Consists of two sets of windings: 3 phase armature winding on the stator distributed with centres 120° apart in space field winding on the rotor supplied by DC Two basic rotor structures used: salient or projecting pole structure for hydraulic units (low speed) round rotor structure for thermal units (high speed) Salient poles have concentrated field windings; usually also carry damper windings on the pole face.Round rotors have solid steel rotors with distributed windings Nearly sinusoidal space distribution of flux wave shape obtained by: distributing stator windings and field windings in many slots (round rotor); shaping pole faces (salient pole)

1. 1539pk
SYNCHRONOUS MACHINES
Copyright © P. Kundur
This material should not be used without the author's consent

2. 1539pk
SM - 1
Synchronous Machines
1. Physical Description
2. Mathematical Model
3. Park's "dqo" transportation
4. Steady-state Analysis
 phasor representation in d-q coordinates
 link with network equations
5. Definition of "rotor angle"
6. Representation of Synchronous Machines in
Stability Studies
 neglect of stator transients
 magnetic saturation
7. Simplified Models
8. Synchronous Machine Parameters
9. Reactive Capability Limits
Outline

3. 1539pk
SM - 2
Physical Description of a
Synchronous Machine
 Consists of two sets of windings:
 3 phase armature winding on the stator
distributed with centres 120° apart in space
 field winding on the rotor supplied by DC
 Two basic rotor structures used:
 salient or projecting pole structure for hydraulic
units (low speed)
 round rotor structure for thermal units (high
speed)
 Salient poles have concentrated field windings;
usually also carry damper windings on the pole
face.
Round rotors have solid steel rotors with
distributed windings
 Nearly sinusoidal space distribution of flux wave
shape obtained by:
 distributing stator windings and field windings in
many slots (round rotor);
 shaping pole faces (salient pole)

4. 1539pk
SM - 3
Rotors of Steam Turbine Generators
 Traditionally, North American manufacturers normally
did not provide special “damper windings” or
“amortisseurs”
 solid steel rotors offer paths for eddy currents,
which have effects equivalent to that of
amortisseur currents
 European manufacturers tended to include damper
windings and negative-sequence current capability
 separate copper rods provided underneath the
wedges
Primary purpose is to meet the negative
sequence current capability
Figure 3.3: Solid round rotor construction

5. 1539pk
SM - 4
Rotors of Hydraulic Units
 Normally have damper windings or amortisseurs
 non-magnetic material (usually copper) rods
embedded in pole face
 connected to end rings to form short-circuited
windings
 Damper windings may be either continuous or non-
continuous
 Space harmonics of the armature mmf contribute to
surface eddy current
 therefore, pole faces are usually laminated
Figure 3.2: Salient pole rotor construction

6. 1539pk
SM - 5
Balanced Steady State Operation
 Net mmf wave due to the three phase stator
windings:
 travels at synchronous speed
 appears stationary with respect to the rotor; and
 has a sinusoidal space distribution
 mmf wave due to one phase:
Figure 3.7: Spatial mmf wave of phase a

7. 1539pk
SM - 6
Balanced Steady State Operation
 The mmf wave due to the three phases are:





 






 


3
2
cosKiMMF
3
2
cosKiMMF
cosKiMMF
cc
bb
aa  





 






 


3
2
tcosli
3
2
tcosIi
tcosIi
sma
smb
sma
 tcosKI
2
3
MMFMMFMMFMMF
sm
cbatotal



8. 1539pk
SM - 7
Balanced Steady State Operation
 Magnitude of stator mmf wave and its relative
angular position with respect to rotor mmf wave
depend on machine output
 for generator action, rotor field leads stator field
due to forward torque of prime mover;
 for motor action rotor field lags stator field due
to retarding torque of shaft load
Figure 3.8: Stator and rotor mmf wave shapes

9. 1539pk
SM - 8
Transient Operation
 Stator and rotor fields may:
 vary in magnitude with respect to time
 have different speed
 Currents flow not only in the field and stator
windings, but also in:
 damper windings (if present); and
 solid rotor surface and slot walls of round rotor
machines
Figure 3.4: Current paths in a round rotor

