Experimental Determination of Parameters of Synchronous Generator
Experimental Determination of Parameters of Synchronous Generator
Experimental Determination of Parameters of Synchronous Generator
DAW GHANIM
EXPERIMENTAL DETERMINATION OF EQUIVALENT CIRCUIT
GENERATOR
by
o DawGhanim
A thesis
MASTER OF ENGINEERING
September 2012
Nearly all of the electric energy used for numerous applications in today' s world,
utmost necessity to pay serious attention to those electric machines, their pcrfonnance
characteristics, various operation conditions and their design parameters. One of the most
reactance lies in the significant role it plays in the quality of the produced electric power
and thus the stability of the voltage obtained across the terminals of a synchronous
values based on the mode of the operation or the loading conditions. The primary purpose
of this thesis is to determine the direct-axis synchronous reactance values as well as the
identification of stator leakage reactance of a real synchronous generator from the so-
called Potier triangle method. A set of standard tests based on the IEEE standard 115 is
set forth and conducted on the machine under test which is a laboratory synchronous
experimental test data and results are provided and analyzed with full details, and then the
Rahman for his patience, kindness and invaluable guidance and encouragement. I am also
truly grateful to the School of Graduate Studies and Faculty of Engineering and Applied
Science at Memorial University of Newfound land for the considerable effort they made to
facilitate my study at MUN. AI this point, special thanks go to Ms. Moya Crocker and
Ms. Colleen Mahoney at the Office of the Associate Dean at the Faculty of Engineering
and Applied Science. 1 would like to acknowledge my fellow graduate students for their
Scientific Research of Libya for financially supporting my study in Canada. Also, the
considerable effort made by the Canadian Bureau for International Education (CBIE),
FinaJIy, my great thanks go to my parents, brother and sisters for being sustainable
iii
Table of Contents
Abstract ........................................................................................................................... ii
3.6.1 Flux and MMF Wavefonns in a Salient-Pole Synchronous Generator ....... .43
X;;).. ........................................................................................... 55
3.7.3 Quadrature-Axis Transient and Subtransient Synchronous Reactances
Fig. 2.1. (a) Typical magnetic circuit elements, (b) Equivalent circuit diagram .............. 17
Fig. 2.2. A conductor moving with an angle (0") through a magnetic field ..................... 20
Fig. 2.4. Balanced 3-phase current applied to the stator of ac machine .. ................. .. ... 25
Fig. 2.5. Production of the armature (stator) mmf wave in ac rotating machines; (a) Cross-
sectional view of2-pole ac machine, (field windings are not shown), (b)The resultant
stator mmfwave with its components . . ...................................... 26
Fig. 3.1. Basic Stator Scheme for a 2-pole 3-phase synchronous generator ..................... 32
Fig. 3.2. Elementary rotor structure of2-pole alternator: (a) cylindrical rotor, (b) salient-
Fig. 3.3. Y-connected 3-phase stator distributed windings of a synchronous machine .... 35
Fig. 3.4. Single-phase equivalent circuit of a cylindrical rotor synchronous generator: (a)
including armature reaction effect, (b) comprising the synchronous reactance .............. 36
Fig. 3.5. Phasor diagram of a cylindrical rotor synchronous generator: (a) lagging power
factor load, (b) unity power factor load, (c) leading power factor load ............................ 40
Fig. 3.6. mmfs distribution in a salient-pole synchronous generator: (a) physical view
(only phase-a is shown on the stator), (b) space-fundamental mmf waves with the
Fig. 3.7. d-q axis steady-state phasor diagram of a salient pole synchronous generator
vii
Fig. 3.8. Steady-state power-angle characteristic of a salient-pole synchronous generator
Fig. 4.2. Open- and Short-Circuit Characteristics for the laboratory synchronous generator
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~M
Fig. 4.3. Typical oscillogram from the slip test.. .... .................... 68
Fig. 4.4. Adjustable 3-phase reactor used as a load in the zero power-factor test ............ 71
Fig. 4.5. Potier triangle method for the laboratory salient-pole synchronous generator .. 72
Fig. 4.6. Test set up for the sudden three-phase short-circuit test .................................. 75
Fig. 4.8. Currents oscillogram from sudden three-phase short-circuit test ....................... 80
Fig. 4.9. Phase-b short-circuit stator current alone with the de field current .................... 80
Fig. 4.10. Polynomial curve fitting ofphase-b short-circuit current envelope ...... .... ... .... 82
Fig. B.I. Typical oscillograms of sudden three-phase short-circuit at 0.6 pu rated voltage
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~n
viii
List of Tables
Table 4.1 Test data from open- and short-circuit tests ... . .................... 64
Table 4.2 Experimental test result from the zero power-factor test ....... .... ........ .. ..... ... .... 72
Table 4.3 Measured constants for the laboratory salient-pole synchronous generator ..... 83
Xc Characteristic reactance
Magnetic flux
Kd Distribution factor
Kp Pitch factor
Xp Potier reactance
Xs Synchronous reactance
Ls Synchronous inductance
Zs Synchronous impedance
Km Machine constant
Power angle
Zb Base impedance
xii
List of Abbreviations
AC Alternating Current
DC Direct Current
MVA Megavolt-Ampere
BP Blondel-Park
AT Ampere-Turns
Wb Weber
VR Voltage Regulation
PF Power Factor
xiii
OCC Open-Ci rcuit Characteristic
xiv
Chapter 1
Introduction
In recent decades, synchronous generators, also called alternators, have become mOTC and
more a sustainable and principal source for the three-phase electric power. Synchronous
generators of large power capacity (50MVA and above) normally have a high efficiency
that can be greater than 98% [1]. The term synchronous stems from the fact that these
machines, under steady-state operation, arc precisely operated at a constant speed referred
to as synchronous speed. They can mainly be classified into two different types; high-
speed generators with a round or non-salient pole rotor structure and low-speed
generators having a salient-pole rotor structure and relatively large nwnber of poles. The
More details about the two generator types will be provided later in chapter 3.
For secure, rel iable power delivery, a synchronous generator, whether it is operated alone
to supply independent loads or interconnected with many others in large power grids,
must be operated within certain operational limits (stability limits) that maintain its
synchronism. Such operation modes are referred to as the steady-state operation of the
condition can be made in a straightforward fash ion. However, this is not always a trivial
task since abnonnal conditions do take place frequently in typical power systems for a
synchronous machines during the abnonnal or transient phenomena for better analysis
saliency, the air-gap inductances and thereby reactances will be time-varying with respect
to the rotor angular position. This fact makes the analysis of such a generator a step more
problematic than that of the cylindrical rotor synchronous generator where a uniform air-
gap does exist. The synchronous reactance of a synchronous generator is defined as the
algebraic sum of the leakage reactance of the stator winding and the magnetizing
reactance due to the air-gap flux [2]. For a cylindrical rotor synchronous generator,
synchronous reactance can be regarded as having one value at any instant of time. In
contrast, the synchronous reactance can be resolved into two dilTerent components based
on the two-axis theory proposed by Park [3] in the case of the salient-pole synchronous
Synchronous reactance is one of the most significant parameters that affects and
conditions. Its value is a function of the armature inductances and the angular frequency
reactance is supposed to have a small value to achieve high output power capability and
thus high steady-state performance. However, as it plays a major role in eliminating the
large currents during transient conditions, a reduction of synchronous reactance should be
limited to the machine's ability to overcome the transient state [4]. Knowing the cutting-
edge methods to determine the various values of this reactance is therefore a very
The focus of attention of this thesis is to find the direct-axis synchronous reactance values
well as the so-called Potier triangle method, which has been found to be a valuable means
to compute the stator leakage reactance and armature reaction voltage drop of
synchronous generators.
The following is a recent review of the previously done work about the studied topic.
Much of the provided literature fOnTIS the major contributions of the IEEE and AlEE
scholars and their associates to the art of synchronous machines analysis and design.
model that satisfactorily simulates their electric and mechanical behaviour during
different operations and can be used with sufficient accuracy in various aspects. In
general, most of the differential equations describing the electric and dynamic behaviour
of a synchronous machine are highly nonlinear because of the existence of the time-
varying inductances due to the rotor position. This fact often leads to more complexity in
the analysis of those machines. A change of machine variables (flux linkages, currents
and voltages) has been found to be an effective way to reduce the complexity in the
In 1913, Andrew Blondel [6] in France suggested a new technique to facilitate the
theory ever since. The two~rcaction theory [6J states that: "When an alternator supplies a
current dephased by an angle tjI with respect to the internal induced E.MF, the armature
reaction may be considered as the resultant of a direct reaction produced by the reactive
current / sintjl and a transverse reaction due to the active current / COStjl". In addition, it
states that: "The two reactions (direct and transverse) and the stray flux rake place in
Ihree difJerent magnetic paths: only the direcl reaclion acts in the main circuit of the field
magnets, while the transverse reaction and Ihe stray fields act. in general, upon circuits
oflow magnelic density" , To sum up, the principle behind Blondel's finding is to resolve
machine into two different space components aligned with two orthogonal fictitious axes
on the rotor [7]. Those axes are referred to as the direct-axis or d-axis, the magnetic axis
of field winding, and the quadrature-axis or q-axis found in the interpolar space between
Blondel's theory opened the door to several new research areas, especially in
presented in the USA by the AlEE members Doherty and Nickle [7] who tried to modify
the two-reaction theory and include some new considerations that Blondel did not account
for in his fundamental proposition, such as the effect of mmfharmonics in both direct and
quadrature axes.
