SYNCHRONOUS MACHINES
Copyright © P. Kundur
This material should not be used without the author's consent
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Synchronous Machines
Outline
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
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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)
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Rotors of Steam Turbine Generators

Traditionally, North American manufacturers normally
did not provide special “damper windings”
 solid steel rotors offer paths for eddy currents,
which have effects equivalent to that of
amortisseur currents

European manufacturers tended to provide for
additional damping effects and negative-sequence
current capability
 wedges in the slots of field windings
interconnected to form a damper case, or
 separate copper rods provided underneath the
wedges
Figure 3.3: Solid round rotor construction
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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 noncontinuous

Space harmonics of the armature mmf contribute to
surface eddy current
 therefore, pole faces are usually laminated
Figure 3.2: Salient pole rotor construction
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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
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Balanced Steady State Operation

The mmf wave due to the three phases are:
MMF
MMF
MMF
a
b
c
 Ki a cos 
i a  I m cos  s t 
2 

 Ki b cos   

3


2 

i b  I m cos   s t 

3 

2 

 Ki c cos   

3


2 

i a  l m cos   s t 

3 

MMF
total
 MMF

3
2
a
 MMF b  MMF
c
KI m cos     s t 
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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
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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
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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
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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
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Review of Magnetic Circuit Equations
(Single Excited Circuit)

Consider the elementary circuit of Figure 3.10
ei 
d
e1 
dt
d
dt
 ri
  Li

The inductance, by definition, is equal to flux linkage
per unit current
L N

i
N P
2
where
P = permeance of magnetic path
> = flux = (mmf) P = NiP
Figure 3.10: Single excited magnetic circuit
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Review of Magnetic Circuit Equations
(Coupled Circuits)

Consider the circuit shown in Figure 3.11
e1 
e2 
d 1
dt
d2
dt
 r1i 1
 r2 i 2
 1  L 11 i 1  L 21 i 2
 2  L 21 i 1  L 22 i 2
with L11 = self inductance of winding 1
L22 = self inductance of winding 2
L21 = mutual inductance between winding 1 and 2
Figure 3.11: Magnetically coupled circuit
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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
I aa  L al  I gaa
 L aa 0  L aa 2 cos 2 
2 

I ab  Iba   L ab 0  L ab 2 cos  2  

3 



  L ab 0  L ab 2 cos  2   
3


The mutual inductances between stator and rotor
circuits vary due to relative motion between the
windings
I afd  L afd cos 
I akd  L akd cos 


I akq  L akq cos       L akq sin 
2

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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)
ea 
da
dt
 R aia  p  a  R aia
 a   l aa i a  l ab i b  l ac i c  l afd i fd  l akd i kd  l akq i kq

Rotor circuit voltage and flux linkage equations
e fd  p  fd  R fd i fd
0  p  kd  R kd i kd
0  p  kq  R kq i kq
 fd  L ffd i fd  L fkd i kd

2 
2


 L afd  i a cos   i b cos   
  i c cos   
3 
3






 kd  L fkd i fd  L kkd i kd

2 
2


 L afd  i a cos   i b cos   
  i c cos   
3 
3






 kq  L kkd i kq

2 
2


 L akq  i a sin   i b sin   
  i c sin   
3 
3



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


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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

 cos 

i d 
2

i  
 sin 
q
 
3 

i0 
1

2

2 

cos   

3 

2 

 sin   

3 

1
2

cos   


 sin   

2 

3 

2  

3 

1

2 
i a 
i 
b
i 
 c
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.
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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
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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)
The P.U. system is referred to as the "Lad
base reciprocal P.U. system"

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
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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
e d  p  d   q  r  R aid
e q  p  q   d  r  R aiq
e 0  p  0  R ai0

Rotor voltage equations
e fd  p  fd  R fd i fd
0  p  1d  R 1d i 1d
0  p  1q  R 1q i 1q
0  p  2 q  R 2 qi 2 q
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P.U. Machine Equations in dqo Reference
Frame (cont'd)

Stator flux linkage equations
 d   L ad  L l  i d  L ad i fd  L ad i1 d
 q   L aq  L l  i q  L aq i1 q  L aq i 2 q
 0   L0 i 0

Rotor flux linkage equations
 fd  L ffd i fd  L f 1d i 1d  L ad i d
 1d  L f 1d i fd  L 11 d i 1d  L ad i d
 1q  L 11 q i 1q  L aq i 2 q  L aq i q
 1q  L aq i 1q  L 22 q L 2 q  L aq i q

