cos(eut + ip)de
at Kg J_e
~NsILlr2^~ [sin(0 vt + ip)]K
irg at 2
^^Ir^NguIz sin(wi ip)
ng
lr2NgU)lLCOs(u)t ip ^r)
Tzg 2
LsU)Ii,cos(u)t ip ^) (3.50)
At
where we define a stator inductance
Ls = ^lT2Nl
irg
(3.51)
To summarize, we have
47
Ea0 = LracoIT cos cut (3.52)
la = II cos(u>t ij)) (3.53)
Eag 7T = Lsu>Il cos(tirt 4> ). (3.54)
We write these as phasors as shown in figure 3 6
Figure 36: Phasor diagram showing the relationships between the emf and current phasors.
EaQ LjqUJI'p (3.55)
ia = ILej^ (3.56)
= LsulLe^Ljn/2 (3.57)
and note that
Eos = jioLgla. (3.58)
Now the voltage at the terminals of the generator, using phasor notation, is the superposition of the emfs due to the rotor and to the stator current:
48
Ea 0 juiLsiaO
(3.59)
This is exactly the relationship we would get from the following circuit model
(figure 37) of an ideal voltage source Eao in series with an inductor Ls
Figure 37: The equivalant circuit to the phasor diagram.
Finally note that the inductance Ls must be modified by adding a con
tribution due to leakage flux that couples through coil aa without going
through the rotor. Details may be found in various references [3] but the final
circuit model remains the same. The key point is that we have embedded all
the details of rotor and winding geometries into a simple model of an ideal
voltage source Eao (given by equation 3.38) in series with an inductance Ls
(given by equation 3.51 modified as noted above).
3.6 Power Transfer
For convenience, and in keeping with the conventions established in the
literature, we shift the origin of time so that the terminal voltage V has zero
phase angle at t = 0. Later we will think of the terminals being attached to
the power grid, so that V is also the power grid voltage at the entrance to
the power plant we axe modeling. Thus, it is reasonable to think of V as our
reference.
Figure 38: Phasor diagram showing the relation between noload emf Eao
and terminal voltage V under load
Figure 38 shows the noload emf Eao leading the terminal voltage V by
an angle 6. This angle is determined by the following construction:
(1) Find the current Ia according to
V_
ZL
V
Rl + jXL
50
(3.60)
{R\ + Xl)he
where Zl is the load impedance, Rl is the load resistance, Xl is the load
reactance (a/L for an inductive load and ^ for a capacitive load) and ,
the phase angle, is given by
, . iXL
Rl
(3.61)
This procedure is valid provided we can characterize the load attached to
the terminals. In other cases (such as when the generator is attached to the
grid) we might instead directly measure Ia and write it as
Ia = Iae~rt, (3.62)
again denoting the phase shift relative to the terminal voltage V by angle .
(2) Using the simple circuit model of the generator shown in figure 37,
we have
Eao Va + ju)Lsia
= Va+uLsIaej4,+j*
= Va + u>LsIa sin + juLsIa cos
Thus the lead angle <5 is given by
c _i IavL cos
o tan .
Va + IauLs\n
We want to know how much power flows out of the circuit on average.. From
equation 2.9 we can write the answer as
51
(3.65)
P = ^/o Va COS (j)
(or P = IaVacos (j) if Ia and Va are nns amplitudes).
However, in describing the dynamics of the rotor below it will prove
necessary to express this power instead in terms of Eao, Va, X$ (where
Xs = juLs) and 8. It can be shown that an expression equivalent to equation
2.9 is
p=\iyz cs
(3.66)
Now
InV* =
ja v a
Eg0 ~ Vg
jXs
Va
^{Ea 0j*Va)Va
Xs
rp (Eao cos 8 + jEao sin 8 Va)Va
Xs
Ea0Va c .V*Ea0Va c
sm 8 + jcos 8.
Xs
Xs
(3.67)
Taking the real part
P= %^sin<5 (3.68)
Xs
This is a very important result, because it contributes a nonlinear term (due
to sin6) in the socalled swing equation for the time dependence of the
rotor angle 8 (see below).
52
3.7 Losses
There is one more topic we need to touch on before we can set up the
swing equation: energy losses. Although there are losses through mutual
and self induction in the rotor coil and armature circuits, I want to talk
about hysteresis and eddy currents.
