synchrounous machine

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Chapter 5 Synchronous Machines

Main features of synchronous machines:

A synchronous machine is an ac machine whose speed under steady-state conditions is

proportional to the frequency of the current in its armature.

The rotor, along with the magnetic field created by the dc field current on the rotor,rotates at the same speed as, or in synchronism with, the rotating magnetic field produced

by the armature currents, and a steady torque results.

Figure 4.12 Schematic views of three-phase generators: (a) two-pole, (b) four-pole, and

(c) Yconnection of the windings.

5.1 Introduction to Polyphase Synchronous Machines

Synchronous machines: Armature winding: on the stator, alternating current.

Field winding: on the rotor, dc power supplied by the excitation system.

Cylindrical rotor: for two- and four-pole turbine generators.

Salient-pole rotor: for multipolar, slow-speed, hydroelectric generators and for most

synchronous motors.

Acting as a voltage source:

Frequency determined by the speed of its mechanical drive (or prime mover).

The amplitude of the generated voltage is proportional to the frequency and the field

current.

tNk

tNk

mepphw

mpphwa

cos

2

poles

cos

=

= (4.45)

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mme2

poles

= (4.46)

tNktdt

dNk

dt

de mepphwmeme

p

phw

a

a

sincos

== (4.47)

tNke mepphwmea sin= (4.48)

pphwmepphwmemax 2 == NkfNkE (4.49)

pphwmepphwmerms 22

2== NkfNkfE

(4.50)

Synchronous generators can be readily operated in parallel: interconnected power

systems.

When a synchronous generator is connected to a large interconnected system containing

many other synchronous generators, the voltage and frequency at its armature terminals

are substantially fixed by the system.

It is often useful, when studying the behavior of an individual generator or group of

generators, to represent the remainder of the system as a constant-frequency,constant-voltage source, commonly referred to as an infinite bus.

Analysis of a synchronous machine connected to an infinite bus.

Torque equation:

RFfR

2

sin2

poles

2

FT

= (5.1)

where

= resultant air-gap flux per poleR

fF = mmf of the dc field winding

RF = electric phase angle between magnetic axes of R and fF

The minus sign indicates that the electromechanical torque acts in the direction to

bring the interacting fields into alignment.

In a generator, the prime-mover torque acts in the direction of rotation of the rotor,

and the electromechanical torque opposes rotation. The rotor mmf wave leads the

resultant air-gap flux.

In a motor, the electromechanical torque is in the direction of rotation, in opposition

to the retarding torque of the mechanical load on the shaft.

Torque-angle curve: Fig. 5.1.

Figure 5.1 Torque-angle characteristics.

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An increase in prime-mover torque will result in a corresponding increase in the

torque angle.

: pull-out torque at . Any further increase in prime-mover

torque cannot be balanced by a corresponding increase in synchronous

electromechanical torque, with the result that synchronism will no longer be

maintained and the rotor will speed up. loss of synchronism, pulling out of step.

maxTT =o90RF =

5.2 Synchronous-Machine Inductances; Equivalent Circuits

Figure 5.2 Schematic diagram of a two-pole,

three-phase cylindrical-rotor synchronous machine.

5.2.1 Rotor Self-Inductance

5.2.2 Stator-to-Rotor Mutual Inductances

5.2.3 Stator Inductances; Synchronous Inductance

5.2.4 Equivalent Circuit Equivalent circuit for the synchronous machine:

Single-phase, line-to-neutral equivalent circuits for a three-phase machine operating under

balanced, three-phase conditions.

sL = effective inductance seen by phase a under steady-state, balanced three-phase

machine operating conditions.

ses LX = : synchronous reactance

aR = armature winding resistance

afe = voltage induced by the field winding flux (generated voltage, internal voltage)

aI = armature current

av = terminal voltage

Motor reference direction:

faasaaa EIjXIRV ++= (5.23)

Generator reference direction:

faasaaa EIjXIRV += (5.24)

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Figure 5.3 Synchronous-machine equivalent circuits:

(a) motor reference direction and (b) generator reference direction.

XXX += als (5.25)

alX = armature leakage reactance

X = magnetizing reactance of the armature winding

RE = air-gap voltage or the voltage behind leakage reactance

Figure 5.4 Synchronous-machine equivalent circuit showing air-gap and

leakage components of synchronous reactance and air-gap voltage.

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5.4 Steady-State Power-Angle Characteristics

The maximum power a synchronous machine can deliver is determined by the maximum

torque that can be applied without loss of synchronism with the external system to which it is

connected. Both the external system and the machine itself can be represented as an impedance in

series with a voltage source.