Taking advantage of the work done by Blondel, Doherty and Nickle, R.H. Park [31 came
up with a new means to analyze synchronous machines in the late 1920's. A traditional
variables). However, Park made an attempt to simplify the analysis by proposing a certain
mathematical transformation that tends to refer three-phase stator variables into new
variables in a frame of reference fixed on a salient-pole rotor whose imaginary axes are
the well-known direct- and quadrature- and zero-axis (dqO coordinate system).
regarded as a means of reducing three ac stator quantities into two dc quantities that are
called dq-axis quantities, and the zero-axis quantities do disappear in such modes of
In 1933 [10], Park once again extended his previous contribution to the two-reaction
theory to include the effect of the synchronizing and damping torque of a synchronous
machine during the transients. Park's two-axis theory was further improved by Crary,
Lewis and Zhang Liu and his associates [I 1,12, 13). In reference [II], Crary extended
Park's previous work to include the effect of armature capacitance in the analysis of a
salient-pole synchronous machine. He applied his new set of equations to solve problems
involving the performance of a synchronous machine with capacitance circuits such as the
problem of self-excitation of a synchronous machine connected to a capacitance load or
to a system through series capacitance. In reference [12], Lewis presented a new concept
seeking to sweep away some further confusion that had been brought to attention due to
Park ' s basic equations, such as having nonreciprocal mutual inductances in the derivation
of an equivalent circuit and the use of per-unit system of notation. Basically, his work
was devoted to the modification of the dq-axis equivalent circuit and corresponding
equations previously obtained by R.H. Park and the others to be as straightforward and
Zhang Liu and his associates [13] proposed another extension of the Blondel-Park (BP)
transformation that was referred to in their work as the EBP transformation. The new
transformation is trying to not only consider the electric subsystem of the ac electric
machines as Park's transformation does, but also to include the mechanical dynamics or
alternative to the two-reaction theory in the electric machinery analysis. His analysis of
synchronous machines included the hunting conditions besides the routine steady-state
and transient analysis. He presented new components, called the forward and backward
components, which are correlated to the familiar direct and quadrature components of the
two-reaction theory. He showed how the new approach of analysis can form a basis of
Emmanuel Del for a permanent magnet synchronous machine [151. In general , Park's dq-
model and its numerous extensions have gained ground as a key tool in several computer
representations and simulation studies on a wide variety of ac machinery ever since [8, 9.
16-25].
The study of the behaviour of synchronous machines during steady-state and transient
importance as the capability chart. power transferred limitations and the short-circuit
During the past several decades, many laboratory test procedures for practical
IEEE standard \\5-\995 [27) and lEe standard 34-4 [28]. The standstill frequency
response test (SSFR) [29-32] and the load rejection test [33, 34] have gained prominence
due to their practically harmless effect on the tested machine and accuracy of the
extracted test measurements. Standstill measurements have an advantage over the other
testing methods, in that they can provide a significant amount of information about the
tested machine. These are applicable to the d-axis and q-axis models and can provide a
complete picture of both while most standard tests do not do so, especially with respect to
tested machine terminal at various levels of frequency, typically in a range from 0.001 Hz
to 200 Hz. Thereafter, the machine parameters both in direct-axis and quadrature-axis can
be evaluated by using the frequency response test data [36J. The load rejection test
requires applying load rejection at two special operating points, in which the current
components appear in the axis of interest only. At this point, for determination of direct-
reactive power load to ensure that the fluxes and thus stator current components exist only
rejection is needed to obtain a proper loading condition in such a way that only the q-axis
stator self-i nductive reactance of synchronous machines and showed that the initial value
of the short-circuit current of a synchronous machine is not only defined by stator self-
induction, as is frequently assumed, but rather by a combination of both stator and field
self-inductive reactances.
In [38], V. KarapetofT outlined the fact that the assumption of a constant armature leakage
reactance in synchronous machines analysis may lead to wide deviation between the
computed and design data. Instead, he showed that the cyclic variation of the leakage
reactance should be taken into account for higher accuracy levels in the analysis.
sequence of stator current that can either be positive, negative or zero phase-sequence
yields sustained, transient and sub-transient reactance. (3)- In the position of the rotor that
gives rise to direct-axis and quadrature-axis reactance. Stator reactance due to the
armature reaction and the stator leakage reactance are combined in one quantity that is
The three-phase short-circuit test has been recognized as a vital test procedure to study
values for specified machine parameters, thus reactances and time constants, of
synchronous generators. Briefly, the sudden three-phase short-circuit test can be carried
generator when it is operated at rated speed and at various percentages of the rated
terminal voltage. The most usual practice has been to Perform the test at a no-load
of the instantaneous stator phase currents following the short circuit can then be recorded
and analyzed by an appropriate method of analysis to extract the relevant subtransient and
load cases, for both a cylindrical rotor and salient-pole synchronous machine, by R. E.
Doherty and C. A. Nickle in [41]. The effect of stator resistance on short-circuit currents
is ill ustrated by two cases; resistance is at first negligible and then it is considered. Also,
the effect of the load's nature on the short-circuit current is absorbed .The investigations
showed that short circuits under load may generate less current that those applied at no-
load. It has also been shown that the second hannonic components in the short-circuit
current depend on the difference between the sub-transient direct-axis and quadrature-axis
obtain the most important machine constants, the machine reactances, resistance and time
constants, are illustrated and tabulated test results are given. Tests included the three-
reactance using the so-called Potier me/hod. They showed how Potier reactance varies
with the increase in the fie ld current for a salient-pole machine and how it can practically
be a constant value for a synchronous machine with a cylindrical rotor. They mentioned
that the Potier reactance when measured at rated tenninal voltage could be much greater
than the actual annature leakage reactance for a synchronous machine. Alternatively, they
suggested that the Potier reactance when measured at a higher value of tenninal voltage
and thus a higher field current can result in a more accurate approx imation of the
10
annature leakage reactance than the classical measuring at nonnal voltage. However, thi s
approach has been found 10 be somewhat risky as the machine under test may not tolerate
the increased high values of field current at which the detennination of armature leakage
leakage reactance that can be applied without any risk to the tested machine, unlike the
preceding approach in [43]. In the proposed method, the curve of the terminal
parameters with a series of important papers [45-48). He obviously focused his attention
Canay showed how one would make an accurate estimation of the synchronous machine
iterations that can be a source of wide discrepancies in the results of the analysis.
It has been realized that the conventional equivalent circuit diagram found in the earlier
theory of the synchronous machine does accurately represent the stator circuit only.
However, Canay revealed that transients for rotor quantities (i.e. field current, field
voltage, etc.) can also be specified if a suitable equivalent circuit diagram that represents
both the stator and rotor circuits is used. For this purpose, he employed a new reactance,
11
machine. The characteristic reactance Xc is independent of rotor reference quantities and
can be found directly from the measurement of the alternating current transfer between
stator and field windings. This can easily be done by measuring field current variations
than the stator leakage reactance whereas it is much lower than the stator leakage
generalized Parks' dq-axis model which has only two rotor circuits (one field winding
and one damper winding), Canay's model can be valid for higher order machine models
with more than two rotor circuits in both direct-axis and quadrature-axis [49J.
that have been set forth in the traditional analysis of synchronous machines and which
analysis with two windings (field and damper) in the direct-axis circuit. Then, they
examined their model with two standard tests; the sudden three-phase short circuit and
stator decrement test with short-circuited field windings. They demonstrated that it is
possible to detennine the model parameters in terms of a set of measurable time constants
regardless of whether equal or unequal direct-axis mutual reactance between the stator,
overview of the most used testing methods for detennination of synchronous machine
parameters that are defined in IEEE standards liS and liS-A, besides some other
11
traditional testing methods. Thereafter, the paper uses the reviewed methods to propose
analogous techniques for estimating machine parameters for linear synchronous machines
(LSM).
given in reference [49J. The paper used a d-axis model that contains more than one
damper winding that is based on the direct-axis equivalent circuit of Canay's generalized
detennined from two-port infonnation. Also, shown in the paper is the noteworthy
observation that only limited d-axis parameters can be found in a unique way for a
synchronous machine model with more than one damper winding in the rotor circuit.
In [52], the authors proposed a new estimation procedure for detennination of direct-axis
parameters of a synchronous machine. In this method, all the parameters derivation was
done using the Matlab derivation program. For more accurate results, a gradient-based
optimisation algorithm was also employed to adjust the machine parameters so that a
close agreement between the simulated tenninal voltage response and the measured
parameters was also presented by a group of IEEE authors [36]. They suggested a simple
and accurate approach to estimate the direct-axis and quadrature-axis stator inductances
of a synchronous machine by applying a hysteresis based current control of the direct and
13
In recent times, the finite element method (FEM) is found to be a valuable means for
estimating most synchronous machine parameters due to its advantages over the classical
testing methods in its abi lity to consider more important facto rs and avoid many
hypotheses by the analysis of the electromagnctic field in the machine's iron core [53·56].