Hysteresis occurs in synchronous machines because of the nature of mag
netic material, especially iron and its alloys. Any magnetic material will have
domains of magnetic dipoles pointing in random directions throughout its
volume and because of this randomness, the overall magnetic moment of
the material is zero. If we pass a current through a wire wrapped around
the bar, the magnetic field, H, will start to cause the domains to line up
increasing the magnetic flux field, B. Two things then start to happen: As
the current increases, more fields line up and the fields that line up reinforce
H to bring more domains into line until almost all are pointing in the same
direction and B is near a maximum. At this point, increases in H do not
mean more increases in B and the material is said to be saturated.
If we now reverse the current through the wire, B does not follow the
same curve down as it had going up as shown in figure 39. This is because
the domains are reinforcing each other and do not immediately want to
return to their preferred state in the material. This refusal to follow the
same path with reversing currents is called hysteresis, and, we can show the
area in the hysteresis curve is equal to the energy lost by the system. Since
we are generating and studying alternating currents in these machines, it is
a very important concept. Where it is important to control the hysteresis,
manufacturers use special steels that have a preferred crystalline structure
[12]
53
Figure 39: A hysteresis loop.
Experiment If you put your hand on a running motor, you will notice
that the stator gets hot. This heat is not, for the most part from friction,but
from I2R (power) losses in eddy currents. Eddy currents occur when a
conducting sheet is subjected to a changing magnetic field and generate little
eddies of current that oppose the motion of the sheet in accordance with
Lenzs law. They are easily experienced by moving a piece of aluminum
or copper between the poles of a strong horseshoe magnet; the drag on
the metal opposing the motion is easy to feel. The teacher can add to
the experiment by cutting slots in another, similarsized sheet and let the
students feel the difference. Alternately, a round magnet can be placed in an
aluminum pop can and rolled across the floor; the can will not go very far.
Most dramatically, you can drop a round magnet into a three or fourfoot
piece of aluminum or copper pipe (the longer the pipe, the better the effect).
A strong magnet will take an appreciably long time to drop through the
tube. The consequence of these damping mechanisms will appear at the end
54
of the next derivation.
3.8 The Swing Equation
We now have enough information to assemble an equation modeling the dy
namics of the generator under load. The generator is supplied with power
from the prime mover (Pm) that spins the rotor at a velocity u>r, and pro
duces electrical power (Pe) to the distribution network. As I pointed out
above, when there is no load on a synchronous machine acting as a genera
tor, the internal emf, Eao, is in phase with the terminal voltage, V. When
a machine is connected to the power grid, Eao is set to the same magnitude
and phase as V so no current flows and the generator is not supplying any
power to the grid. To start power flow, either more torque is supplied by the
prime mover, or the excitation current is increased in the rotor increasing
the flux to the stator.
In the first instance, one would assume the frequency would increase but
because frequency must remain constant, the angle 6 increases. Real power
is fed to the system and we have the situation shown in figure310; the
generated voltage (dashed curve) leads the reference voltage (solid curve) by
the load angle 6. In the second case, 6 does not increase but reactive power
is fed into the system.
To understand the dynamics of the generator we first must look at the
mechanics of the machine. The rotor in the machine has moment of inertia
J and turns at an angular velocity uiT powered by the torque of the prime
mover, Tm, and opposed by the torque produced by the generator, Te, or,
from first principles:
55
I
Figure 310: Phase shift due to power angle 6.
Y,T* = Jj
JJ = Tm Te. (3.69)
But u> is the timerate of change of the rotor with respect to the stator or,
w = f which imP!ies ^ so
J9" =Tm Te.
(3.70)
Under steadystate conditions, 6 is steadily advancing according to
d33 = waat. (371)
We are interested in deviation in rotor speed, so we can change reference
frames from the stator to the rotor by defining
0m ^sst "I"
(3.72)
56
where 6m is the angular displacement of the rotor from the rotating reference
frame. Differentiating 3.72 twice with respect to time gives
C = C (3.74)
Equation 3.73 shows O' is constant when S' is zero (no shift in relative rotor
position) which we expect and 3.74 represents the rotor acceleration in me
chanical radssec2, or the rotor is accelerating in both frames of reference.
We now substitute equation3.74 into 3.70
J6'^ = TmTe. (3.75)
We are actually interested in the power of the system, so we multiply both
sides by uj
JuO" = u(TmTe), (3.76)
note that Ju M, the angular momentum, and Toj P or
MC = Pm Pe (3.77)
Now we use equation 3.68 for Pe :
Pe=EaoVasinS (3.78)
and note that there is also a damping term, D oc <5', that will act against the
57
mechanical power2 and equation 3.77 becomes
M8" = Pm
EgaVg
X
sin 8 D8r.
