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1550 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 3, MAY 2004
Modeling and Simulation of a Salient-Pole
Synchronous Generator With Dynamic Eccentricity
Using Modified Winding Function Theory
Iman Tabatabaei, Jawad Faiz, Senior Member, IEEE, H. Lesani, and M. T. Nabavi-Razavi
Abstract—This paper models and simulates a salient-pole syn-
chronous generator using a modified winding function theory and
more precise stator and rotor winding distribution with dynamic
eccentricity between the stator and rotor. Air-gap permeance is
also computed more accurately compared to currently available
methods. Inductances with this method are compared to those ob-
tained from other methods and it is shown that the results are
closer to those obtained from finite element computations. Finally,
the calculated inductances are used in a coupled electromagnetic
model for simulation and studying the frequency spectrum of the
stator line current in the presence of dynamic eccentricity.
Index Terms—Eccentricity, line current frequency spectrum,
salient-pole synchronous generator, winding function.
LIST OF SYMBOLS
, , Self-inductance of stator phase windings.
, , Mutual-inductance of stator phase windings.
Self-inductance of excitation winding
(including leakage flux).
, , Mutual-inductances between stator phases
and excitation winding.
V, I and
Voltage, current and flux-linkage of wind-
ings.
Developed electromagnetic torque.
Co-energy.
Degree of dynamic eccentricity.
I. I
NTRODUCTION
S
YNCHRONOUS machines are the most important and
valuable devices in power systems. These generators are
generally well constructed and robust, but the possibilities of
incipient faults are inherent due to stresses involved in the
electromechanical energy conversion process. Fault diagnosis
can produce significant cost saving by allowing for the sched-
uling of preventive maintenance, thereby preventing extensive
downtime periods caused by extensive failure [1]. In addition
to bad performance, faults reduce the life span of synchronous
generators. Fault diagnosis of large and costly generators in
Manuscript received December 17, 2002; revised December 29, 2003.
I. Tabatabaei was with the Department of Electrical and Computer Engi-
neering, Faculty of Engineering, University of Tehran, Tehran, Iran.
J. Faiz, H. Lesani, and M. T. Nabavi-Razavi are with the Department of Elec-
trical and Computer Engineering, Faculty of Engineering, University of Tehran,
Tehran, Iran (e-mail: jfaiz@ut.ac.ir).
Digital Object Identifier 10.1109/TMAG.2004.826611
power stations, oil refineries and petrochemical industries is
thus required for preventive maintenance of the machines.
Various fault diagnosis techniques have been proposed in-
cluding expert system [2], neural networks (NN) [3], fuzzy logic
approaches [2] and fuzzy neural networks [4]. The expert sys-
tems and fuzzy logic approaches have some intrinsic shortcom-
ings, such as the difficulty of acquiring knowledge and main-
taining fault databases. In NN approaches, the training data must
be sufficient and compatible to ensure proper training.
A synchronous generator fault not only damages the machine
itself but may also cause an interruption in power and hence
loss of revenue. It is therefore of great importance to recog-
nize imminent failures in the machines as early as possible, so
that they can be corrected, thereby improving the reliability of
power system. In various diagnostic techniques, monitoring and
measuring electrical, magnetic, chemical, acoustic and thermo-
dynamic quantities as well as measuring partial discharge are
required.
About 60% of faults in electrical machines are caused by me-
chanical parts such as bearings, shaft and coupling. Nearly 80%
of these faults result in the displacement of the axis of symmetry
or the rotating axis of the rotor. Therefore, existing asymmetry
between rotor and stator cause 50% of faults in motors [5]. Fur-
thermore, if these faults have not been diagnosed and prevented,
the rotor may touch the stator and result in irreparable damage
of machines.
In the case of static eccentricity, the rotating axis of the rotor
coincides with its axis of symmetry, but these are displaced with
respect to stator axis of symmetry. In the dynamic eccentricity
condition, the stator axis of symmetry coincides with the ro-
tating axis of the rotor, but the rotor axis symmetry is displaced
with respect to the two former axes. Finally, in the mixed ec-
centricity condition, all three axes are displaced with respect to
each other.