Air-gap torque
T e   di q   qi d
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Steady State Analysis Phasor
Representation
For balanced, steady state operation, the stator voltages may
be written as:
e a  E m cos  t   
e b  E m cos  t  2  3   
e c  E m cos  t  2  3   
with
ω = angular velocity = 2πf
α = phase angle of ea at t=0
Applying the d,q transformation,
e d  E m cos  t     
e q  E m sin  t     
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,
e d  E m cos    0 
e q  E m sin    0 
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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,
e d  E t cos    0 
e q  E t sin    0 

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
e d  E t sin  i
e q  E t cos  i
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Steady State Analysis Phasor
Representation (cont'd)

The phasor terminal voltage is given by
~
E t  e d  je q
 E R  jE l
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
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Internal Rotor Angle

Under steady state
e d   
q
 i dR a
  L qi q  i dR a  X qi q  i dR a
Similarly
e q  
d
 i qR a
  X d i d  X ad i fd  i q R a

Under no load, id=iq=0. Therefore,
 q   L qi q  0
 d  L ad i fd
ed  0
e q  L ad i fd
~
E
and t  e d  je q  jL ad i fd

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
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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
e L
di
solving
i  Ke
 Lt
 R

Em
Z

 iR
dt
sin  t     
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
Figure 3.24: RL Circuit
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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
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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
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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
e d  p  d    q  i dR a
e q  p  q    d  i qR a
transformer voltage terms: pψd, pψq
speed voltage terms:   q ,   d

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.
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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
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Synchronous Machine Representation in
System Stability Studies

Stator Transients (pψd, pψq) are usually 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   1 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)
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Equation of Motion (Swing Equation)

The combined inertia of the generator and primemover is accelerated by the accelerating torque:
J
dm
dt
 T a  Tm  T e
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
=
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Equation of Motion (cont'd)

The above equation can be normalized in terms of
per unit inertia constant H
1 J  0m
2
H 
2 VA base
where
a0m = rated angular velocity of the rotor in
mechanical radians per second

Equation of motion in per unit form is
2H
d r
dt
 Tm  T e
where
r 
Tm 
Te 

m
0m
Tm  0 m
VA
= per unit mechanical torque
base
Te  0 m
VA
= per unit rotor angular velocity
= per unit electromechanical torque
base
Often inertia constant M = 2H used
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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 semiheuristic 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)
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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
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
The effects of saturation is represented as
L ad  K sd L adu
(3.182)
(3.183)
L aq  K sq L aqu
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,
 I   at 0   at
K sd 

(3.186)
 at
(3.187)
 at   I
For the nonlinear segment of OCC,  I can be
expressed by a suitable mathematical function:
 I  A sat e
B sat
  at
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  TI

(3.189)
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Open-Circuit Characteristic (OCC)

Under no load rated speed conditions
id  iq   q  e d  0
E t  e q   d  L ad i fd

Hence, OCC relating to terminal voltage and field
current gives saturation characteristic of the d-axis
Figure 3.29: Open-circuit characteristic showing effects of
saturation
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
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 daxis, 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
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
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
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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 E3.1

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 shows results with distinct d- and q-axis
saturation representation
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
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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)
E´

Et
x d
Steady-state Model
 constant field current
 neglect saliency (xd = xq = xs)
Et
Eq
xs
Eq = Xadifd
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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
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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
~ *
S  P  jQ  E t ~I t  E t I t cos   j sin  
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
Fig 5.12: Armature current heating limit
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Field Current Limit

Because of the heating resulting from RfdI2fd 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 E t
X ad i fd sin  i  X s l t cos 
X ad i fd cos  i  E t  X s l t sin 
Therefore
P  E t l t cos  
Q  E t l t sin  
X ad
Xs
X ad
Xs
E t i fd sin  i
2
E t i fd cos  i 
Et
Xs

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
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Field Current Limit
Fig. 5.13: Steady state phasor diagram
Fig. 5.14: Field current heating limit
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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
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End Region Heating Limit
Fig. 5.15: Sectional view end region of a generator
Fig. 5.16: End region heating limit
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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
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Effect of Changes in Terminal Voltage Et
Fig. 5.17: Effect of reducing the armature voltage on the
generator capability curve
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