(3.79)
3.9 Scaling
Before analyzing the swing equation in the next chapter, we want to choose
appropriate reference (base) scales to reduce the equation to dimensionless
(per unit) form. Let SB be the voltampere rating of the generator (say
1000 MVA) and VB be the rated output voltage (say 22 kV). We will find it
convenient to define a reference value of reactance
Vt
XB = ^ (0.48 Henries). (3.80)
SB
A time scale TB (usually given the symbol H in power system literature)
may be defined as
\usM
Sb
(3.81)
where M is the rotor angular momentum and u>s is the angular frequency.
This time scale is suggested by the fact that the kinetic energy stored in the
rotor, \Mlosi will drain away due to a constant electrical draw SB in a time
Tb unless mechanical power continues to be provided.
Suppose the rotor is steel (density 7800 kgm3) with radius 1.0 m, length
10 m, and rotational frequency 60 Hz. Then
2The damping arrises primarily from eddy currents in the rotor of a generator. Modem
machines are specifically designed to take advantage of damping to keep oscillations at a
minimum during power fluxuations.
58
Ju> (3.82)
^TnR22irf (3.83)
pK2lR^f (since m = pirR2l) (3.84)
(7800)7r2(l0)(l.0)460 wattsecond2 (3.85)
46 MWs2. (3.86)
Then Tb ( H ) evaluates to \ ^^ooo^46^ = 8.6708 ~ 8.7 seconds.
The original swing equation (3.79) is
M^S = P^sm6D^6. (3.87)
ctt A at
Note that M has units of (seconds)2 and D has units of seconds, following
conventions in the power systems literature. Equation 3.87 becomes
= P sin 6 D^8. (3.88)
dt2 X dt y J
in per unit form. For further analysis, we follow Canizares [5] and select
M = .1 sec2, V = E = 1, X = .5, and D = .1 sec. We will also find it
convenient to drop the tildes over the parameters for the rest of this paper
but understand equation 3.88 is dimensionless. Equation 3.88, then, is the
swing equation that I will use in the next chapter.
59
Chapter 4
Analysis of the Model
4.1 Introduction
Equations like 3.88 are known as differential equations and are one of the
primary way we study dynamic systems (those systems that evolve through
time). Sometimes we are lucky and we can solve the system in closed form,
i.e. the solutions can be expressed as an explicit function in time.
Most differential equations, however, are not that nice. To see how these
equations evolve in time, we use computers to show how the system grows
in time and plot the solutions as a function of time or plot one variable
as a function of another. In this section, I will use the graphical analysis
techniques used by Strogatz [9] where we will look for fixed points (points
where the solution does not change over time) and analyze their behavior.
Figure 41 shows one of the results of this analysis; the fixed points of the
swing equation with the power angle, S, plotted as a function of the power,
P. The curve shows a bifurcation curve, and area where there are two distinct
solutions to the system of equations (see next section).
60
Figure 41: Bifurcation curve for the swing equation in the u> = 0 plane. The
lower branch (dark curve) are all the stable fixed points, the upper branch
are all unstable fixed points.
4.2 Fixed Points
Equation 3.88 is a second order differential equation in the variable <5. It may
be written as two dimensionless first order equations in variables 6 and u>:
8' = = f(uj) (4.1)
1 EV
J (Psm8Duj)=g(S,Lj). (4.2)
where the numerical values of M = .1, V = X = 1, X = .5, and D = .1 are
from the previous chapter [5].
The first task is to identify the fixed points by setting 8' and
of the above equations equal to zero and solving for the variables (6 and ui).
61
In the first equation, $ = 0 when a> = 0 for all values of u> which means the
fixed points always live on the horizontal (5) axis. When
P_
M
PX
EV
f>o
0 =
EV
IM
sin 8
EP
M
V D
Vsin(5)^(0)
XM w M
sin 6
8,11 (w)
(4.3)
(4.4)
(4.5)
(4.6)
thus the system has one fixed point at /f = (<5o, 0) where So falls in the range
0 < <$o < Â§. We note there is another fixed point at / = (7r <5o, 0) by
noting the trigonometric relationship
sin(7r S0) = sin(^o) (4.7)
(it doesnt matter which way we go to get to 7r; see below). We should also
note that the second fixed point is really (rwr5o) where n is an odd integer
since the sine function is periodic over 27T. We can use this and the fact that
u> is an angular velocity and S is expressed in radians to wrap the S u>
plane onto a cylinder 27T radians in circumference making the cylinderspace
we see in the classic driven pendulum [9]. This means we can disregard all
other points (that is, we only have to evaluate the points at n = 1 and even
then, 7r are the same point in cylinderspace).