10. 1539pk
SM - 9
Direct and Quadrature Axes
 The rotor has two axes of symmetry
 For the purpose of describing synchronous
machine characteristics, two axes are defined:
 the direct (d) axis, centered magnetically in the
centre of the north pole
 The quadrature (q) axis, 90 electrical degrees
ahead of the d-axis
Figure 3.1: Schematic diagram of a 3-phase synchronous
machine

11. 1539pk
SM - 10
Mathematical Descriptions of a
Synchronous Machine
 For purposes of analysis, the induced currents in
the solid rotor and/or damper windings may be
assumed to flow in two sets of closed circuits
 one set whose flux is in line with the d-axis; and
 the other set whose flux is along the q-axis
 The following figure shows the circuits involved
Figure 3.9: Stator and rotor circuits

12. 1539pk
SM - 11
Review of Magnetic Circuit Equations
(Single Excited Circuit)
 Consider the elementary circuit of Figure 3.10
 The inductance, by definition, is equal to flux linkage
per unit current
where
P = permeance of magnetic path
> = flux = (mmf) P = NiP
Li
ri
dt
d
e
dt
d
e
1
i






PN
i
NL 2



Figure 3.10: Single excited magnetic circuit

13. 1539pk
SM - 12
Review of Magnetic Circuit Equations
(Coupled Circuits)
 Consider the circuit shown in Figure 3.11
with L11 = self inductance of winding 1
L22 = self inductance of winding 2
L21 = mutual inductance between winding 1 and 2
2221212
2211111
22
2
2
11
1
1
iLiL
iLiL
ir
dt
d
e
ir
dt
d
e








Figure 3.11: Magnetically coupled circuit

14. 1539pk
SM - 13
Basic Equations of a Synchronous Machine
 The equations are complicated by the fact that the
inductances are functions of rotor position and
hence vary with time
 The self and mutual inductances of stator circuits
vary with rotor position since the permeance to flux
paths vary
 The mutual inductances between stator and rotor
circuits vary due to relative motion between the
windings





 






 



3
2cosLL
3
2
2cosLLII
2cosLL
ILI
2ab0ab
2ab0abbaab
2aa0aa
gaaalaa





 



sinL
2
cosLI
cosLI
cosLI
akqakqakq
akdakd
afdafd

15. 1539pk
SM - 14
Basic Equations of a Synchronous Machine
 Dynamics of a synchronous machine is given by the
equations of the coupled stator and rotor circuits
 Stator voltage and flux linkage equations for phase a
(similar equations apply to phase b and phase c)
 Rotor circuit voltage and flux linkage equations
kqakqkdakdfdafdcacbabaaaa
aaaaa
a
a
ilililililil
iRpiR
dt
d
e




kqkqkq
kdkdkd
fdfdfdfd
iRp0
iRp0
iRpe














 





 













 





 













 





 


3
2
sini
3
2
sinisiniL
iL
3
2
cosi
3
2
cosicosiL
iLiL
3
2
cosi
3
2
cosicosiL
iLiL
cbaakq
kqkkdkq
cbaafd
kdkkdfdfkdkd
cbaafd
kdfkdfdffdfd

16. 1539pk
SM - 15
The dqo Transformation
 The dqo transformation, also called Park's
transformation, transforms stator phase quantities from
the stationary abc reference frame to the dqo reference
frame which rotates with the rotor
The above transformation also applies to stator flux
linkages and voltages
 With the stator quantities expressed in the dqo
reference frame
 all inductances are independent of rotor position
(except for the effects of magnetic saturation)
 under balanced steady state operation, the stator
quantities appear as dc quantities
 during electromechanical transient conditions,
stator quantities vary slowly with frequencies in
the range of 1.0 to 3.0 Hz
The above simplify computation and analysis of results.































