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:

MMFa Ki a cos

ia Im coss t

MMFb Ki b cos

ib Im cos s t

MMFc Ki c cos

ia lm cos s t

MMFtotal MMFa MMFb MMFc

3
KI m cos s t
2

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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
dt

e1

d
ri
dt

Li

The inductance, by definition, is equal to flux linkage

per unit current
LN

N2P
i

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

d1
r1i1
dt

e2

d2
r2i2
dt

1 L11i1 L 21i2
2 L 21i1 L 22i2

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
Iaa L al Igaa
L aa0 L aa2 cos 2
2

Iab Iba L ab0 L ab2 cos 2

L ab0 L ab2 cos 2

3

The mutual inductances between stator and rotor

circuits vary due to relative motion between the
windings
Iafd L afd cos
Iakd L akd cos

Iakq 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

da
R aia pa R aia
dt

a laaia labib lacic lafdifd lakdikd lakqikq

Rotor circuit voltage and flux linkage equations

e fd pfd R fdifd
0 pkd R kdikd
0 pkq R kqikq

fd L ffdifd L fkdikd

2
2

L afd ia cos ib cos

ic cos

3
3

kd L fkdifd Lkkdikd

2
2

L afd ia cos ib cos

ic cos

3
3

kq Lkkdikq

2
2

L akq ia sin ib sin

ic sin

3
3

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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

2
2

cos
cos

cos
3
3

id
i 2 sin sin 2 sin 2
q 3
3
3

i
1
0
1
1

2
2
2

ia
ib
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 qr R aid
e q p q dr R aiq
e 0 p 0 R ai0

Rotor voltage equations

e fd p fd R fdifd
0 p 1d R 1di1d
0 p 1q R 1qi1q
0 p 2q R 2qi2q

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P.U. Machine Equations in dqo Reference

Frame (cont'd)

Stator flux linkage equations

d Lad Ll id Lad ifd Lad i1d

q Laq Ll iq Laqi1q Laqi 2 q
0 L0 i0

Rotor flux linkage equations

fd L ffdifd L f 1di1d L adid

1d L f 1difd L11di1d L adid
1q L11qi1q L aqi2q L aqiq
1q L aqi1q L 22qL2q L aqiq

Air-gap torque

T e diq qid

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Steady State Analysis Phasor

Representation
For balanced, steady state operation, the stator voltages may
be written as:
e a Em cost

eb Em cost 2 3
e c Em cost 2 3

with
= angular velocity = 2f
= phase angle of ea at t=0
Applying the d,q transformation,
ed Em cost
eq Em sint

At synchronous speed, the angle is given by = t + 0

with = value of at t = 0

Substituting for in the expressions for ed and eq,

ed Em cos 0
eq Em 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 is the same as the angular frequency
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,
ed Et cos 0

eq Et sin 0

The above quantities can be represented as phasors with

d-axis as real axis and q-axis as imaginary axis

Denoting i, as the angle by which q-axis leads E

ed Et sin i
eq Et cos i
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Steady State Analysis Phasor

Representation (cont'd)

The phasor terminal voltage is given by

~
in the d-q coordinates
E t ed jeq
E R jEl

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 = t applies to both

in the case of machines, = rotor speed
in the case of a.c. circuits, = angular frequency

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Internal Rotor Angle

Under steady state

ed q idR a
Lqiq idR a X qiq idR a

Similarly
e q d iqR a
X did X adifd iqR a

Under no load, id=iq=0. Therefore,

q Lqiq 0
d L adifd
ed 0
e q L adifd

~
E
and t e d je q jL adifd

Under no load, Et has only the q-axis component

and i=0. As the machine is loaded, i increases.
Therefore, i 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 (t+). If switch "S" is closed at t=0,
the current is given by

eL
solving

Lt

i Ke

di
iR
dt

Em
sint
Z

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 idR a
e q p q d iqR a
transformer voltage terms: pd, pq
speed voltage terms: q , d

The transformer voltage terms represent stator

transients:
stator flux linkages (d, q) cannot change
instantaneously
result in dc offset in stator phasor current

If only fundamental frequency stator currents are of

interest, stator transients (pd, pq) 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 (pd, pq) 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:

dm
Ta Tm Te
dt

where

Tm =

mechanical torque in N-M

Te =

electromagnetic torque in N-m

combined moment of inertia of generator

and turbine, kgm2

am =

angular velocity of the rotor in mech. rad/s

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 J20m
H
2 VAbase
where
a0m = rated angular velocity of the rotor in
mechanical radians per second

Equation of motion in per unit form is

2H

d r
Tm Te
dt

where

m
0m

Tm

Tm0m
VAbase

= per unit mechanical torque

Te

Te 0m
VAbase

= per unit electromechanical torque

= per unit rotor angular velocity

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 sdL adu
L aq K sqL aqu

(3.182)
(3.183)

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 at0 at

(3.186)

at
at I

(3.187)

K sd

For the nonlinear segment of OCC, I can be

expressed by a suitable mathematical function:

I AsateBsat at TI

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(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 adifd

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

i (deg)

ifd (pu)

1.0

0.889

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

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

i (deg)

ifd (pu)

0.667

0.889

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

0.660

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)

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 It cos j sin

where 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 Et
X adifd sin i X slt cos
X adifd cos i E t X slt sin

Therefore
X ad
E tifd sin i
Xs
X
E2
Q E tlt sin ad E tifd cos i t
Xs
Xs

P E tlt cos

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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