The analogy to the pendulum problem leads us to assume /* = (Sq, 0)
is stable and the other is not. In fact when we compute the Jacobian of the
system we get:
62
(4.8)
fs i 4 0 l
gs 9u> _ _ ~wkcos^o D M
which gives the eigenvalues
d)_ D \!d2~ 4^^^
!2 2M 2M
With the given values of D, M, E, V, and X we have
(4.9)
= 5 5.0%/(.01 .8cos50). (4.10)
This will result in a stable spiral for /* for Sq between 0 and a value just
smaller than ^ (this is affected by the damping term inside the radical; see
below) [9] .
The second fixed point has a Jacobian of
J =
0
wk cos
that changes the eigenvalues to
(4.11)
(2) D V/Â£>2+4i^Zc^o
x2 2M 2M
which result in a saddle point ((the square root of D2 plus something)/2M)
is larger than D/2M, therefore, Ai is positive and have different signs) [9]
It is useful to look at the eigenvalues as a function of a control parameter,
which in this case, we take to be the mechanical input power P. Using
equation 4.6 to express in terms of P, we rewrite equations 4.9 and 4.12
63
as
(1) D \jD2 4MjF cos(sin 1(^))
1>2 2M 2 M
for fi and
(2) D \/D2 + 4MF cos^m"1^))
1>2 2M 2M
for / where 0< P < 2. Recall that fixed point solutions are only obtained
when 0 < ^ < 1 (P, X, P and V are physical quantities that are by
definition positive). For V = E = 1 and X = 0.5 this implies that fixed
point solutions are only found when the central parameter P lies in the
range 0 < P < 2. However, other time varying solutions can and do exist
when P exceeds this range (see below).
Figure 42(a) shows both the stable fixed point eigenvalues only and
figure 42(b) shows the stable and unstable fixed point eigenvalues with the
real part of the eigenvalues plotted horizontally and the imaginary parts
plotted vertically. Note that the last two plots in (a) (blue and red points
on the real line) and the first two points of the unstable eigenvalue (purple
and light green at (0,0) and (1,0)) are the same.
We also note that at some point P < 2 the stable fixed point changes
from a stable spiral to a stable node (both eigenvalues become real and
negative). To find this point it is necessary to find the value of 6q where the
radical becomes zero or where
2 aMEV .
D = 4 cos Sq
(4.15)
Solving for Sq and plugging in the values for constants equation 4.15 evaluates
64
Figure 42: Eigenvalue plots a) Stable eigenvalues and b) Stable and unstable
eigenvalues.
50 ~ 89.28 = 1.56 rad.
Substituting <5o = sin1 (^) = sin_1(.5P) and solving for P (since P is our
control parameter) results in
P~ 1.99984.
This means the bifurcation diagram will have three types of fixed points for
0 < P < 2. A stable spiral and a saddle node from 0< P < 1.99984 and a
stable node and a saddle node from 1.99984 < P < 2. Note that the size of
the stable node region increases with the square of the damping term; the
more damping in the system, the larger the range in P in the bifurcation
curve that will be covered by the stable node.
Figure 43 shows the vector field for the stable eigenvalues clearly showing
the stable spiral at 6 =  with P = 1 and the saddle node at <5 = 7r.
65
1 t i 4 S 6
Figure 43: Vector field of the system with P = 1. The colors denote vector
length with red on the 8 axes the smallest. Note the evidence of the spiral
at  and the saddle point at
When P in equation 4.2 is 0, the fixed points live at the origin and (0,
7r) with the origin the stable fixed point. As P increases in value, the fixed
points move toward each other until they meet at S =  and all points in
the phase space move instead to a limit cycle (shown in figure44). It is
important to note that this limit cycle is a completely separate solution,
which numerical investigations reveal exists even for P < 2 (see below).
That the system goes to a limit cycle can be seen by computing the
vector field by dividing d6 by du> as shown in Strogatz. Thus,
d8 ui co
du ~ sin (5 Du>) P 2s'm6 .luj
Consider the 8 ui plane. Noting that u is positive in the first quadrant
4.16
66
u
4 r
4
r <
4
4 <
4 4
srSSffSS*
sS/Z/SS*
' / / / / / X '
'////// '
*'////<
* * t t 1 t * < .
* < 1 f t f * t
4 < 1 1 t f
t * i f t t *
4 i t t t *
* M \ *
tr
* >.
V '
k k
V
1
\ \ \ \ y
m h
6
Figure 44: Vector field of the system with P = 2.5. Note there is no evidence
of a spiral or a saddle point and all vectors are pointing to the limit cycle.