 





 






 





 










c
b
a
0
q
d
i
i
i
2
1
2
1
2
1
3
2
sin
3
2
sinsin
3
2
cos
3
2
coscos
3
2
i
i
i

17. 1539pk
SM - 16
Physical Interpretation of dqo
Transformation
 The dqo transformation may be viewed as a means
of referring the stator quantities to the rotor side
 In effect, the stator circuits are represented by two
fictitious armature windings which rotate at the
same speed as the rotor; such that:
 the axis of one winding coincides with the d-axis
and that of the other winding with the q-axis
 The currents id and iq flowing in these circuits
result in the same mmf's on the d- and q-axis as
do the actual phase currents
 The mmf due to id and iq are stationary with respect
to the rotor, and hence:
 act on paths of constant permeance, resulting in
constant self inductances (Ld, Lq) of stator
windings
 maintain fixed orientation with rotor circuits,
resulting in constant mutual inductances

18. 1539pk
SM - 17
Per Unit Representation
 The per unit system is chosen so as to further
simplify the model
 The stator base quantities are chosen equal to the
rated values
 The rotor base quantities are chosen so that:
 the mutual inductances between different
circuits are reciprocal (e.g. Lafd = Lfda)
 the mutual inductances between the rotor and
stator circuits in each axis are equal (e.g., Lafd =
Lakd)
 One of the advantages of having a P.U. system with
reciprocal mutual inductances is that it allows the
use of equivalent circuits to represent the
synchronous machine characteristics
The P.U. system is referred to as the "Lad
base reciprocal P.U. system"

19. 1539pk
SM - 18
P.U. Machine Equations in
dqo reference frame
 The equations are written with the following
assumptions and notations:
 t is time in radians
 p = d/dt
 positive direction of stator current is out of the
machine
 each axis has 2 rotor circuits
 Stator voltage equations
 Rotor voltage equations
0a00
qardqq
darqdd
iRpe
iRpe
iRpe



q2q2q2
q1q1q1
d1d1d1
fdfdfdfd
iRp0
iRp0
iRp0
iRpe





20. 1539pk
SM - 19
P.U. Machine Equations in dqo Reference
Frame (cont'd)
 Stator flux linkage equations
 Rotor flux linkage equations
 Air-gap torque
 
 
000
21
1
iL
iLiLiLL
iLiLiLL
qaqqaqqlaqq
dadfdaddladd






qaqq2q22q1aqq1
qaqq2aqq1q11q1
dadd1d11fdd1fd1
dadd1d1ffdffdfd
iLLLiL
iLiLiL
iLiLiL
iLiLiL




dqqde iiT 

21. 1539pk
SM - 20
Equivalent Circuits for Direct and
Quadrature Axes
 Equivalent circuits representing the complete
machine characteristics including the effect of
voltage equations are shown in Figure 3.18 where:
aqq22q2aqq11q1
d1fd11d1d1fffdfd
LLLLLL
LLLLLL


Figure 3.18: Complete equivalent circuits

22. 1539pk
SM - 21
Steady State Analysis Phasor
Representation
For balanced, steady state operation, the stator voltages may
be written as:
with
ω = angular velocity = 2πf
α = phase angle of ea at t=0
Applying the d,q transformation,
At synchronous speed, the angle θ is given by θ = ωt + θ0
with θ = value of θ at t = 0
Substituting for θ in the expressions for ed and eq,
 
 
 


32tcosEe
32tcosEe
tcosEe
mc
mb
ma
 
 

tsinEe
tcosEe
mq
md
 
 0mq
0md
sinEe
cosEe



23. 1539pk
SM - 22
Steady State Analysis Phasor
Representation (cont'd)
 The components ed and eq are not a function of t because
rotor speed ω is the same as the angular frequency ω
of the stator voltage. Therefore, ed and eq are constant
under steady state.
In p.u. peak value Em is equal to the RMS value of terminal
voltage Et. Hence,
 The above quantities can be represented as phasors with
d-axis as real axis and q-axis as imaginary axis
Denoting δi, as the angle by which q-axis leads E
 