The major contribution of this research work is to determine the steady-state, transient,
synchronous generator. Moreover, the Potier triangle method is developed and presented
for a laboratory salient-pole synchronous generator in this thesis. This method makes it
possible to compute the Potier reactance, the drop in the terminal voltage due to annature
reaction and the required excitation corresponding to the rated voltage for any specified
load current of a synchronous generator. For those objectives, certain standard tests
corresponding to the IEEE standard 115-1995 [27] are carried out on the studied
short-circuit test.
This thesis is broken down into five chapters that can be stated as follows:
Chapter 1 provides a general introduction and a recent review of the previous literature
14
Chapler 3 gives a full explanation about the theory of synchronous generators within the
Chapler 4 provides the experimental test results that were obtained in the laboratory by
Chapler 5 is the conclusion and recommendations of the work that may be done in the
15
Chapter 2
The rotating ae electric machines can mainly be divided into two types: synchronous
machines and induction machines. These machines are widely used in the industry for
both generator and motor applications. As its name implies, the rotating machinery
principle is based on the production of the rotating magnetic field inside the machine. The
knowledge of magnetic circuits. The principle of such circuits is in fact similar to that of
electric circuits. An electric circuit provides a path for the electric current to flow through.
Simi larly, a magnetic ci rcuit yields a closed path to flow for the magnetic fl ux traveling
through it. The electric current flowing in an electric circuit is generated by the voltage
source or the electromotive force (em!) of that circuit. By analogy, the magnetomotive
force (mmf) is responsible for establishing the magnetic flux in a magnetic circuit. For
proceeding with this discussion, let us refer to the magnetic circuit in Fig.2.1.
16
Iron Core
--rp---il>--~
I
N I
I
L.:_--<---- _ _ J
(a) (b)
Fig.2.t (a) Typical magnetic circuit clcments, (b) Equivalent circuit diagram
Fig.2.l shows a simple magnetic circuit with its equivalent circuit. It can be noted that
this is basically identical to the transformer configuration except for the absence of the
secondary winding. As a current (I) flows through a coil of (N) turns around the iron core,
a significant amount of the net flux established by this current will be confined by the iron
core due to the very high permeability of the iron. There is still a small portion of the flux
that spreads out in the air around the iron core. This flux is commonly known as the
leakage flux. The leakage flux might have a noticeable impact on the performance of such
electromagnetic devices.
The magnetic flux density is defined as the ratio of the total flux (ftI) to the area (A) of the
The unit (Wb/m2) is now denoted by the IEEE standard symbol (T) or Tesla.
to the product of the current in the electrical conductor times the number of its turns. Thus
17
mmf = :F =N x I Ampere-Turns (AT) (2.2)
In terms of the magnetic field intensity (H), the above equation can be rewritten as:
:F= Hx l (2.3)
It is evident from Fig.2.1 (b) that there is an analogy between electric and magnetic
circuits regardless of the difference in the nature of their parameters. The magnetomotive
force, mmf, can now be expressed in terms of the equivalent magnetic circuit parameters
as follows:
(2.4)
1? is called the reluctance of the magnetic circuit that corresponds to the resistance in an
J? = ~ Ampere-tumsperWeber(AT/Wb) (2.5)
where,
).I: The permeability of the magnetic material in Henrys per metre (HIm).
18
2.2 Induced Voltage Equation
the conductor will then experience a force (F) that tends to rotate it. The direction of the
rotation is given by Fleming's left-hand rule [57]. This is actually the basis of a molor
In contrast. the basic principle of an electric generator is based on the flux-cuffing action
of Faraday's law, which states that whenever a conductor cuts constant magnetic field
lines, a voltage will be induced across the conductor terminals. The voltage will also be
induced if the flux is time-varying while the conductor is held stationary. The polarity of
this voltage is determined by the right-hand rule of Fleming. The induced or generated
voltage is referred to as the counter emf for the mOlar case. The induced voltage is thus
defined as:
v: The velocity by which the conductor crosses the flux lines in (mls).
19
Generally, if the conductor moves at an angle (eO) with respect to the flux lines as seen in
Fig.2.2 A conductor moving with an angle (9°) through a magnetic field [57J
This is only the emf induced in a si ngle wire; let us now find the expression of the
equation governing the induced emf in a set of three-phase windings such as those
[n the three-phase rotating machines, three-phase windings are placed on the stator 120
electrical degrees apart in the space from each other in order to produce as sinusoidal as
possible induced voltage. Consequently, the induced voltages in the stator windings have
an equal magnitude and they are 120° out of phase from each other. Fig.2.3 shows a
cross-sectional view of a typical 2-pole 3-phase ac machine. The rotor poles are assumed
20
Field Winding
Axis
Induced Voltage
Phase a
'. Stator Winding Axis
Phase a
The nux density wave distributing in the air-gap of the machine is assumed sinusoidal in
The air-gap nux per pole can be obtained by the linear integral of the flux density over
(2.10)
where (I) is the axial length of the stator, (r) is the radius of the stator inner surface.
The above equation compules the per pole flux for a two-pole ac machine. More
generally, for a machine with p-number of poles, the flux equation can be rewritten as:
(2.11)
21
In accordance with Faraday's law, the induced emf in one phase of the stator windings
where A is the flux linkage for a stator phase in (Weber-turns) and is equal to
A = Ntpcoswt (2.13)
whcre ()) =27tf is the angular velocity of the rotor (rad/s), N: number of turns per stator
phase.
from which the maximum value of the induced voltage per phase is
(2.15)
In a practical ac machine, the armature windings on the stator are often distributed in slots
and made short-pitch to improve the sinusoidal shape of the induced voltage. Therefore,
the distribution and pitch factors denoted by Kd and Kp , respectively must be taken into
22
E,nu {per phase) = 4. 44 Nf rp KdKp = 4.44 Nfrp Kw (2.17)
Kwis known as the winding factor for the stator distributed windings which equals the
Eventually, the three-phase set of the induced voltage can be expressed as:
(2. 19)
flux-cutting action or the presence of the magnetic field in the machine air-gap. Similarly,
a balanced three-phase set of currents can produce a rotating magnetic field when flowing
through a set of three-phase windings. Hence, the revolving magnetic field is created by
the stator supply current in a motor action ac machine, whereas it is established by the
load current in a generator action ac machine. Knowing the nature of the wavefonn of this
phase windings, each 120 electrical degrees apart, are placed on the stator as shown in
Fig.2.5(a). These windings are commonly distributed in the stator slots. A balanced three-
phase current is then applied to the stator distributed windings so that a revolving
magnetic field is established in the air-gap. It is actually the sinusiodally distributed mmf
23
induced in the stator windings that is the source whereby the rotating magnetic field is
created. Its amplitude and direction depend on the instantaneous current in each phase of
the stator windings. The currents' reference directions are indicated by dots and crosses in
Fig.2.S(a). The balanced three-phase stator current, as shown in Fig.2.4, may be defined
as follows:
(2.22)
(2.23)
Each of these currents wil l produce a sinusoidally distributed mmf wave in the individual
phase winding through which it is flowing . Therefore, the resultant mmf distributing in
the air-gap of the machine is the sum of the three mmf waves from each phase winding
alone as seen in Fig.2.S(b). Since the applied three-phase currents are balanced and the
stator windings are shifted by 120° in the space from each other, the three components of
the resultant mmfwave also have a 120° displacement in time from each other.
Let us now investigate the amplitude of the travelling mmf wave in ac machines. From
Fig.2.4, at instant of time (rot= 0°), the phase a current is at its maximum value. As a
result, the mmf wave of phase a, represented by the vector (F.) in Fig.2.S, also has its
maximum value (Fma.~ - Nlm) and is aligned with the magnetic axis of phase a. At the same
24
moment, the corresponding mmf waves of the other two phases have equal magnitude
(2.24)
The resultant stator mmf (FT) can now be found by the vector sum of its three
25
Stator
<--+--4+-I- ~~ Axis of
phase a
(.)
(b)
Fig.2.S Production of the annaturc (stator) mmfwave in ac rotating machines; (a) Cross-sectional
view of2-pole ac machine, (field windings arc not shown) [58], (b)The resultant stator mmfwave
with its components [591
It is obvious from Eq. (2.26) that the net mmf wave is a sinusoidally distributed wave
moving in the positive direction along the magnetic axis of phase a with an amplitude
26
At a later time (wt= 90°), the instantaneous value of the phase currents are
i.=O (2.27)
Therefore,
F.= 0 (2.28)
Sim ilarly, the total mmfwave is the vector sum of the three components as in Fig. 2.6 (b).
FT=O+q.. Fma~ cos 30° +q.. Fma~ cos 30° =~ Fmox - ~N 1m (2.29)
Again, the resultant mmfwave has the same amplitude and sinusoidal wavefonn as those
~':1 fT
F.~ j..