(and if ui is sufficiently large), the denominator is negative and the vectors
point down (the slope is negative). In the first quadrant below the limit
cycle we see that the maximum value for 2 sin <5 = 2 when 8 = . Thus, the
numerator in equation 4.16 has a maximum value of P 2 Au> which will
be positive when
P > 2 + .Iuj (4.17)
and u) is sufficiently small. To find out how small we can solve equation 4.17
for ui and get
u < 10(P 2).
67
If, for example, P = 2.5, then u> < 5 guarantees the numerator in equation
4.16 will be positive. If w is negative (quadrant IV) then the vectors will
be up to the left when the denominator is positive (negative slope; as seen
in figure 44) or up and to the right if the denominator is negative (positive
slope). Thus, we have constructed a trapping zone in a band in phase space
to which all trajectories point and a limit cycle must exist by the Poincar5
Bendixson theorem [9].
Such arguments about existence of a limit, cycle are inconclusive when
P < Pjaax < where Pmax is the maximum mechanical input power for
which the generator maintains uniform rotation (<$o = Â§ and u> = 0). Here we
resort to numerical investigations, looking at the asymptotic behavior (Â£ >
oo) of trajectories starting from large initial values of u>. For a considerable
range of P < 2 limit, cycles are found to exist. There appears to be a
minimum P below which stable limit cycle solutions are not found, but this
remains open to further investigation.
There is an interesting mechanical analog to this system. Consider a
damped pendulum of length l and whose axle is subject to a fixed torque
Fixed points occur when the gravitational torque mgl sin S exactly opposes
the imposed torque. Here mgl plays the role of If the imposed torque
exceeds mgl, there is no fixed point and the pendulum rotates continuously.
Even when the imposed torque lies below the value mgl, the pendulum can be
set into motion with large enough initial u> that continuous rotation persists.
It would be interesting for a school class to build a model that demonstrates
this.
The pendulum analogy leads to seeing the fixed points as points on a
circle as shown in figure 45. When P = 0 the system starts out with the
two fixed points at zero and k (zero defined as the bottom of the circle)
68
Unstable fixed point
Stable fixed point Annihilation
Figure 45: Fixed points annihilating on the circle.
shown in figure45(a). In figure45(b) the fixed points are moving together
as P increases and finally at 45(c) they meet and annihilate at <5 =
The power generator lives or dies by the parameter 6 and the range of
the parameter depends on the amount of power demanded by the outside
system. If the instantaneous demand on the generator is not too great, the
rotor will oscillate around a new value of 6 until equilibrium is established.
All modern power plants are designed to shut down if the demands are too
great rather than damage the equipment.
4.3 Basins of Attraction
The preceding analysis considered stability of fixed point solutions to infin
itesimal pertubations (referred to in the power systems literature as steady
state stability). What happens if a large pertubation occurs instead? One
approach is to use the full nonlinear equations and identify numerically the
domains of initial conditions that are attracted to the different solutions
(stable fixed point, limit cycle, etc.). These domains are called a basins of
attraction, and to find them, we have used software from Nusse and Yorke
69
Dynamics, Numerical Explorations [13].
As we can see from figures 46, 47, 48, and 49, there are two basins
of attraction. The basin at the bottom of the graph (the magenta blob) is
associated with the stable fixed point when 0 < P < 2. Note that it moves
to the right and gets smaller as P increases in value which corresponds to
the stable fixed point sliding up the circle and annihilating with the unstable
fixed point (figure 45).
The interesting feature in the figures is the second basin (the cyan area)
of initial conditions that asymptotically approach the yellow curve. At P =
0.18 this curve coincides with the boundary of the basin of attraction of
the stable fixed point (magenta region). This, then, appears to be a semi
stable limit cycle, where trajectories on one side (cyan region) attract to
it but trajectories on the other side (magenta region) repel from it. For
larger values of P (0.6, 1.2 and 2.4) the yellow curve now is surrounded
on both sides by initial conditions that evolve towards the curve, indicating
that it has now become a stable limit cycle. If we keep with the pendulum
analogy, the limit cycle happens when the system goes over the top. The
sudden appearance of the second attractor probably is a result of the systems
inability to get over the top unless it has enough initial energy.
70
Figure 46: Basin of attraction with P = .18
Figure 47: Basin of attraction with P = .6.