 0tq
0td
sinEe
cosEe


itq
itd
cosEe
sinEe



24. 1539pk
SM - 23
Steady State Analysis Phasor
Representation (cont'd)
 The phasor terminal voltage is given by
in the d-q coordinates
in the R-I coordinates
 This provides the link between d,q components in a
reference frame rotating with the rotor and R, I
components associated with the a.c. circuit theory
 Under balanced, steady state conditions, the d,q,o
transformation is equivalent to
 the use of phasors for analyzing alternating
quantities, varying sinusoidally with respect to
time
 The same transformation with θ = ωt applies to both
 in the case of machines, ω = rotor speed
 in the case of a.c. circuits, ω = angular frequency
lR
qdt
jEE
jeeE
~



25. 1539pk
SM - 24
Internal Rotor Angle
 Under steady state
Similarly
 Under no load, id=iq=0. Therefore,
and
 Under no load, Et has only the q-axis component
and δi=0. As the machine is loaded, δi increases.
Therefore, δi is referred to as the load angle or
internal rotor angle.
 It is the angle by which q-axis leads the phasor Et
adqqadqq
adqd
RiiXRiiL
Rie


aqfdaddd
aqdq
RiiXiX
Rie


fdadq
d
fdadd
qqq
iLe
0e
iL
0iL




fdadqdt ijLjeeE
~


26. 1539pk
SM - 25
Electrical Transient Performance
 To understand the nature of electrical transients, let
us first consider the RL circuit shown in Figure 3.24
with e = Emsin (ωt+α). If switch "S" is closed at t=0,
the current is given by
solving
 The first term is the dc component. The presence of
the dc component ensures that the current does not
change instantaneously. The dc component decays
to zero with a time constant of L/R
iR
dt
di
Le 
 
 

tsin
Z
E
Kei mt
L
R
Figure 3.24: RL Circuit

27. 1539pk
SM - 26
Short Circuit Currents of a Synchronous
Machine
 If a bolted three-phase fault is suddenly applied to
a synchronous machine, the three phase currents
are shown in Figure 3.25.
Figure 3.25: Three-phase short-circuit currents

28. 1539pk
SM - 27
Short Circuit Currents of a Synchronous
Machine (cont'd)
 In general, fault current has two distinct
components:
a) a fundamental frequency component which
decays initially very rapidly (a few cycles) and
then relatively slowly (several seconds) to a
steady state value
b) a dc component which decays exponentially in
several cycles
 This is similar to the short circuit current in the case
of the simple RL circuit. However, the amplitude of
the ac component is not constant
 internal voltage, which is a function of rotor flux
linkages, is not constant
 the initial rapid decay is due to the decay of flux
linking the subtransient circuits (high resistance)
 the slowly decaying part of the ac component is
due to the transient circuit (low resistance)
 The dc components have different magnitudes in
the three phases

29. 1539pk
SM - 28
Elimination of dc Component by
Neglecting Stator Transients
 For many classes of problems, considerable
computational simplicity results if the effects of ac
and dc components are treated separately
 Consider the stator voltage equations
transformer voltage terms: pψd, pψq
speed voltage terms:
 The transformer voltage terms represent stator
transients:
 stator flux linkages (ψd, ψq) cannot change
instantaneously
 result in dc offset in stator phasor current
 If only fundamental frequency stator currents are of
interest, stator transients (pψd, pψq) may be
neglected.
dq, 
aqdqq
adqdd
Ripe
Ripe



30. 1539pk
SM - 29
Short Circuit Currents with Stator Transients
Neglected
 The resulting stator phase currents following a
disturbance has the wave shape shown in Figure
3.27
 The short circuit has only the ac component whose
amplitude decays
 Regions of subtransient, transient and steady state
periods can be readily identified from the wave shape
of phase current
Figure 3.27: Fundamental frequency component of short
circuit armature current

31. 1539pk
SM - 30
Synchronous Machine Representation in
System Stability Studies
 Stator Transients (pψd, pψq) are neglected
 accounts for only fundamental frequency
components of stator quantities
 dc offset either neglected or treated separately
 allows the use of steady-state relationships for
representing the transmission network
 Another simplifying assumption normally made is
setting in the stator voltage equations
 counter balances the effect of neglecting stator
transients so far as the low-frequency rotor
oscillations are concerned
 with this assumption, in per unit air-gap power
is equal to air-gap torque
(See section 5.1 of book for details)
1