~--------~
Positive sequence \ Fe
of phase c axis \
\"
Positive sequence
of phase b axis
(a) col= O°
Another example demonstrating this fact is the instant of time (cot= 120°) when phase b
current has its positive peak value as illustrated in Fig.2.4. The instantaneous value of the
phase currents and the associated mmf components are now as follows:
27
ia "' -~ Im . ,
F '" _ "max (2 .30)
ic "' -~ 1m ,
F, '" _ "max (2.32)
FT '" "~n cos 60° + Fmax + F~>x cos 60° = ~ Fmax '" ~ N 1m (2.33)
Thus, the resultant mmf wave is once again sinusoidally distributed havi ng the same
amplitude, but this time is centred on the magnetic axis of the phase b winding.
It can clearly be seen that as time passes, the resultant mmf wave distri buting in the air-
gap of an ac machine retains its sinusoidal fonn with a constant amplitude of (~Fm,..,),
where Fmax is the maximum value of the mmf wave produced by any individual phase
winding. It is also moving around the air-gap at a constant angular velocity Ws '" 2nf(rad
Is).
In general , for a p-pole machine, the mechanical speed of rotation for the travelling mmf
wave is:
28
where
In fact, in the synchronous machine theory, the speed at which the rotating magnetic field
rotates in the air-gap is precisely equal to the rotor speed and is referred to as Ihe
29
Chapter 3
Over the years, synchronous generators, also called alternators, have been used as a
perfonnance characteristics. For this object, this chapter is developed. The focus of
input power into electric output power and rotates at a constant speed referred to as the
synchronous !!'peed. The word 'synchronous' comes from the ract that in this kind or ac
machine, the rotor and its associated dc magnetic field must have precisely the same
speed as the rotating magnetic field produced by the stator currents. Thus, the rotor speed
is a runction or the number or poles orthe machine and the electric rrequency (Hz) or the
stator current. Synchronous generators operate a large number or various loads with wide
rating ranges. They have been extensively studied and examined in the literatures.
Synchronous generators, based on their speed of rotation, can be divided into two
categories: high-speed and low-speed generators. The high speed generators are often
30
driven by means of steam or gas turbines. They are knov.rn as turbo or turbine-generators.
The low speed generators, on the other hand, are driven by means of waterwheel
generators. The actual distinction between the two types of synchronous generators lies in
built of thin iron laminations of highly permeable steel core. The laminated core is to
reduce the magnetizing losses such as the eddy current and hysteresis losses. The stator
accommodates the three-phase armature windings which are the distributed stator
windings. The distributed windings are embedded in the slots inside the stator core 120
electrical dcgrees apart in the space to minimize the space harmonics in the resultant air-
gap flux waveform. They are also made short-pitch in order to produce a smooth
sinusoidal voltage waveform at the stator terminals. The reason behind having the
annature windings fixed on the stator is that they need to be well-insulated due \0 the high
voltagcs and transient currents they may experience during different operation modes. A
31
Stator core
The rotor of the synchronous generator hosts windings which carry the dc field current
and are connected to the excitation source of the generator via brushes and the slip rings
assembly. The synchronous generator has two different rotor configurations based on its
speed. Turbo- generators used for high speed operation have a round rotor structure.
synchronous generators. They have a uniform air-gap and normally have either two or
four field poles, depending on the required speed of operation. Hydro-generators used for
low speed operation have rotors with salient poles structure. These generators are known
gap is formed between the rotor and stator inner surface. The salient-pole rotor has a
amortisseur windings equipped to the rotor of this type in order 10 damp out the speed
oscillation in the rotor. Damper windings are copper or brass bars embedded in the salient
32
pole faces with both their ends shorted-circuited by means of shorting rings to fonn a
cage structure similar to that for a squirrel cage rotor of an induction machine. These
windings playa major role in retaining the generator synchronism during the dynamic
transient or hunting. The cross-sectional view of two rotor structures of the synchronous
~
N ~ Pole Face ~crDamperWinding
. . "..--..
,
oN 0
I 0, .
, • 0
Slot~ .
~FiCld
/"
. s / I S
Winding 0 0
(a) (b)
Fig.3.2 Elementary rotor structure of2-pole alternator: (a) cylindrical rotor, (b) salient-pole rotor
1601
The field winding on the rotor of a synchronous generator is supplied by dc field current
from an external dc source which is called the exciter for the generator. Based on the
exciter arrangement, use of sl ip rings mayor may not be ,needed for this purpose. The
field current will then produce a constant magnetic flux around the rotor in the air-gap.
The rotor is rotated by means of a prime mover at the synchronous speed (ns) of the
distributed stator (annature) windings in the stator due to the flux-cutting action or
33
will flow through the three-phase stator windings causing a synchronously revolving
magnetic field to be established in the air-gap. For maintaining the synchronism, the
rotating magnetic field must be locked in or synchronized with the mechanical constant
speed of the rotor magnetic field. In other words, the revolving magnetic field produced
by the stator rotates at the same synchronous speed as the rotor and is therefore stationary
with respect to it. The synchronous speed can be defined by the following relationship:
120r
n s= - p - ('Pm) (3.1)
or
4rrf
WS= p (cadis) (3.2)
where
As pointed out earlier, the stator of a synchronous generator has three-phase armature
windings distributed inside the slots of its core. These stator windings can be either Y- or
Ct.-connected. However, they are often connected in Y-form for convenience. Fig.3 .3
indicates the typical Y-connected stator windings with their parameters. The stator
windings have a resistance (Rs) and reactance due to the leakage flux linking the winding
34
called leakage reactance of the stator winding (XI =21[1 L), where L is the leakage
inductance of the armature winding in Henry. The armature resistance is very small
x,
R, v.
v,
The value of the per phase stator resistance (Rs) can easily be found by a simple dc
measunnent of the resistance between any two phases at a time. The line-to-line average
dc resistance (RIc) is obtained. For Y-connected stator windings, the per phase effective
(3.3)
where K is a factor which rises due to the sk in effect caused by the alternating (ac) stator
current and is approximated to be 1.6 for 60 Hz operation [59]. The thermal effect of the
35
3.4.1 Armature Reaction
The load current when flowing in the stator windings of a synchronous generator
produces its own magnetic field. This magnetic field will distort the main air-gap flux set
up by the rotor field winding. In other respects, the magnetic field due to the load current
the power factor of the load. The magnetizing effect of the annature or stator flux on the
air-gap fl ux is referred to as the armature reaction. The effect of the annature reaction in
reactance (Xar)'
For the sake of deriving the steady state machine equations for the synchronous generator,
~
jX' + jX, R' ~+
~ +
' ~'
(.) (b)
Fig.3.4 Single-phase equivalent circuit of a cylindrical rotor synchronous generalOr: (a)
including annature reaction efTect, (b) comprising the synchronous reactance
36
Note that this representation only includes the stator circuit. Since it carries the dc field
current, the rotor circuit may however be modeled simply by a single coil of (Nj) turns
Ear is caJ1ed thc air-gap vollage or the vollage behind Ihe armature leakage reactance. It
The reactance due to the armature reaction Xar and armature leakage reactance XI are
The synchronous reactance can also be expressed in terms of the synchronous inductance
Lsas:
(3.5)
The synchronous inductance (Ls) is defined as the effective inductance as viewed from
the armature windings under steady-state operation. Neglecting the effects of saliency, as
with the cylindrical rotor synchronous machine, the synchronous inductance and thereby
the reactance of the armature winding are independent of the rotor position (8 m ). Thus,
37
The synchronous impedance is defined as
(3.6)
Rs is often omilled for its relatively very small value for large synchronous machine
analysis.
The per-phase equivalent circuit comprising the synchronous reactance of the stator
windings in series with the stator resistance is shown in Fig.3.4 (b). Note that this model
For a salient-pole synchronous generator, the synchronous reactance has two different
Using the developed equivalent circuit, let us now find the expression of the air-gap
where the script (-) denotes the phasor form of the associated quantities.
The change in the terminal voltage magn itude from the no-load to full-load operation of a
value known as the vollage regula/ion, VR(%). The expression of voltage regulation is
given as follows:
38
VR(%) = (Vt(n!) - VtCf!)) x 100% (3.9)
Vt(fl)
where
Vt(nl) is the no·load terminal voltage (equals the induced voltage Eg)
The voltage regulation (VR%) may be a negative or positive value based on the nature of
the load current. For a synchronous machine operating as a generator, the percentage
voltage regulation is normally a large positive value for a lagging load, a small positive
value for a unity power factor load and a negative value for a leading power factor load.
induced voltage because there will be no voltage drop in the armature windings.
However, for a loaded generator, due to the armature reaction associated with the load
current, there will be either a reduction or increase in the terminal voltage. Fig.3.5 shows
the phasor diagram of a cylindrical rotor synchronous generator with various loads
39
~~
~ Is VI IsRs
Ib)
Ie)
Fig.3.5 Phasor diagram of a cylindrical rotor synchronous generator: (a) lagging power factor
load, (b) unity power factor load, (e) leading power factor load [61]
In the above phasor diagram, the angle G5°) between the induced voltage and terminal
voltage is referred to as the lorque or power angle. The angle (0°) between the load
current and the terminal voltage is the power factor (PF) angle of the load.
value, a course of action must be taken. Thus, the induced voltage must be changed in
correspondence with the varialions in the terminal voltage to overcome the armature
reaction voltage drop. Referring to Eq. (2.17) in chapter 2, the rms per-phase induced
where
40
Km is known as machine constant, ({J is the flux per stator pole in (Weber).