71
Figure 48: Basin of attraction with P = 1.2
Figure 49: Basin of attraction with P = 2.4
Chapter 5
Applications to Science
Teaching
5.1 Equipment
In making equipment for school use, our original design goals were to produce
a cheap, robust (able to withstand abuse from a typical 14 yearold middle
school student) device that could readily demonstrate AC and DC voltage,
current, relative phase composition (with the aid of an oscilloscope) and
the relative power to run simple, everyday electrical equipment. To meet
that goal, we built a bicycle generator using an exercise stand, a modified
automobile alternator, and a heavyduty DC motor to use as the 12volt
exciter source for the alternator rotor and voltage source for DC experiments.
Figure 51 illustrates our concoction. The exercise stand originally had a
squirrelcage resistance device driven by the rear wheel of the bicycle (the
squirrelcage drive bearing). We removed the squirrel cage fans and used the
remaining axle to power the 12volt generator off one shaft and the alternator
off the other side.
73
Figure 51: The main components of the bicyclegenerator system (rear view)
Since a car alternator is designed to produce 12volt DC we modified ours
to provide threephase AC current and voltage. We removed the stator, and
soldered leads to the three coil ends where they attach to the diode bar.
We then led the ends through the case and reassembled the alternator. We
used red, white, and black as arbitrary colors for the phases and a fourth,
green, wire was attached to the neutral lug on the case (figure 52). This
setup allows measurements phasetophase and phasetoneutral. No special
modification was needed with the 12 volt source except to put a plate with
banana plugs for voltage takeoff for DC experiments.
Experiment In the initial setup for the system we had to choose either
specifying a voltage or a frequency with a (young and eager) cyclist pedaling
away at about 60 RPM. We chose a frequency, set our target at 60 Hz, and
74
Figure 52: Generator stator showing the 3phase leads still attached to the
diode bar.
let the voltage fall where it may. The synchronous speed,ns, we needed at
the shaft is set by the formula:
n3
120/
P
RPM
(5.1)
where / is the frequency in hertz and p is the number of poles. Substituting
in the required numbers gives na = 12(^60) = 600 RPM.
But the gearing on the bicycle would not give the correct shaft speed
when directly coupled to the alternator. We then tried for 12 volts RMS
output when the field coils were fed by the 12 volt DC source. In this
case we found the shaft speed by calculating the ratios in the drive train
and found the shaft speed to be 2400 RPM which results in a frequency of
240 Hz (actual frequency varies due to the cyclist and the coupling we use
to drive the alternator shaft). The alternator was then hooked up to the
test bench and driven at the target speed. The output voltage was plotted
75
against field current to find the hysteresis curve and thereby, the optimal
field current. However, measured voltages were erratic, especially when we
tried to reverse the field current. We corrected this problem by separating
the stator leads from the diode bar in the alternator.
A similar problem occurred with the DC source which was putting out
90 volts in initial tests when coupled directly to the squirrelcage shaft. To
find the correct speed for the generator, two motors were coupled shaftto
shaft. The driving motor speed was varied and output voltage and current
were recorded from the generator as a function of shaft speed that led to a
gear reduction of 12:1.
These experiments should be done by the teacher to set up the system.
They could also be assigned to students to help them learn about mechan
ical advantage, magnetic saturation in the rotor coil, voltage output and
frequency as a function of shaft speed and as a function of rotor current.
We have been using the system for over a year in the lab and have two
identical systems in the public schools, and we have found some weak points.
The most obvious is the tire slipping against the rotor under heavy loads
such as using this apparatus to heat a cup of water with an immersion heater.
Also, the coupling from the drive shaft to the alternator shaft needs to be
more robust. The piece of reinforced rubber hose currently in use needs to be
redesigned, or a new mounting scheme developed. Because it is torsionally
flexible, the hose will transmit an oscillating frequency to the shaft of the
alternator that will effect the final frequency of the generator. We would
also like to add some sort of transducer on the pedals so we could calculate
the power applied by the rider.
Overall, though, the system is a success. It is relatively inexpensive (less
76
than $300 complete with used bicycle, excluding machining costs) and does
an adequate job of demonstrating AC circuit phenomena. And, as a practical
demonstration of the amount of work needed to heat a cup of coffee, it cant
be beat!
A second demonstration device we made is called The tower of power
(figure 53). It is a simple tower made from dexion (an anglesteel material
with premade holes and slots) about seven feet tall with a nine inch boom
projecting from the top and a 12 volt DC motor attached near the base.
The boom supports a doubledouble pulley system for raising and lowering
masses with the motor. With this system we have been able to raise a
bowling ball the entire seven feet with the bicycle generator as the 12volt
source.