This paper investigates the dynamic eccentricity condition
in a salient-pole synchronous machine caused by geometrical
asymmetry between its rotor and stator. The contributions of the
present paper includes: 1). Accounting for linear rise of mmf
across the slots, 2). Approximation of air-gap with three terms
instead of two terms as reported in [11].
II. A M
ODIFIED WINDING FUNCTION THEORY
Modeling faulty electrical machines requires long computa-
tion using techniques that are normally more complicated than
0018-9464/04$20.00 © 2004 IEEE
TABATABAEI et al.: MODELING AND SIMULATION OF A SALIENT-POLE SYNCHRONOUS GENERATOR 1551
that of healthy machines. To date, finite element (FE), equiva-
lent magnetic circuit [7] and winding function [6] methods have
been used for modeling, simulation and analysis of electrical
machines with different types of fault. Other methods using
some simplifying assumptions have been proposed to solve this
problem for particular cases [8]. In this paper, a machine is mod-
eled by using a modified winding function theory for computing
magnetic inductances and electromagnetic coupling equations
between the stator phase windings and the excitation winding.
One of the advantages of this method is that it is possible to
predict transient and steady-state performance of any machine
with any type of winding distribution and air-gap length, while
taking into account the effect of all spatial and time harmonics.
This means that all faults occurring in the stator windings, rotor
turns and air-gap eccentricity can be included in the model ob-
tained using this theory.
This theory was initially presented for the computation
of single-phase induction motors [7]. The theory was then
extended for the steady-state equivalent circuits of two-phase
asymmetrical induction motors, including odd- and even-har-
monics of the magnetic field. This method was also presented to
analyze linear induction motors [8]. By extension of the theory
and combining it with concepts of coupled circuits, equivalent
d-q-o models for synchronous and induction machines were
proposed and used in the analysis of induction machines with
concentrated windings [10]. In recent years, this theory has
been widely used to study the transient behavior of induction
motor under inner faults such as rotor broken bars, stator turns
short-circuit and different types of the air-gap eccentricity.
Although this theory has been considerably used in the anal-
ysis of induction motors [10], it is considered to be less attrac-
tive for synchronous machines. The most important work on this
machine is modeling a salient-pole synchronous machine under
dynamic eccentricity [9].
The principal equation of the theory which presents the mu-
tual inductance of two arbitrary windings x and y in respect to
the winding distribution
and is as follows [12]:
(1)
where operator
is defined as the mean of function f over [0,
] and P is the permeance distribution of the air-gap. If is an
arbitrary angle in the stator reference frame, it follows that:
(2)
The derivation of the equations (1) and (2) is discussed in [12].
These equations have been developed by taking into account a
more precise distribution of stator phases and rotor excitation
windings and also a more precise computation of the air-gap
permeance. Its application results are compared with that ob-
tained by the winding function and FE method [9]. This com-
parison shows that the result obtained is closer to that of the FE
computation than that of the normal winding function.
Fig. 1. Positions of rotor and stator air-gap dynamic eccentricity.
III. AIR-GAP PERMEANCE
COMPUTATION
Air-gap permeance is proportional to the inverse of the
air-gap length. This quantity is thus neglected for the points far
from the poles shoes air-gap, and taken into account only for
the air-gap between the salient-poles and the stator. Therefore,
the air-gap permeance distribution, between the salient pole of
the rotor and stator, is as follows:
(3)
where and are the mean radius of air-gap and air-gap dis-
tribution, respectively. These two quantities are constant for the
all points between the salient-pole and stator, in the symmetrical
case. These two geometrical quantities of the machine are cal-
culated in the dynamic air-gap eccentricity presented in Fig. 1.
It is noted that
and are the centerline of rotor and stator
respectively.
Vector
is called the dynamic eccentricity vector. The
eccentricity factor is the ratio of the length of this vector, to the
symmetrical air-gap length over the pole shoes
. This vector
rotates around stator symmetrical axes with angular speed equal
to the mechanical speed of the rotor. Therefore, once the motor
starts up, it is assumed that this vector coincides with the ref-
erence axes of the mechanical angle, and moreover, is always
equal to the mechanical angle of the rotor. Fig. 2 shows the po-
sition of an arbitrary point M on the rotor pole shoes in the dy-
namic eccentricity condition. The distance of this point from the
rotor center is equal to
(rotor radius), then
(4)
For the inner radius of stator,
, the mean length and radius of
the air-gap above the poles shoes are as follows:
(5)
(6)
Since the air-gap length above the poles shoes is much smaller
than the rotor radius, the second term in (4) may be approxi-
mated with
. Equations (5)–(6) can therefore be rewritten as
(7)
(8)
1552 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 3, MAY 2004
Fig. 2. Position of a point on the rotor pole shoes during air-gap dynamic
eccentricity.