32. 1539pk
SM - 31
Equation of Motion (Swing Equation)
 The combined inertia of the generator and prime-
mover is accelerated by the accelerating torque:
where
Tm = mechanical torque in N-M
Te = electromagnetic torque in N-m
J = combined moment of inertia of generator
and turbine, kg•m2
am = angular velocity of the rotor in mech. rad/s
t = time in seconds
ema
m
TTT
dt
d
J 


33. 1539pk
SM - 32
Equation of Motion (cont'd)
 The above equation can be normalized in terms of
per unit inertia constant H
where
a0m = rated angular velocity of the rotor in
mechanical radians per second
 Equation of motion in per unit form is
where
= per unit rotor angular velocity
= per unit mechanical torque
= per unit electromechanical torque
 Often inertia constant M = 2H used
base
2
m0
VA
J
2
1
H


em
r
TT
dt
d
H2 

m
m
r
0



base
m0m
m
VA
T
T


base
m0e
e
VA
T
T



34. 1539pk
SM - 33
Magnetic Saturation
 Basic equations of synchronous machines
developed so far ignored effects of saturation
 analysis simple and manageable
 rigorous treat a futile exercise
 Practical approach must be based on semi-
heuristic reasoning and judiciously chosen
approximations
 consideration to simplicity, data availability,
and accuracy of results
 Magnetic circuit data essential to treatment of
saturation given by the open-circuit characteristic
(OCC)

35. 1539pk
SM - 34
Assumptions Normally Made in the
Representation of Saturation
 Leakage inductances are independent of saturation
 Saturation under loaded conditions is the same as
under no-load conditions
 Leakage fluxes do not contribute to iron saturation
 degree of saturation determined by the air-gap
flux
 For salient pole machines, there is no saturation in
the q-axis
 flux is largely in air
 For round rotor machines, q-axis saturation
assumed to be given by OCC
 reluctance of magnetic path assumed
homogeneous around rotor periphery

36. 1539pk
SM - 35
 The effects of saturation is represented as
Ladu and Laqu are unsaturated values. The saturation
factors Ksd and Ksq identify the degrees of
saturation.
 As illustrated in Figure 3.29, the d-axis saturation is
given by The OCC.
 Referring to Figure 3.29,
 For the nonlinear segment of OCC, can be
expressed by a suitable mathematical function:
aqusqaq
adusdad
LKL
LKL


I
I




at
at
sd
at0at
K
 TIatsatB
satA 
 eI
I
(3.182)
(3.183)
(3.186)
(3.187)
(3.189)

37. 1539pk
SM - 36
Open-Circuit Characteristic (OCC)
 Under no load rated speed conditions
 Hence, OCC relating to terminal voltage and field
current gives saturation characteristic of the d-axis
fdaddqt
dqqd
iLeE
0eii


Figure 3.29: Open-circuit characteristic showing effects of
saturation

38. 1539pk
SM - 37
 The d- and q-axis air-gap flux linkages are given by
Therefore, total air-gap flux in per unit is equal to
the air-gap voltage
 The saturation factor Ksd can thus be determined,
for given values of terminal voltage and current by
first computing Ea and then using Equations 3.187
and 3.189.
Figure 3.31 Equivalent circuits identifying nonlinear elements
and air-gap flux linkages
 
  qldadqlqaq
dlqaqdldad
iLiReiL
iLiReiL

 (3.192)
(3.193)
(3.194)  tlata I
~
jXRE
~
E
~


39. 1539pk
SM - 38
 For salient pole machines, since q-axis flux is
largely in air, Laq does not vary significantly with
saturation
 Ksq=1 for all loading conditions
 For round rotor machines, there is saturation in
both axes
 q-axis saturation characteristic not usually
available
 the general industry practice is to assume
Ksq = Ksd
 For a more accurate representation, it may be
desirable to better account for q-axis saturation of
round rotor machines
 q-axis saturates appreciably more than the d-
axis, due to the presence of rotor teeth in the
magnetic path
 Figure 3.32 shows the errors introduced by
assuming q-axis saturation to be same as that of
d-axis, based on actual measurements on a 500
MW unit at Lambton GS in Ontario
 Figure shows differences between measured
and computed values of rotor angle and field
current
 the error in rotor angle is as high as 10%, being
higher in the underexcited region
 the error in the field current is as high as 4%,
being greater in the overexcited region