Experiment The most interesting visual demonstration we did with the
tower was set up with the circuit shown in figure 54. The left half of the
circuit is simply the bicycle generator powering the tower of power motor to
raise the mass. If pdealing stops and the mass starts to fall, the motor acts
like a generator. However, current through the auto brakelight generates
enough field in the stator of the towers motor/generator to produce a strong
countertorque against the permanent magnet rotor to stop a two kg mass
from falling at any appreciable speed. After the mass is raised, the switch
is thrown to the right side of the circuit and the falling mass turns the shaft
of the motor which acts as a generator to light the flashlight, bulb. This
demonstration was a big hit with younger students who could readily see
the conversion of mechanical to electrical to potential to electrical energy.
Since the automobile brakelight will glow when the mass is being raised,
but not while it is holding the mass at the top of the tower, high school
77
Figure 53: The infamous TowerofPower
78
DC Generator/Motors
Figure 54: Schematic for Towerofpower experiment. The motor/generator
on the left is on the tower, the one on the right is on the bycyclegenerator.
students should be able to see a demonstration of countertorque when a
generator is put under load.
5.2 Using the Equipment in the Science Classroom
The bicycle generating station could be very successful when
used in the physics or physical science classroom, either as a part of an ex
isting curriculum or as a basis for a new one. Its strength is in its versatility;
it can provide intuitive as well as technical demonstrations. A demonstra
tion at the intuitive level could be showing that power demands increase
significantly when a resistance is put into the circuit. A more technical
demonstration would be to measure that resistance that leads to the current
going through the circuit. Another demonstration is: When connected to
a coil (a reactance), the student can feel there is no significant difference
Experiment
79
between that and a pure resistance but, hooked up to an oscilloscope, he
can measure the difference in the phase angle between the voltage and the
current. The system is designed to lead the student from easy investigations
to research in an intuitive way.
As stated earlier, the purpose of Energy 2020 is to let middle and high
school students have the opportunity to do real research. But we also have
to see how this overall goal will fit into the Colorado Standards for Teaching
Science. Since I am not out to assess the overall E2020 program, I will
only look at the projects we have done in the Physics Department and fit
those to the Standards. We have two bicycle generation stations operating
in the school system at this time at Kepner Middle and North Side High in
Denver. The Kepner teacher used the bike in her science club which met
after school but did not integrate it into the classroom. The North teacher,
despite late delivery, was able to use the equipment for part of two physics
classes.1
The biggest problems at North were trying to fit the bicycle activities into
the existing curriculum and the teachers unfamiliarity with the equipment.
The first problem is going to be encountered by any change in an existing
program; there is only so much time in a semester. And, despite double
the class time per session, there is a lot of material to cover in high school
physics. The second problem feeds off the first; in fact, together the two
problems can form a feedback loop that can exceed a teachers tolerance
level in a very short period of time.
Norths solution to these problems was to use the equipment very spar
1 North uses a one semester block scheduling that compresses one academic year into
one semester. The first class to use the bicycle was a standard physics class; the second
was tin advanced class.
80
ingly in the first semester, and try more for research in the second. Ac
cordingly, the first semester students used the equipment to design labs to
explore electricity. All of these were rather simple, but did get the students
involved in learning how electricity works. During the second semester, how
ever, students had a full four weeks at the end of the semester to complete
a project that would focus on research. Both classes avoided the second
problem, for the most part, by having Dr. Tagg attending on a regular basis
to help them (and the teacher) with their particular problems.
The results were mixed. Dr. Tagg saw the second semester more suc
cessful than the first, while the teacher had the opposite opinion. This was
understandable; from the perspective of E2020 the second class did do much
better research than the first. On the other hand, the teacher believed the
first class learned more than the second. The final answer is still up in the
air, but the teacher is back this summer working on E2020 and plans to
write her masters thesis on a new curriculum based on the bicycle.
Another way to use the equipment in teaching is to take the approach
of presenting less material and covering it more thoroughly. This would fit
very well into the singlesemester block scheduling, as at North, because I
dont believe there is enough time in one semester to allow the students to
fully understand the concepts presented in one full year.
When we look at how the E2020 project would fit the Colorado Stan
dards, we see right off that the first standard (how does science and scientists
work) is met automatically by the very nature of the program. By working
with the faculty at UCD, the student will learn very quickly how a research
scientist works. Likewise, standard two (...know and understand common
properties, forms, and changes in matter and energy) is met by working
with the bicycle generator and tower converting mechanical to electrical en
81
ergy. The bicycle and tower also meets standard five (Students know and
understand interrelationships among science, technology and human activity
and how they can affect the world.) Six (Students understand that science
involves a particular way of knowing and understand common connection
among scientific disciplines.) is also covered very well in the program by
working on research projects of their own choosing that connect basic physics
to engineering systems that meet important social needs.