It is evident that the induced voltage is proportional to the resultant flux in the machine
and speed of operation. The induced voltage can thereby be changed either by controlling
the flux or the speed of rotation of the generator. The speed must be held constant for a
constant frequency operation. The induced voltage may then be changed by varying the
net flux in the air-gap. This can be made achievable by a proper adjustment to the dc field
current fed into the rotor winding. This approach to maintain the tenninal voltage at the
desired level is known as excitation control of the synchronous generator. This discussion
It can be seen from Fig.3.5(a) that for a lagging PF load (inductive load), the terminal
voltage will be reduced. Therefore, the induced voltage must be increased to keep the
rated terminal voltage. The reason for this reduction in the terminal voltage is that the
needed to overcome the air-gap flux demagnetization. Ir more loads are now added to the
generator at the same PF, then the armature voltage drop IsXs will be even greater,
causing a significant reduction in the terminal voltage. In this respect, the voltage
regulation has a large positive value for a synchronous generator with a lagging PF load.
41
3.5.2 Unity Power Factor Load
In the case of unity power factor loads (zero reactive power), Fig.3.S(b), there will be a
slight reduction in the terminal voltage due to the armature reaction effect of the unity PF
load current. Therefore, the induced voltage of the generator should also be increased to
retain the rated terminal voltage. In fact, the increase in the induced voltage needed here
is less than that for the lagging PF case. As a result, the voltage regulation under such an
From the phasor diagram in Fig.3.S(c), it can be observed that the terminal voltage can be
greater than the induced voltage of a synchronous generator operating with a leading PF
load (capacitive load). The increase in the terminal voltage can be justified by the fact that
the armature reaction due to the leading PF load current is said to be magnetizing or
strengthening the total air-gap flux. That is the reason that less amount of excitation
(under-excitation) is required to return the terminal voltage to the desired rated value.
Based on that, the voltage regulation has a negative value for a synchronous generator
It can be concluded that the growth of unity and lagging power factor loads at
the generator's terminal voltage within the desired limits. In contrast, the field excitation
may need to be reduced to keep the required terminal voltage for leading power factor
loads.
42
It should be emphasized that the opposite of the above discussion is true for a motor
action synchronous machine. Thus, for a synchronous motor, the armature reaction
increases the main air-gap flux for the lagging power factor stator current (under-
excitation mode) and decreases the flux for the leading power factor stator current (over-
So far, initial consideration has been given to the cylindrical rotor theory where all the
constant, regardless of the rotor position, due to the uniform air-gap geometry. In
addition, both the armature and main-field flux waves were to act on the same magnetic
circuit for the same reason. In a salient-pole generator, this, however, cannot be the case
because of the saliency effect of the rotor. Thus, the non-uniform air-gap of the salient-
pole generator will cause the air-gap inductances and reactances to be time-varying based
on the rotor angular position with rcgard to the armature rotating mmf wave.
Consequently, the armature reaction flux created in the air-gap by the stator current
cannot be the same at any instant of time. This fact makes the analysis of such salient-
pole electric machines even more complicated and needs careful consideration.
analyze the theory of salient-pole synchronous machines [6]. Its principle is to separate all
the machine variables (fluxes, currents and voltages) into two mutuaJly perpendicular
43
components acting along two different axes on a salient·pole rotor. The two relevant axes,
as shown in Fig.3.6, are commonly called the direct axis and quadrature axis, and are
denoted by d·axis and q-axis, respectively. The fonner (d·axis) is aligned with the axis of
symmetry ofa field pole while the other (q-axis) exists in the region midway between the
salient poles (inter·polar air-gap space). It is arbitrarily chosen, based on the IEEE
standard definition [60], that the q·axis leads the d·ax is by 90°. With this in mind, the
armature mmf (r.) due to the stator current can be resolved into two different
components as indicated in Fig.3.6(a); one acts along the d·axis (.:Fd) whereas the other
Since it coincides with the field pole axis and is thereby assumed to act over the same
magnetic circuit as the fie ld mmf, the (.:Fd) component of the armature mmf distributes
similarly to the main field mmf and may result in either a magnetizing or demagnetizing
effect. The (rq) component, as its magnetic effect is found in the space between the
salient poles, distributes quite differently from the field mmf and may therefore cause a
d-Axis
"
44
..
d-Axis
Stator Surface
~~--~~~~~~~
(b) I
I
Sllf1Jl
______~,-rr__~I~__~,O________~I_
rr ____~) wt
Rotor
Fig.3.6 mmfs distribution in a salient-pole synchronous generator: (a) physical view (only phase-
a is shown on the stator), (b) space-fundamental mmfwaves with the respective d-q axes [60]
The space-fundamental annature mmf and field mmf waves distributing in a salient-pole
generator are shown in FigJ.6(b) with both the stator and rotor structure rolled out for
clarity.
Accordingly, the annature reaction effect can be accounted for by two corresponding
magnetizing reactances; d-axis annature reaction reactance (Xad ) and q-axis annature
reaction reactance (X aq ). The leakage reactance of the armature winding (XI) is the same
for both the direct and quadrature axis (since it is independent of the rotor angle) and
45
therefore can be equally added to the armature reaction reactance to obtain the
Xq = Xaq + XI (3.12)
where
Because of the saliency of the rotor, the established air-gap fl ux due to a given annature
mmf (:F.) will be created more at some points than others. In other words, the annature
flux is greater in the short air-gap along the d-axis and smaller in the longer ai r-gap
between the salient poles along the q-axis, (qJd> qJq). In fact, this aspect takes place
because the air-gap magnetic reluctance of the flux path is smaller for the polar or d-axis
than that fo r the inter-polar or q-axis, (R.J < Rq). This also explains the reason why the
direct axis synchronous reactance (X d ) is larger than the quadrature axis synchronous
The steady-state phasor diagram for a salient-pole synchronous generator based on two-
axis theory is set forth and shown in Fig.3.7. The diagram considers the Jagging power
factor load case. It is of interest at this stage to derive the steady-state voltage equation
46
Fig.3.7 d-q axis steady-state phasor diagram of a salient pole synchronous generator with lagging
As seen in Fig.3.7, the stator current (I.) has two orthogonal components centred on the d-
fa = lssin (6 + 8) (3.13)
Iq = lscos (6 + 8) (3.14)
Thus, (3.15)
It must also be noticed that the q-axis component of the stator current fq is in phase with
the induced voltage vector (8g ) whereas the d-axis component fa is in time-quadrature
(90°) with it. In the case of no-load, the only generated voltage in the stator wind ing is
that induced by the field excitation (8g ) [62]. However, if a load is connected, the
armature reaction associated with d-axis and q-axis annature currents will cause two
47
relevant components of voltage to be induced in the stator windings. Each of these
voltages lags by 90° behind the stator current component producing it and can be given
(J.16)
(J.17)
As a result, the net generated voltage per one phase in the stator winding is the phasor
sum of the induced emf due to the field excitation plus both the components of the
Again, the total generated voltage is also equal 10 the phasor sum of the tenninal voltage
Comparing Eq.(3.18) with Eq.(3.19), the phasor excitation voltage or induced emf can be
found as:
Noting the Eqs.(3.11 and 3.12); this can be written in tenns of the synchronous reactance
48
For more clarity, a similar expression for the induced voltage equation can be derived
The magnitude of the induced voltage phasor (E9 ) in Fig.3.7 is equal to the length (OF).
Thus,
(3.22)
Let the length ( DC) be equal to the magnitude of the vector ( Eo ) and defined from the
phasor·diagram as:
Moreover, from the phasor diagram the length (Be) equals the quantity (Is Xq) , and
similarly (liD) equals (Is Xd). Now, the length (CF) can be found as:
CF = liiG - BH I (3.24)
substituting the Eqs. (3.23 and 3.26) into Eq. (3.22) so:
49
Note that the above equation calculates only the magnitude or nns value of the induced
voltage. The direction of Eg or the power angle (5°) can be obtained from the phasor
diagram as follows:
Thc stator current for the lagging pfload can be expressed as:
Since Eo is in phase with Eg , the power angle (5°) may be found substituting the above
value of the stator current into Eq.(3.23) as:
Neglecting the stator resistance for large synchronous generators. the power angle can be
rewritten as:
(3.32)
The phasor diagram provided in thc previous section can be used for the sake of deriving
curve that describes the real output power as a function of the power angle (5°) is called
50
The total three-phase output power delivered by a salient-pole synchronous generator
consists of two components resulting from the d-q axis components of the stator current
Alternatively, the same power equation can be obtained from the phasor diagram in
Fig.3.7 as:
(J.J5)
However,
Also,
and (3.39)
For large synchronous generators, Rs « (Xa.Xq) and thus it is ignored, then we have:
la = E9 - Vtcoso (3.40)
X,
51
Substituting Eqs.(3.40 and 3.41) into Eq.(3.37) and using the trigonometric identity:
Finally, the output power relation for a salient-pole synchronous generator is thus:
The above equation is the expression of the power-angle characteristic curve of a salient-
pole synchronous generator. It gives the three-phase output power of the generator when
the voltage and reactance quantities are used as the per-phase basis. The first part of the
right-hand side of Eq.(3.44) represents the developed electric power due to the field
excitation of the synchronous generator, whereas the second part, which is totally
independent of the field excitation, is known as the reluctance power of the salient-pole
synchronous generator. The resultant power is thereby the algebraic sum of the electric
and reluctance power as shown in Fig.3.8. The reluctance power is basically an additional
power generated due to the variations in the air-gap magnetic reluctances as a function of
the rotor position (i.e. the power developed due to the saliency effect). This power will in
tum develop a torque of its own known as the reluctance torque. Only at small excitation
does the reluctance power or torque have a significant impact. If that is not the case, the
analysis of a salient-pole synchronous machine may otherwise be made on the same basis
52
The machine is still able to develop power or torque even when the excitation is removed.
