electric machine Chapter 5 Synchronous Machines

electric machine

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) Y connection 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

t N

k

t N

k

me p

ph w

m p

ph w a

ω

ωλ

cos

2

polescos

=

(4.45)

m me

2poles ω

k t

dt

d N k dt

d

e a w ph p me me w ph p me

a = λ = Φ cosω −ω Φ sinω (4.47)

t N

k

e a =−ωme w phΦpsinωme (4.48)

p ph w me p

ph w me max = k N Φ =2 f k N Φ

E ω π (4.49)

p ph w me p

ph w me

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:

RF f

R

2

sin2

δ = electric phase angle between magnetic axes of ΦR 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

2

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

max

T

⇒

§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

s

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

machine operating conditions

s e

a ˆ ˆ ˆ

ˆ R I jX I E

V = + + (5.23) Generator reference direction:

f a a s a a

ˆ R I jX I E

V =− − + (5.24)

Figure 5.3 Synchronous-machine equivalent circuits:

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

ϕ

X X

E = 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

4

§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

Eˆ1 = 1 (5.36)

2 2

ˆ E

E = (5.37)

Z j

e Z X j R

Z = + = φ (5.38)

Z

j j

j

j j

e Z

E e

Z

E e

Z

E e E Ie

E

Icosφ = 1 cosδ −φ − 2 cos−φ ) (5.40)

2 1

2 cos

Z

R E Z

E E

P = δ −φZ − (5.41)

2 1

2 sin

Z

R E Z

E E

P = δ +αZ − (5.42) where

1 sin

Z

R E Z

E E

P = δ −φZ − (5.44) Frequently, R<< Z, Z ≈ X andαZ ≈0,

δsin

2 1 2 1

X

E E P

P = = (5.45)

Equation (5.45) is commonly referred to as the power-angle characteristic for a

synchronous machine

The angle δ is known as the power angle

Note that E1 and E2 are the line-to-neutral voltages

For three-phase systems, a factor “3” shall be placed in front of the equation

The maximum power transfer is

X

E E P

max , 2 max ,

1 = = (5.46) occurring when δ =±90o

If δ >0, Eˆ1 leadsEˆ2 and power flows from source Eˆ1 to Eˆ2

When δ <0, Eˆ1 lagsEˆ2 and power flows from source Eˆ2 to Eˆ1

Consider Fig 5.12 in which a synchronous machine with generated voltage

and synchronous is connected to a system whose Thevenin equivalent is a voltage source in series with a reactive impedance The power-angle characteristic can be written

EQ f a

X X

V E P

+

= (5.47)

6

Figure 5.12 Equivalent-circuit representation of

a synchronous machine connected to an external system

Note that P∝E1E2, P ∝ X− 1, Pmax ∝E1E2, and Pmax ∝ X−1

In general, stability considerations dictate that a synchronous machine achieve steady-state operation for a power angle considerably less than 90o

Figure 5.14 Equivalent circuits and phasor diagrams for Example 5.7

8

10

§5.3 Open- and Short-Circuit Characteristics

§5.3.1 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

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

a f

Eˆ = ˆ + (5.26)

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

Figure 5.8 Phasor diagram for short-circuit conditions

ag a u s

I

V X

,

, , = (5.28)

a

rated a s

I

V X

′

= ,

(5.29)

12

Figure 5.9 Open- and short-circuit characteristics showing equivalent magnetization line for saturated operating conditions

f O

f O

′′

′

=SCR (5.30)

AFSCAFNLSCR = (5.31)

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

t

T R

5.234

(5.32)

eff ,

ntaturecurrecircuitarm

short

lossloadcircuit short

§5.5 Steady-State Operating Characteristics

Figure 5.15 Characteristic form of synchronous-generator compounding curves

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

a

a I V Q

P + =

powerApparent (5.48)

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

a s

a jX I V

Q j

P− = ˆ + ˆ (5.49)

a s a f

Eˆ = ˆ + ˆ (5.50)

2 2

2 2

s

f a a s

a

X

E V X

V Q

P (5.51)

Figure 5.18 Typical form of synchronous-generator V curves

16

Figure 5.19 Losses in a three-phase, 45-kVA, Y-connected, 220-V, 60-Hz, six-pole synchronous machine

§5.6 Effects of Salient Poles; Introduction to Direct-And

Quadrature-Axis Theory

§5.6.1 Flux and MMF Waves

18

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

3 ,

3 = 2V cos3ω t−120 +φ = 2V cos3ω t+φ

E c e o e (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

Figure 5.23 Phasor diagram for a synchronous generator showing the relationship between the voltages and the currents

d l a

X = + ϕ (5.55)

q l a

q q

I X I I

X I

ˆˆ

ˆ

ˆoaba

aba

Eˆ = ˆ + ˆ + ˆ + ˆ (5.59)

20

Figure 5.25 Generator phasor diagram for Example 5.9

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