40. 1539pk
SM - 39
Figure 3.32: Field current and internal angle errors with
conventional saturation representation

41. 1539pk
SM - 40
 The q-axis saturation characteristic is not readily
available
 It can, however, be fairly easily determined from
steady-state measurements of field current and
rotor angle at different values of terminal
voltage, active and reactive power output
 Such measurements also provide d-axis
saturation characteristics under load
 Figure 3.33 shows the d- and q-axis saturation
characteristics derived from steady-state
measurements on the 500 MW Lambton unit
Figure 3.33: Lambton saturation curves derived from
steady-state field current and rotor angle measurements

42. 1539pk
SM - 41
Example 3.3
 Considers the 555 MVA unit at Lambton GS and
examines
 the effect of representing q-axis saturation
characteristic distinct from that of d-axis
 the effect of reactive power output on rotor angle
 Table E3.1 shows results with q-axis saturation assumed
same as d-axis saturation
 Table E3.2 shows results with distinct d- and q-axis
saturation representation
Table E3.1
Pt Qt Ea (pu) Ksd δi (deg) ifd (pu)
0 0 1.0 0.889 0 0.678
0.4 0.2 1.033 0.868 25.3 1.016
0.9 0.436 1.076 0.835 39.1 1.565
0.9 0 1.012 0.882 54.6 1.206
0.9 -0.2 0.982 0.899 64.6 1.089
Table E3.2
Pt Qt Ksq Ksd δi (deg) ifd (pu)
0 0 0.667 0.889 0 0.678
0.4 0.2 0.648 0.868 21.0 1.013
0.9 0.436 0.623 0.835 34.6 1.559
0.9 0 0.660 0.882 47.5 1.194
0.9 -0.2 0.676 0.899 55.9 1.074

43. 1539pk
SM - 42
Synchronous Machine Parameters
 Synchronous machine equations and equivalent
circuits we have developed are in terms of
inductances and resistances of stator and rotor
circuits
 fundamental parameters
 Fundamental parameters specify electrical
characteristics completely
 however, they cannot be directly determined
from measured machine responses
 Traditional approach to assigning machine data
 in terms of derived parameters related to
observed behaviour from the terminals under
specified conditions

44. 1539pk
SM - 43
Standard Parameters
 Traditional approach to assigning machine data.
Derived parameters related to time responses of
terminal quantities
 Following a disturbance:
 currents are induced in rotor circuits
 currents in some circuits decay faster than others
 Parameters associated with
 rapidly decaying components (subtransient
constants)
 slowly decaying components (transient constants)
 sustained components (synchronous constants)
 Standard parameters
 effective inductances seen from stator terminal
during sustained, transient, and subtransient
conditions
Ld, Ld
', Ld
'' ; Lq, Lq
', Lq
''
 time constants associated with the decay of
transient and subtransient currents
T '
d0, T ''
d0 ; T '
q0, T ''
q0 (open circuit)
T '
d, T ''
d ; T '
q, T ''
q (short circuit)

45. 1539pk
SM - 44
Relationship Between Standard and
Fundamental Parameters
 d-axis parameters
 Similar expressions apply to q-axis

































ladlfddafd
adfdl
d1
d1
d
lad
lad
fd
fd
d
adfd
adfd
d1
d1
0d
fd
fdad
0d
fdd1d1adfdad
adfdd1
ld
fdad
fdad
ld
ladd
LLLLLL
LLL
L
R
1
T
LL
LL
L
R
1
T
LL
LL
L
R
1
T
R
LL
T
LLLLLL
LLL
LL
LL
LL
LL
LLL