Fig. 3. Polar distribution of air-gap magnetic permeance of a healthy
synchronous machine.
where the term has been ignored due to
the small air-gap length compared to the rotor radius. The mean
radius of the air-gap poles is therefore almost equal to this ra-
dius in the healthy machine. Moreover, the magnetic permeance
distribution of the air-gap above the pole shoes is obtained by
substituting (7), (8) into (3) as follows:
(9)
Figs. 3 and 4 present the polar distribution of air-gap magnetic
permeance of a synchronous machine in the healthy condition
and 25% dynamic eccentricity. In such eccentricity, these distri-
butions rotate with mechanical speed of the rotor. In [6], this
distribution has been approximated by the first ten terms of
its Fourier series, leading to a reduction in the accuracy of the
computations.
IV. I
NDUCTANCE COMPUTATIONS
All synchronous machine inductances can be computed by
substituting the magnetic permeance distribution of the air-gap
and the excitation of the rotor into (1). In [11], a linear rise of
mmf of the air-gap across the stator slots has been considered
and the first three terms of its Fourier series has been used. In
this paper, the mmf rise is taken as shown in Fig. 5. The stator
winding of a typical synchronous machine is shown in Fig. 6,
with its specifications given in Table I [9].
Fig. 7 shows the turn functions of stator phases winding.
Fig. 8 presents a similar function for the excitation winding.
Fig. 4. Polar distribution of air-gap magnetic permeance of a synchronous
machine with 25% dynamic eccentricity.
Fig. 5. Turn function of a winding with mmf rise across the slots.
Fig. 9 shows the self-inductance of the stator phase winding with
25% dynamic eccentricity.
V. C
OMPARISON WITH
FINITE ELEMENTS
COMPUTATION RESULTS
The FE method was used to determine the stator self-induc-
tances of a faulty synchronous machine with 25% dynamic ec-
centricity The results from the FE method, shown in Fig. 10,
can be compared with those in Fig. 9 obtained by the modi-
fied winding function method. This comparison indicates a good
agreement between these two results. A comparison of self-in-
ductance of phase A of the stator winding of a machine with
25% dynamic eccentricity is shown in Fig. 11. The self-induc-
tances were obtained using the FE method and the unmodi-
fied winding function theory. This comparison indicates that the
peak inductance values from the FE method are slightly dis-
torted, while the unmodified winding function method results
are relatively smooth. Fig. 11 depicts that the proposed modifi-
cation on the winding function theory results in slightly distor-
tion in the peak inductance values. The reasons for good agree-
ment between the proposed method results and the FE results
are as follows.
1) A precise distribution of air-gap magnetic permeance was
included for the dynamic eccentricity condition.
2) A more precise distribution of the turn function of wind-
ings and linear mmf rise across the slots was considered.
Although it has not been proved here, the main reason for the
difference between the results from the proposed method and FE
method is probably due to neglecting saturation in the winding
function. This claim needs more detailed study.
TABATABA EI et al.: MODELING AND SIMULATION OF A SALIENT-POLE SYNCHRONOUS GENERATOR 1553
Fig. 6. Distribution of stator phases windings of a synchronous machine.
TABLE I
S
PECIFICATIONS OF A 480 V, 60 Hz, 475 kW SYNCHRONOUS GENERATOR
MACHINE [9]
Fig. 7. Turn functions of stator phase windings taking into account the mmf
rise across the slots.
VI. SIMULATION
RESULTS
In order to simulate performance of a synchronous machine
with a geometrical asymmetry condition, the electromagnetic
coupling model of the machine circuits is solved using a 4th
and 5th order Runge-Kutta method. One of the important and
basic stages of this modeling technique is the calculation of in-
ductances of the machine. All the inductances are computed at
several rotor angular positions and stored within a computer file.