power would disappear and one would end up with the fi rst term of Eq.(3.44) alone,
which is in fact the same as the power developed by a cylindrical rotor (nonsalient-pole)
synchronous generator.
pew)
Resultant power
L-____~~----~~1=
80~
· ----~ O'
disturbances or abnormal conditions. Disturbances may occur for many reasons such as
the mechanical torque on the generator shaft or in the electric load applied to the
generator termi nals and so forth. A synchronous generator is most likely to lose its
53
synchronism during these awkward conditions unless certain procedures are quickly
done.
electric transient or mechanical (dynamic) transient in nature. The time needed by the
generator to overcome the transient operation and return to its steady·state operation is
called the transient period which may last for a finite period of time. The principle reason
for the increased chance of losing the generator synchronism during the transient period is
that the speed oscillation, also called hunting, would take place on the rotor shaft due to
the fluctuation of the rotor around its new angular position (torque-angle) making the
rotor speed up and thus pulling it out of the synchronism. Various techniques are applied
them are the use of damper windings and fly wheels [57].
It has already been noted that the direct and quadrature axis synchronous reactance
mentioned in the preceding sections were intended only for the steady-state operation
where a positive sequence annature current was to flow in the stator windings after
reaching its sustained val ue. At this point, it is desirable to visualize the physical
sequence (steady-state) stator current is applied at an instant of time when the rotor is
54
synchronously driven and in a position so that its polar or direct axis is directly in line
with the peak of the rotating armature fundamental mmf. Also, assume that this current is
acting alone (all the field circuits are opened). As a result of this operation mode, a path
of high penneance will be offered in the air-gap and thereby the flux linkage of the
annature winding is the greatest for any rotor position. Under this condition, the ratio of
the net armature flux linkage per one phase winding to the current through the same
phase, times the angular frequency (2Tif) yields the steady-state value of the direct-axis
Similarly, the same stator current is now applied but at a different instant, at which the
rotor has its interpolar or quadrature-axis aligned with the axis of the annature mmf wave.
With this in mind, minimum air-gap penneanee is therefore presented resulting in much
smaller flux linkage of the stator windings. Under this condition, the offered air-gap
During the transient conditions, the components of the synchronous reactance would not
have the same values as those for the steady-state conditions. Associated with transient
value, suppose that a positive-sequence current is now suddenly applied to the stator
windings. Furthermore, the rotor field winding is short-circuited and not excited (also, the
damper circuit, if found, is somehow opened). The rotor is synchronously driven and in a
position such that its direct-ax is is in line with the stator mmf wave. The current will be
55
induced in the closed field ci rcuit opposing the suddenly appearing annature mmf and
tending to keep the field linkage at almost zero (constant flux linkage law: the tota/flux
linkage in any closed e/eclrical circuit cannot be changed instantly) at the first short
duration [63]. Therefore, the annature flux will be alternatively restricted to the low
penneance path offered by the interpolar space. The net annature flux linkage is thereby
less than that in the steady-state condition. If that is the case, the ratio of the total
annature flux linkage to the annature current per phase times the angular frequency (2Jrj)
gives the direct-axis transient synchronous reactance (X~). It should be emphasized that
the transient current induced in the field windings will decay to zero after just a short time
from the appearance of the suddenly applied annature current due to the fi eld winding
resistance.
For the subtransient direct-axis reactance, the exact same conditions specified for defining
(X~) are still valid except that the damper circui t effect would definitely take place. Thus,
during the subtransient interval, besides the current induced in the closed field windings,
an extra current will also be induced in the shorted damper windings. Because those
wind ings are in a position that is, compared to the field windings, closer to the armature
structure, the annature flux linkage per phase winding is much less than thai which
corresponds to the transient period. The associated annalure winding reactance due to
Again, the subtransienl current induced in the damper windings will die out more rapidly
than that induced in field windings due to the comparatively higher resistance of the
damper windings. It is evident that for a salient-pole synchronous generator with damper
56
or amortisseur windings (Xd > Xd > Xd'), whereas (Xd == X;j) for one without damper
windings [64].
X~ )
The conditions for defining the transient and subtransient quadrature-axis synchronous
reactance are quite analogous to those specified above for both transient and subtransient
direct-axis synchronous reactance. The only difference between the two cases lies in the
rotor position at the time the sudden positive-sequence annature current is applied.
Hence, for X~ and X~' , it is assumed that the annature current is suddenly applied at such
an instant that the rotor is synchronously rotated so that its quadrature-axis (interpolar
space) is now aligned with the axis of the rotating annature mmf wave. Likewise, a
For the transient value, consider the same assumptions that the field windings are
unexcited and closed and the damper or amortisseur windings, if found, are imagined to
be open. In this case, unlike the direct-axis transient reactance case, the current will not be
induced in the closed field windings as the field linkage due to the annature flux is
already zero. Therefore, the annature flux will distribute through the same low penneance
path between the field poles as that presented in the definition of the quadrature-axis
written that (Xq ~ X~ ) for a laminated salient-pole synchronous generator. For a solid
57
rotor synchronous machine, this comment is incorrect, because the transient currents
For the subtransient value, the damper windings are assumed to be close-circuited in
addition to the closed field windings. In this respect, a subtransient current will flow in
the damper circuit trying to maintain its flux linkage at approx imately zero and thereby
mak ing the armature flux pass through the minimum permeance path of the interpolar
space. The result of this is much lower net armature flux linkage in one phase winding per
Potier reactance, denoted by (Xp), is a fictitious reactance that comprises the armature
leakage flux and the extra leakage flux of the field windings due to the fi eld current of a
synchronous machine [28]. It is sometimes used instead of the armature leakage reactance
(XI) in the phasor representation of synchronous machines. For a solid (cylindrical) rotor
synchronous machine, the Potier reactance is almost equal to the stator leakage reactance.
However, it is onen greater than the stator leakage reactance for one of a salient-pole
type. The purpose of employing this reactance is to determine the armature reaction
voltage drop for any given toad current. The Potier reactance cannot be directly
measured. Nevertheless, it can be obtained graphically from both the no-load saturation
58
curve and the zero-power factor rated current saluration curve; (Polier Triangle Method)
machine, a variety of tests may be conducted. The IEEE and lEe organizations have
published a standardized guide to describe those tests in great detail; [27] and [28],
respectively. The most commonly used test among the group is the sudden three-phase
shon circuit. This test can provide accurate enough results that help to investigate the
mach ine behaviour during the transient condit ions. The sudden three-phase shon circuit
59
Chapter 4
In the foregoing part of this thesis, a reasonable effort has gone into providing a clear
picture about the synchronous machine theory, especially the theory of a salient-pole
synchronous mach ine. Now, it is time to introduce the essential work which has been
done in this research. This chapter presents the experimental test results that were
parameters of a salient-pole synchronous generator. The main aim is to find the steady-
state and transient direct-axis synchronous reactance as well as the armature reaction
voltage drop (Potier reaclance milage drop) for the synchronous generator to be tested.
60
The machine being treated, as shown in FigA.I, is a salient-pole synchronous generator
with damper bar windings. The nameplate quantities of this synchronous generator are:
3-phase 2 KVA, 208 V (line-to-line), 4-pole, 1800 rpm, 60 Hz, 5.5 A, PF: 0.8
In FigA.I, the machine to the left-hand side is a dc motor (shunt connection) that is used
as a prime mover for the synchronous generator on the right-hand side. Also, it can be
seen that a rheostat is connected to the armature circuit of the dc motor to achieve more
been the usual practice in electric machinery design and analysis. Therefore, before
proceeding in this chapter, it is better to first determine the base impedance (Zb) for the
studied generator. This can be done using the given rated values as follows:
Z _ Vb rated (L-L)
(4.1)
b - .f3 X Ib rated
208
Zb =";3 X 5.5 = 21 .83 n
As outlined earlier in chapter three (See.3A) the per-phase ac stator resistance for Y-
61
Rs = iRdC(L-L) x 1.6 n (4.3)
where RdC is the average line-to-line dc resistance that was measured to be (2.2 Q) for the
tested generator. It is to be noted that 1.6 is the skin effeet of the alternating stator current
1
R, ~"2 x (2.2) x (1.6) ~ 1.76 n ~ 0.081 pu
The tests conducted on the chosen electric machine in this thesis are:
1- Open-Circuit Test.