Figure 5.11 (a) Impedance interconnecting two voltages; (b) phasor diagram.

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cos22 IEP = (5.34)

Z

EEI 21

= (5.35)

jeEE 11

= (5.36)

22 EE = (5.37)

ZjeZXjRZ =+= (5.38)

( ) ZZZ

jj

j

j

je

Z

Ee

Z

E

eZ

EeEIeI

=

== 2121 (5.39)

( ) ( ZZZ

E

Z

EI = coscoscos 21 ) (5.40)

( )2

2

2212 cos

Z

RE

Z

EEP Z = (5.41)

( ) 22

221

2 sin Z

RE

Z

EE

P Z += (5.42)

where

==

X

RZZ

1tan90 o (5.43)

( )2

2

1211 sin

Z

RE

Z

EEP Z = (5.44)

Frequently, 0and, , leads and power flows from source to .1E 2

E 1E 2

E

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Figure 5.12 Equivalent-circuit representation of

a synchronous machine connected to an external system.

Note that ,21EEP 1 XP , 21max EEP , and .

1

max

XP

In general, stability considerations dictate that a synchronous machine achieve

steady-state operation for a power angle considerably less than .o90

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Figure 5.14 Equivalent circuits and phasor diagrams for Example 5.7.

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5.3 Open- and Short-Circuit Characteristics5.3.1 Open-Circuit Saturation Characteristic and No-Load Rotational

Losses

Figure 5.5 Open-circuit characteristic of a synchronous machine.

5.3.2 Short-Circuit Characteristic and Load Loss

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Figure 5.6 Typical form of an open-circuit core-loss curve.

( )saafa jXRIE += (5.26)

Figure 5.7 Open- and short-circuit characteristics of a synchronous machine.

Figure 5.8 Phasor diagram for short-circuit conditions.

( )laaaR jXRIE += (5.27)

aca

aga

usI

VX

,

,

, = (5.28)

a

ratedas

I

VX

=, (5.29)

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Figure 5.9 Open- and short-circuit characteristics showing

equivalent magnetization line for saturated operating conditions.

fOfO

=SCR (5.30)

AFSC

AFNLSCR = (5.31)

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Figure 5.10 Typical form of short-circuit load loss and stray load-loss curves.

t

T

R

R

t

T

+

+=

5.234

5.234(5.32)

( )2eff,

ntaturecurrecircuitarmshort

lossloadcircuitshort

=aR (5.33)

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5.5 Steady-State Operating Characteristics

Figure 5.15 Characteristic form of synchronous-generator compounding curves.

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Figure 5.16 Capability curves of an 0.85 power factor, 0.80 short-circuit ratio,

hydrogen-cooled turbine generator. Base MVA is rated MVA at 0.5 psig hydrogen.

aaIVQP =+=22powerApparent (5.48)

Figure 5.17 Construction used for the derivation of a synchronous generator capability curve.

asa IjXVQjP += (5.49)

asafa IjXVE += (5.50)

222

2

=

++

s

faa

s

a

X

EV

X

VQP (5.51)

Figure 5.18 Typical form of synchronous-generator V curves.

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Figure 5.19 Losses in a three-phase, 45-kVA, Y-connected,

220-V, 60-Hz, six-pole synchronous machine.

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5.6 Effects of Salient Poles; Introduction to Direct-And

Quadrature-Axis Theory5.6.1 Flux and MMF Waves

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Figure 5.20 Direct-axis air-gap fluxes in a salient-pole synchronous machine.

( )33,3 3cos2 += tVE ea (5.52)

( )( ) ( )3333,3 3cos21203cos2 +=+= tVtVE eeb o (5.53)

( )( ) ( )3333,3 3cos21203cos2 +=+= tVtVE eeco

(5.54)

Figure 5.21 Quadrature-axis air-gap fluxes in a salient-pole synchronous machine.

Figure 5.22 Phasor diagram of a salient-pole synchronous generator.

5.3.2 Phasor Diagrams for Salient-Pole Machines

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Figure 5.23 Phasor diagram for a synchronous generator showing

the relationship between the voltages and the currents.

dlad XXX += (5.55)

qlaq XXX += (5.56)

Figure 5.24 Relationships between component voltages in a phasor diagram.

ba

ab

oa

ao =

(5.57)

aqa

q

qq

IXII

XI

oaba

abao ==

= (5.58)

qqddaaafa IjXIjXIRVE +++= (5.59)

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Figure 5.25 Generator phasor diagram for Example 5.9.

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