The matrix form of the fundamental equations of this model is
as follows:
(10)
(11)
(12)
(13)
Fig. 8. Turn functions of excitation winding with 25% dynamic eccentricity.
Fig. 9. Calculated self-inductance of stator phases winding with 25% dynamic
eccentricity using modified winding function.
Fig. 10. Self-inductance of stator phases winding with 25% dynamic
eccentricity using FE method [9].
By solving the above equations, the frequency spectrum of the
stator phase current is estimated. Fig. 12 shows the frequency
spectrum of the stator phase current for a healthy machine, as
well as, 15% and 25% dynamic eccentricity conditions. Fig. 13
presents the variation of the amplitude versus eccentricity level.
1554 IEEE TRANSACTIONS ON MAGNETICS, VOL. 40, NO. 3, MAY 2004
Fig. 11. Calculated self-inductance of phase a of stator with 25% dynamic
eccentricity: (a) Unmodified and (b) modified winding function.
Fig. 12. Frequency spectrum of stator phase current (17th and 19th
harmonics) for: (a) healthy machine, (b) 15% dynamic eccentricity, (c) 25%
dynamic eccentricity.
VII. CONCLUSION
Modeling and simulation of a salient-pole synchronous gen-
erator has been carried out using a precise distribution of air-gap
magnetic permeance present in the dynamic eccentricity condi-
tion. A more precise winding function and linear rise of mmf
across the slots has also been taken into account. Probably the
discrepancy between the winding function and FE method re-
sults is due to neglecting the saturation effect. However, the
Fig. 13. Amplitude versus eccentricity level for the 17th and 19th harmonics.
results presented in this paper are closer to the FE method re-
sults than that of other available methods. The simulation results
show that the 17th and 19th harmonics can be employed to di-
agnose the dynamic eccentricity of the machine.
R
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Iman Tabatabaei received the B.Sc. and M.Sc. degrees from the Department
of Electrical and Computer Engineering, University of Tehran, Tehran, Iran,
in 2000 and 2003, respectively. He is working toward the Ph.D. degree at the
University of Ryukyus, Japan.
His research interests include electrical machine modeling, analysis under
fault conditions, and control.
TABATABA EI et al.: MODELING AND SIMULATION OF A SALIENT-POLE SYNCHRONOUS GENERATOR 1555
Jawad Faiz (M’90–SM’93) received the B.S. and M.S. degrees in electrical
engineering from the University of Tabriz, Iran, in 1974 and 1975, respectively,
graduating with First Class Honors. He received the Ph.D. degree in electrical
engineering from the University of Newcastle upon Tyne, U.K., in 1988.
Early in his career, he served as a Faculty Member at the University of Tabriz
for ten years. After obtaining the Ph.D. degree, he rejoined the University of
Tabriz where he held the position of Assistant Professor from 1988 to 1992, As-
sociate Professor from 1992 to 1997, and has been a Professor since 1998. Since
February 1999, he has been a Professor with the Department of Electrical and
Computer Engineering, Faculty of Engineering, University of Tehran, Tehran,
Iran. He is the author of 145 publications in international journals and confer-
ence proceedings. His teaching and research interests are switched reluctance
and VR motors design, design and modeling of electrical machines and drives.
Dr. Faiz is a Senior Member of Power Engineering, Industry Applications,
Power Electronics, Industrial Electronics, Education, and Magnetics Societies
of the IEEE. He is also a member of Iran Academy of Science.
H. Lesani received the M.S. degree in electrical power engineering from the
University of Tehran, Tehran, Iran, in 1975, and the Ph.D. degree in electrical
engineering from the University of Dundee, U.K., in 1987.
Early in his career, he served as a Faculty Member with Mazandaran Univer-
sity. After obtaining the Ph.D. degree, he joined the Department of Electrical
and Computer Engineering, Faulty of Engineering, University of Tehran, where
he is an Associate Professor. His teaching and research interests are design and
modeling of electrical machines and power systems.
M. T. Nabavi-Razavi received the M.Sc. degree in electrical engineering from
Sharif University of Technology.
He is currently a Lecturer with the Department of Electrical and Computer
Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran. His
interests include electrical machines modeling and analysis.