82
Appendix A
Symbols
8  Power angle. The amount in radians that the generated
emf leads the bus voltage or rotor reference frame leads or lags
the reference frame of the stator.
ip The flux coupling or linkage.
4> The amount in radians the current lags or leads the bus
voltage.
9  The angle in radians or degrees (specified in the text) the
rotor makes with respect to the stator frame of reference.
7 The angle made by the rotor with respect to the stator in
a rotating magnetic circuit.
T An arbitrary surface.
$ Magnetic flux.
uj Rotational speed in rad/sec
qo Test charge.
H The permeability, or the magnetization developed in a
material in a magnetic field. It is the ratio of the flux density to
the magnetic intensity. fi0 is the permeability of air.
83
B Magnetic flux.field
E Electric field.
e The instantaneous value for the emf in a circuit. See V.
H Magnetic field.
I Current. When capitalized it is the maximum current in
the circuit; small case i is the instantaneous value.
N Number of turns in a coil.
P The real power supplied to the grid by a generator mea
sured in watts.
Q The reactive power part of the generated power.
R The resistance of a circuit that disapates real power.
S The total power generated, P + jQ.
V Voltage. V is the maximum voltage while v is instants
neous voltage.
X The reactance of a circuit that is associated with the
imaginary part of the load.
Z The total real and imaginary resistance in a circuit, R +
jX.
84
Appendix B
Glossary
Armature See Phase.
Basin of attraction The set of points in the phase plane that are at
tracted to one particular fixed point or limit cycle.
emf The electormotive force.
Field coils The windings around the rotor in a synchronous machine
that produce the magnetic flux B.
Fixed point A solution to a dynamic system that does not change as
t > oo.
Flinging The tendency of a electrical or magnetic field to spread out
in free space. In a magnetic circuit, these effects can be disreguarded if the
size of the gap is small compared to the area of the poles (figure ??).
Generator A devise used for converting mechanical, chemical, nuclear,
or other forms of energy into electrical energy.
Grid The distribution network for electrical power including power sta
tions, substations, transmission lines, etc. The most important fact about
the grid is that the frequency and voltage are invariant. That is, when a
generator is connected to the grid its voltage and frequency output cannot
85
Figure Bl: Fringing at the poles of a magnet showing the spreading of B.
be changed unless it comes off line.
Hysteresis In an electromagnet, it is the tendency for the value of the
magnetic flux B to depend on whether the magnetic field H is increasing or
decreasing. As H increases, we will reach a point where B does not increase
proportionately and the material is said to be saturated.
mmf The magnetomotive force.
Phase One of a balanced set of inductive coils on the stator of a syn
chronous machine. Also called an armature. It also refers to comparing
sinusoidal wave forms: Waves are said to be in phase when they have ex
actly the same frequency and match peaktopeak.
Prime mover The devise that supplies energy to the generator. In figure
1.1, the prime mover is a steam turbine.
Root mean square (rms or RMS) A typical value of the continuously
varying quantity of voltage or amperage in an AC circuit. It can be shown
to be the value equivalant to the DC current that would dissapate the same
power through a resister in the circuit.
Rotor The rotating central core of the machine that is magnetized and
provides the flux linkage for enerating the emf. When the machine acts like
86
an induction motor, the rotor might be a permanent magnet.
Salient Literally means protruding or sticking up. They are found on
synchronous machines that use multipole rotors that cannot be wound on
a cylindrical (round) rotor (figure B2).
Figure B2: A fourpole synchronous machine with a salient rotor.
Saturated See hysteresis.
Stator The windings or coils wound around the periphery of a synchro
nous machine imbedded in a highly permeable magnetic material.
Threephase The total number of inductor circuits in a synchronous ma
chine. They are connected together in either a Y or Delta configuraton
(see figureB3). In the Y configuration (figureB3(a)), one end of each ciol
is connected in common with the other end free; the Delta configuation
(figureB3 (b)) connects each end of the coil to and end of a different coil.
Transmission fines The primary means to move and deliver electrical
power, especially high voltages. They are strung in groups of three, one for
each phase, and are made of aluminum rather than copper because of the
87
Figure B3: a) Y and b) Delta coil connections.
weight factor.
88
Bibliography
[1] Grainger, J J and William D Stevenson, Jr. 1994. Power system analy
sis, New York: McGraw_Hill.
[2] Nasar, S A. 1996. Electrical Energy Systems, New Jersey: Prentice Hall.
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