2- Sustained Short-Circuit Test.
3- Slip Test.
4- Zero Power-Factor Test.
5- Sudden Three-Phase Short-Circuit Test.
All five tests are discussed in full detail with an analytical approach in the next sections.
The test resulls from the first two tests are provided together since they are
complementary to each other and need to be simultaneously presented. The rest of the
The open-circuit test is carried out on a synchronous generator at no load. The reason is to
Characleri~'lic (OCC) for the tesled synchronous generator. Thereafter, the ace is used
62
graphically to determine the saturated and unsaturated steady·state values of the
synchronous reactance as well as the machine Short-Circuit Ratio (SCR). The short-
circuit ratio (SCR) for a synchronous generator is defined as the rat io of the fi eld current
required to obtain the rated terminal voltage at open circuit to the field current required
Test Procedures
During the open-circuit test, the rotor of the unloaded generator is first run at its rated
synchronous speed of 1800 rpm with no excitation being appl ied to the rotor (field)
windings. The field current is then gradually supplied to the fie ld windings while
measuring the open-circuit terminal voltage till reaching its saturation condition (i.e. the
generated voltage increases very slowly or not at all with the increase in the field current).
The readings of the no-load terminal voltage versus the field current are recorded. After
fini shing, a curve of the no-load terminal voltage with the change in the field current is
plotted using the test data as shown in FigA.2. This curve represents the no-load
saturation curve or the open-circuit characteristic (acC) for the tested generator. The no-
load saturation curve obeys a linear relationship (straight line) as long as the machine's
magnetic core is not yet saturated. The extension of this straight portion for higher values
The object of the sustained short-circuit test is to find the short-c ircuit saturation curve or
the so-called Short-Circuit Characteristic (SeC) for a synchronous generator. This curve
is then used with the no- load saturation curve for the same purpose. One has to keep an
63
eye on the value of the stator current not to exceed its rated value while performing this
test. Generally, during the test, the rated value of the stator current of the tested generator
Test Procedures
The generator under test is operated at the rated synchronous speed with its stator
current is then gradually increased while measuring the short-circuit current flowing in
the stator windings. The variation of the short-circuit stator current with the field current
yields the short-c ircuit characteristic (SCC) of the tested generator. The field current
reading at which the rated stator current is obtained should be precisely recorded for
finding the value of the short circuit ratio (SCR) as will be provided later on. The see for
the tested generator is shown below in Fig.4.2.
The following table provides the test data obtained from the open- and short-circuit tests:
Field Current (A) 0.1 0.2 0.3 0.4 0.5 0.7 0.81 0.9 1.5 1.6
L-L Terminal Voltage
(Vi 30 56 82 111 137 187 208 223 274 279
OCTes/
Field Current (A) 0.1 0.2 0.3 0.4 0.53 0.6 0.7
Armature Current (A)
SC Test 1.2 2.2 3.25 4.26 5.5 6.25 7.28
64
Air-Gap Line
/
/
300 /
/
/
/
H 250
, /
/
/
Fig.4.2 Open- and Short-Circuit Characteristics for the laboratory synchronous generator
The direct-axis saturated synchronous impedance Za(sat) can be detcnnined from point A
with reference to FigA.2. This point corresponds to the no-load rated voltage on the open-
the acc (taken at point A) and the short-circuit armature current on the sec
corresponding to the field current at the same point as can be seen in FigA.2. Thus.
z _ VT rated (L - L)
(4.4)
d(sat) - {3x Isc corresponding to the field current at rated voltage
Vr 208
thus, Zd(sat) = ~ = v'3x8.32 = 14.43 n
The direct-axis saturated synchronous reactance Xd(sat) is therefore:
In per-unit,
xd(sat) -_ -Xd(sat)actllal
-Z-,-- (4.6)
14.32
Xd(sat) = 21.83 = 0.656 pu
way, though from any point on the air-gap line, such as point B in Fig.4.2 which
corresponds to the field current for rated short-circuit armature current. Then,
z _V air-gap (L-L)
(4.7)
d(unsat) - ..fjx 101 SC rated
147
Zd(unsat) = ../3 X 5.5 = 15.43 n
The direct-axis unsaturated synchronous reactance Xd(unsat) is:
66
It should be mentioned that both of the reactance values, just calculated above, are only
0.81
SCR = 0.53 = 1.528
One should note that the reciprocal of the SCR is equal to the saturated value of the d-axis
The objective of the sl ip test is to find the value of the quadrature-axis synchronous
reactance Xq by finding the saliency ratio (Xq/Xd,). This ratio is then to be multiplied by
the value of the direct-axis synchronous reactance Xd obtained from the open-circuit and
saturated value of Xd, is used, the result will be a saturated value of Xq. A sim ilar
statement can be made if an unsaturated value of Xd, from the open-circuit and short-
Test Procedures
During the slip test, the generator being tested is driven by means of a prime mover at a
speed slightl y different from the synchronous speed, about 1% more or less, to achieve a
67
very small slip. The field windings on the rotor must be kept open-circuited so that
across the generator terminals. At this point, the applied voltage should not be more than
approx imately 25% of the rated voltage of the generator [57]. An oscillogram of the
reduced armature (stator) voltage and current can then be recorded as shown in FigA.3.
Fig.4.3 shows the oscillation of the stator current and voltage for the tested generator
during the slip test. It can be clearly noticed from this figure that both the stator voltage
and current oscillate, having minimum and maximum values as time passes.
d-axis q-axis
68
At an instant of time when the stator voltage has its maximum amplitude, the stator
current has its minimum value. This is the time when the rotor has its direct-axis in line
with the armature magnetomotivc force mmf and thereby the offered air-gap reactance
when the stator voltage is minimum, the stator current is maximum. The rotor quadrature-
axis at this specific time is aligned with the armature mmf and thus the presented air-gap
From the stator current and voltage waveforms found from the slip test, the following
x . _ Vt(L-Llmax (4.10)
d(sllp) - ..fj x ' min
63
Xd(slip) = ..[3 x 9.5 = 3.83 n
52.2
Xq(slip) = ..[3 X 12.5 = 2.41 n
Xq] 2.41
[j( . = 3.83 = 0.63 (4.12)
d slip
69
Finally, as mentioned earlier, the saliency ratio can be used with the Xd value previously
obtained from DCC and SCC to compute the actual steady·state saturated and unsaturated
value of the quadrature·axis synchronous reactance for the tested generator. That is,
(4.13)
This test is perfonned to obtain the zero power· factor saturation curve or, as it is called,
the Zero Power·Facfor Characteristic (ZPFC) that is used with the DCC to detennine the
generator at any given load current. The ZPFC is a graph of the tenninal voltage against
the field current with the stator current held constant (mostly at the rated value) and with
zero lagging power factor operation (9=90°). Consequently, the Potier reactance (defined
in Ch.3, Sec.3.8) is found using any point on the ZPFC. It is often sufficiently accurate to
use the point that corresponds to the rated voltage and rated current, as is the case in this
70
Fig.4.4 Adjustable 3-phase reactor used as a load in the zero power-factor test
Test Procedures
The zero power-factor test is a straightforward test carried out by loading the tested
method is used to ensure zero delivered real power from the generator to the load during
the test. The reactor used as a load for the tested generator is shown above in FigAA. By
a proper adjustment to the field excitation and the load, both al the same time, readings of
the tenninal voltage variation with the field current when a constant rated armature
current is drawn can be measured in steps. From those readings, the zero power-factor
noticed that the ZPFC is nearly parallel to the acc and shifted by a constant distance to
the right.
71
}:> Experimental Results from the Zero Power-Factor Test
Provided below is the test data from the zero power-factor test conducted at rated stator
current.
The zero power-factor rated current saturation curve or (ZPFC) is plotted on the same
graph with the previously obtained open-circuit characteristic (OCC) as shown in Fig.4.5
below.
Air-Gap Line
ace
~
"0 260 R
ZPFC
> Ruter/ Cllrre,,'
]200
~
..J
..J
The Potier reactance can be calculated in reference to Fig.4.S by the following steps:
• First, the OCC and ZPFC are plotted against the dc excitation current on the same
graph.
Point A, the ordinate of which is the rated voltage, is detennined on the ZPFC.
To the left of A, a horizontal line BA equals OH is drawn towards the OCC. The
the ZPFC. This field current is also equal to that needed to circulate the rated
Line BC parallel to the air-gap line is drawn upwards from point B. The
From C, a perpendicular line is drawn downwards till it intersects with the line
BA, at the point F. The fonned triangle AFC between the two curves is known as
• Finally, the vertical leg CF of the Potier triangle represents the leakage or Potier
reactance drop (laX,,) at the rated annature current. Thus, for the laboratory
and therefore, the Potier reactance X" (stator leakage reactance XI) is equal to:
12.7
Xp = X, = 55 = 2.31 n = 0.106 pu
73
Adding the Potier reactance drop UaXp) to the rated terminal voltage gives the
length OR which represents the voltage behind Potier reactance Ep. Thus,
(4.15)
With reference to FigA.5, the distance BA represents the total field current required to
circulate the rated stator current at short-circuit condition. This current can be resolved
into: the field current BF necessary to overcome the leakage reactance drop, and the
current.