46. 1539pk
SM - 45
Simplified Models for Synchronous
Machines
 Neglect of Amortisseurs
 first order of simplification
 data often not readily available
 Classical Model (transient performance)
 constant field flux linkage
 neglect transient saliency (x'
d = x'
q)
 Steady-state Model
 constant field current
 neglect saliency (xd = xq = xs)
E´
dx
Et
Eq
Et
xs Eq = Xadifd

47. 1539pk
SM - 46
Reactive Capability Limits of Synchronous
Machines
 In voltage stability and long-term stability studies,
it is important to consider the reactive capability
limits of synchronous machines
 Synchronous generators are rated in terms of
maximum MVA output at a specified voltage and
power factor which can be carried continuously
without overheating
 The active power output is limited by the prime
mover capability
 The continuous reactive power output capability is
limited by three considerations
 armature current limit
 field current limit
 end region heating limit

48. 1539pk
SM - 47
Armature Current Limit
 Armature current results in power loss, and the
resulting heat imposes a limit on the output
The per unit complex output power is
where Φ is the power factor angle
 In a P-Q plane the armature current limit, as shown
in Fig. 5.12, appears as a circle with centre at the
origin and radius equal to the MVA rating
  sinjcosIEI
~E
~
jQPS tt
*
tt
Fig 5.12: Armature current heating limit

49. 1539pk
SM - 48
Field Current Limit
 Because of the heating resulting from RfdI2
fd power
loss, the field current imposes the second limit
 The phasor diagram relating Et, It and Eq (with Ra
neglected) is shown in Fig. 5.13
Equating the components along and perpendicular to
the phasor
Therefore
 The relationship between P and Q for a given field
current is a circle centered at on the Q-axis and with
as the radius. The effect of the maximum field current
on the capability of the machine is shown in Fig. 5.14
 In any balanced design, the thermal limits for the field
and armature intersect at a point (A) which represents
the machine name-plate MVA and power factor rating
tE


sinlXEcosiX
coslXsiniX
tstifdad
tsifdad
s
2
t
ifdt
s
ad
tt
ifdt
s
ad
tt
X
E
cosiE
X
X
sinlEQ
siniE
X
X
coslEP



50. 1539pk
SM - 49
Field Current Limit
Fig. 5.13: Steady state phasor diagram
Fig. 5.14: Field current heating limit

51. 1539pk
SM - 50
End Region Heating Limit
 The localized heating in the end region of the armature
affects the capability of the machine in the underexcited
condition
 The end-turn leakage flux, as shown in Fig. 5.15, enters
and leaves in a direction perpendicular (axial) to the
stator lamination. This causes eddy currents in the
laminations resulting in localized heating in the end
region
 The high field currents corresponding to the
overexcited condition keep the retaining ring saturated,
so that end leakage flux is small. However, in the
underexcited region the field current is low and the
retaining ring is not saturated; this permits an increase
in armature end leakage flux
 Also, in the underexcited condition, the flux produced
by the armature current adds to the flux produced by
the field current. Therefore, the end-turn flux enhances
the axial flux in the end region and the resulting heating
effect may severely limit the generator output,
particularly in the case of a round rotor machine
 Fig. 5.16 shows the locus of end region heating limit on
a P-Q plane

52. 1539pk
SM - 51
End Region Heating Limit
Fig. 5.15: Sectional view end region of a generator
Fig. 5.16: End region heating limit

53. 1539pk
SM - 52
Reactive Capability Limit of a 400 MVA
Hydrogen Cooled Steam Turbine Generator
 Fig. 5.18 shows the reactive capability curves of a 400
MVA hydrogen cooled steam turbine driven generator
at rated armature voltage
 the effectiveness of cooling and hence the
allowable machine loading depends on hydrogen
pressure
 for each pressure, the segment AB represents the
field heating limit, the segment BC armature heating
limit, and the segment CD the end region heating
limit
Fig. 5.18: Reactive capability curves of a hydrogen cooled
generator at rated voltage

54. 1539pk
SM - 53
Fig. 5.17: Effect of reducing the armature voltage on the
generator capability curve
Effect of Changes in Terminal Voltage Et