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Chapter 4

Synchronous Generators

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• In stator, a three-phase winding similar to the one described in chapter 4. Since the main voltage is

induced in this winding, it is also called armature

winding.

• In rotor, the magnetic field is generated either by a

permanent magnet or by applying dc current to rotor winding. Since rotor is producing the main field, it is also

called field winding. Two rotor designs are common:

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Basic Topology

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o Salient-pole rotor with “protruding” poles

o Round or Cylindrical rotor with a uniform air gap

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Sliprings

S

N

S

N

S

N

(a)

NN

S

End View Side View

BR

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The Speed of Rotation of a Synchronous Generator

4

120m

e

n Pf =

Where

fe = electrical frequency, in Hz

nm = mechanical speed of magnetic field, in rpm

= rotor speed, in rpm

P = number of poles

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The Internal Generated Voltage of a Synchronous Generator

• It was shown previously, the magnitude of the voltage induced in a given stator phase was found to be

�� = 2��∅ = ∅�

• The induced voltage is proportional to the rotor flux for a given rotor angular frequency in electrical Radians per

second.

• Since the rotor flux depends on the field current IF, the

induced voltage EA is related to the field current as

shown below. This is generator magnetization curve or the open-circuit characteristics of the machine.

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(a) Plot of flux versus field current for a

synchronous generator. (b) The magnetization curve for the synchronous generator.

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jX���� RA

+

Vϕ

-

ARjX

+ EAR

- +

Enet

-E

A

IA

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The Equivalent Circuit of a Synchronous Generator

3 Reasons why �� ≠ ∅

1. Armature Reaction distortion of the air gap flux

produced by the stator created magnetic field.2. Self Inductance of the armature coils (X).3. Resistance of the armature coils (R).

The Equivalent Circuit of a Synchronous Generator

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Armature Reaction: If a load is connected to the

stator windings a current will flow resulting in a magnetic field Bs. This varying field produces a voltage in the stator windings Es so that

∅ = �� + ��

The net magnetic field in the air gaps is ���� = �� + ��

We can represent the armature reaction by a reactance X of the form

�� = −����

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• The two reactances (self + armature) may be combined into a single reactance called the synchronous reactance

of the machine:

The per phase equivalent circuit of a synchronous generator.

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S ARX X X= +

����

The Equivalent Circuit of a Synchronous Generator The Equivalent Circuit of a 3Φ Synchronous Generator

The Phasor Diagram of a Synchronous Generator

• The Kirchhoff’s voltage law equation for the armature circuit is

• The phasor diagrams for unity, lagging, and leading

power factors load are shown here:

Unity Lagging Leading

The angle between ∅�����is known as the

torque angle δ

< � =< �� −< ∅

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A S(R + jX )

A AE V I

ϕ= +

Power and Torque in Synchronous Generators

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Figure 4-15: The power-flow diagram of a synchronous generator.

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4

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• The input mechanical power is given by

• The power converted from mechanical to electrical

power is given by

�� ! " = ��#$ % &� ��������

• The real and reactive electrical output power is given by

( )conv ind m A A

P E I Cosτ ω γ= =

in app mP τ ω=

3 ( )

3 ( )

OUT A

OUT A

P V I Cos

Q V I Sin

φ

φ

θ

θ

=

=

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• If the armature resistance is ignored (Since RA << XS),

Note: 1.Pout = Pconv for Ra = 0.

2. Pout (max) occurs for δ = 90°.

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( )( )

3 ( )

A

A

S

A

CONV OUT

S

E SinI Cos

X

V E SinP P

X

δθ

φ δ

=

=����

• Induced torque of the generator is given by

• Note that this equation offers an alternative form for the

induced torque presented before by

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3 ( )A

ind

m S

V E Sin

X

φ δτ

ω=

( )ind net R

KB B Sinτ δ=

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The Effect of Load Changes on a Synchronous Generator Operating Alone

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The behavior of a synch. Gen under load depends on

the power factor of the load and whether the generator is acting alone or in parallel with other synchronous generators.

An increase in load is an increase in the real and/or

reactive power supplied by the generator. Such a load increase results in an increase in ��. With constant field resistance the field current and field flux will remain constant. If also the prime mover maintains constant speed ω, then the internal voltage �� =

∅�is also constant.

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The Effect of Load Changes on a Synchronous Generator Operating Alone

• At constant field current and rotor speed

a. Lagging p.f. with load at same p.f.

�� (�)! �* *�&*�+ ��#$ ,�. !. &. ∅

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The Effect of Load Changes on a Synchronous Generator Operating Alone

• At constant field current and rotor speed

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The Effect of Load Changes on a Synchronous Generator Operating Alone

• At constant field current and rotor speed

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The Effect of Load Changes on a Synchronous Generator Operating Alone

• At constant field current and rotor speed

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The Effect of Load Changes on a Synchronous Generator Operating Alone

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Synchronous Generator Example 1

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Synchronous Generator Example 1

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Synchronous Generator Example 1

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Synchronous Generator Example 1

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Synchronous Generator Example 1

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Synchronous Generator Example 2

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Ex 2

Synchronous Generator Example 2

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Ex 2

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Synchronous Generator Example 2

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Ex 2

Synchronous Generator Example 2

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Ex 2