The voltage drop due to the Potier reactance is laXp. If the Potier triangle is slid down
(i.e. Potier triangle is developed at a voltage value which is less than the rated terminal
vo ltage), the Potier reactance voltage drop laXp would obviously be larger than that
obtained at the rated terminal voltage which in tum results in less accurate approximation
of the stator leakage reactance XI' In fact, the quantity UaXp) is comprised of both the
annature reaction voltage drop and voltage drop due to stator leakage reactance. If the
ZPFC is developed at value of stator current which is difTerent from the rated value, the
voltage drop due to annature reaction would vary, whereas that due to stator leakage
74
4.4.5 Sudden Three-Phase Short-Circuit Test
One important aspect to visualize the behaviour of the synchronous generators during the
dynamic and electric transients is the sudden three-phase short-ci rcuit. Many of the
oscillogram of a short-ci rcuit current suddenly applied to the generator terminals. The
synchronous reactances are often of little significance during the short-circuit conditions
[66J, and therefore they are beyond the scope of this thesis.
Test Procedures
75
The sudden three·phase short·circuit test is perfonned on the laboratory generator at rated
tenninal voltage and with no· load operation. That is, the tested salient·pole generator is
driven at the synchronous speed and is excited so that the rated tenninal voltage (208V) is
Subtransienf
Pre/aull Period
---H-++H-++H-++Hr':f++-'\-~ time
Fig.4.7 shows a typical short·circuit current wavefonn suddenly nowing in one stator
into three distinct periods of time, associated with them are three different components of
76
the short·circuit current. Those are the subtransient component /" , transient component /' ,
and sustained or steady·state component /s- The subtransient period results in the largest
initial value of the short·circuit current, mainly caused by currents induced in damper
windings, and lasting for a very short time (only the first few cycles) from the instant of
the short·circuit. The transient period immediately follows the subtransient period and
spans a relatively longer time. Right after and ultimately, the steady·state period does
appear during which the short·circuit current approaches its sustained value [60J.
11 should be noticed from Fig.4.7 that the current waveform envelope obeys an
exponential decay. Moreover, the decay is very fast at the first couple of cycles (the
subtransient period) and gradually slows down during the following few cycles (transient
period) till it becomes almost constant at the steady·state period. The time constant
controlling the rapid decay during the subtransient period is called the direct·axis
subtransient short·circuit time constant T~' while that controlling the slower decay during
the transient period is called the direct·axis transient short-circuit time constant T~. It is
decaying, though to zero, with a time constant known as the short·circuit armature time
constant Ta. The initial value of each component depends on which point of the cycle the
circuit current (thus, the current waveform has either a negative or positive dc-offset)
77
In some cases, second hannonic components do exist (the value of which depends on the
difference between X~' and X~' ) in the short-circuit current, especially for a salient-pole
generator without damper windings. For a generator with damper windings, subtransient
currents would flow in them preventing the existence of the second hannonic armature
currents. However, those components are nonnally very small and can be regarded as
negligible [64].
From the above discussion, the nns amplitude of the total ac component of the short-
circuit current in one phase at any instant of time can be defined as follows:
The initial value of the subtransient short-circuit current (I") can be estimated by
extending the line connecting the rapidly decaying peaks of th9 ac component back to the
zero-time point or the instant of the short circuit. In a similar approach, the initial value of
the transient short-circuit current (I') is found, although it neglects the rapidly decaying
reactance X~ and X~' can now be found, with reference to Eqs. (4.16 and 4.17), as the
78
ratio of the prefault open-circuit voltage to the associated components of short-circuit
X'd ~ (4.18)
I'
X"d ~ (4.19)
I"
The time constants Td and Td' are defined as the time requi red for the related current
component to decrease to (l ie) or 0.368 times its initial value [65] , as is clear from
Fig.4.10.
The field current following the sudden short-circuit also has both ac and dc components.
The ac component of the field current is said to be produced by the dc component of the
stator current and thereby decays with the same time constant Ta. Conversely, the dc
component of the fie ld current can be regarded as the reason fo r the appearance of the ac
component in the armature current [65]. The dc component of the field current, like the ac
component of the armature current, also is compri sed of transient, subtransient and
steady-state components with the same associated time constants, Td and rd'.
}I;o Experim ental Results from the Sudd en Three-Phase Short-Circuit Test
The fo llowing test results were experimentally obtained from a sudden three-phase short-
circuit developed at the term inals of the laboratory salient-pole synchronous generator at
rated voltage for no load. Fig.4.8 shows the stator phase currents with the dc fie ld current
after the short circuit while Fig.4.9 indicates phase-b short-c ircuit stator current alone
79
Phase a
I!!A,
i l"
Phau b
- -
PI/usee
_ - _I\VNNW\I'JVo W/l.VNJ\NtNIIIWf>N\.NI.!IN.JI/
p'
-
Field Current I,
Fig.4.8 Currents oscillogram from sudden three- phase short-circuit test al I pu rated vo ltage
Phase b
Field Current I,
-t----_
Time Scale (5) ---+
Fig.4.9 Phase-b short-circuit stator current alone with the de field current at I pu rated voltage
80
Referring to the obtained test results as in Figs.4.8 and 4.9 the following discussion can
be provided. Before the instant of short-circuit, the flux linkage of the excited field
Immediately after the short-circuit happens and by the principle of constant flux linkage,
which states: "the flux linkage of any closed circuit offinite resistance and e.mf cannot
change instantly" (8], the suddenly appearing annature mmfwill be resisted by ajump or
sudden increase in the field current over its initial value before the short circuit as seen in
Fig.4.8. In other words, the field current will be increased simultaneously at the short-
circuit moment to overcome the demagnetizing effect of the large current and maintain
the air-gap flux linkage constant for a short time following the short-circuit [65].
Subtransient current flows in the damper windings for similar reasons. This current would
however die out very rapidly due to the high resistance of damper windings. Again, as per
the theorem of constant flux linkage, the terminal voltage is zero at the short circuit. The
change of flux linkage with the stator current must also be zero [63]. This fact justifies the
large initial values of the short-circuit current; that is, current components must be
induced in the stator windings in order for the air-gap flux linkage to remain constant for
The phase-b current waveform in Fig.4.9 is used to obtain the exponential decay or the
envelope of the short-circuit current. A set of points which arc the positive peaks of this
phase current is estimated whereby the short-circuit current envelope is drawn as shown
accurate shape of the current envelope around the selected points indicated by the nodes
81
or script "0" in FigA.l0. As pointed out earlier, by extrapolating the subtransient and
transient current envelope back to zero-time or the time of the short-circuit occurrence,
the subtransient and transient current components can be evaluated, whereby X~ and
1.4 , - - - _ - _ - - _ - _ - - _ - _ - - _ - ,
Amperes(%)
1.2
0.368 (I"-I')
/"
0.2
---------------------------------------------------;;-1
----00~-0~.0~2~~0~.0~4-~0~.0~
. -~0~.0~
. -~0~.,~~0.~,2~~0.~'4~~O.16
Time(s)
Therefore, X~ and X:; are determined using Eqs. (4.18 and 4.19 ) as follows:
208
Xd = J3 x 53.03 = 2.2611 = 0.104 pu
208
X~ = J3 x 80.61 = 1.4911 = 0.068 pu
The time constants T~ and T~' arc estimated to be 50ms and 15ms, respectively.
82
The total rms ac component of the short·circuit current can now be expressed, referring to
4.5 Summary
The following table summarizes the parameters that have been determined for the
R, 0.081
Xa(sat) 0.656
Xa(unsat) 0.702
Xq(sat) 0.413
Xq(unsac) 0.442
X; 0.104
X"
d 0.068
Xp 0.106
T; 0.05 (s)
T"
d 0.015 (s)
83
Chapter 5
5.1 Conclusion
This research work has been devoted to the experimental determination of equivalent
during steady-state and transient operation is, in particular, largely affected by these
parameters. The investigation starts by providing a clear insight into the scope of
synchronous machines theory. At this point, special attention has been given to the
synchronous generator of a salient-pole type as it is the machine being dealt with in this
thesis. Thereafter, a set of standard tests as defined by the IEEE standard 11 5· 1995 is
conducted in the laboratory for the purpose of the estimation of the machine parameters.
The experimental test results are then analyzed, whereby the required machine parameters
The further work that would be suggested in regard to this thesis is that veri fication of the
84
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94
Appendix A
Table A.I Typical constants of th ree- phase synchronous mach ines (Adapted from Refs. 42 and
65)
IType X. X. X. X;; X,
2- Po le tu rbine 1.10 1.07 O. I S 0.09
(= X;)
generators 0.95 - 1.45 0.92 - 1.42 0.12 - 0.21 0.07 - 0.14
0.35 0.55
Salient-pole 1.15 0.75
(= X~)
generators (without 0.60 - lAS 0.40 - 0.95 0.20 - 0.45 0.30 - 0.70
dampers)
95
Table A.I Cont.
(.) Xo varies so critically with stator winding pitch that an average value can hardly be given.
(A) For single-phase machines (or three-phase machines designed for single-phase operation),
T~' may have much higher values than listed.
96
Appendix 8
Field Current If
Field Current I f
97