Dr. Amr AbdAllah 1

Electric Machines IIIA

COURSE EPM 405A

FOR

4th Year Power and Machines

ELECTRICAL DEPARTMENT

Lecture 05

Introduction:A synchronous machine is a doubly excited machine. Its rotor poles are

excited by a dc current and its stator winding are connected to the ac supply*.

Structure:- Three phase distributed windings on the stator (similar to the IM) and is

called armature windings.

- Field winding on the rotor carrying direct current fed normally from an external dc source through slip rings and brushes.

Dr. Amr AbdAllah 2

Synchronous Machine Modeling

a‘

b c

ab‘c‘

Stator

N

S

Salient Pole rotor

N

S

Non-salientpole rotor

a b

Slip rings

Bru

shesc

3 phase AC + -DC excitation

Introduction:Synchronous machines are broadly divided into two groups:

1- High speed machines with cylindrical (non-salient) rotorsUsed in large generators (several hundred mega watts) with two or

sometimes four poles and are usually driven by steam turbines. The rotors are long and have small diameters. Non-salient pole rotor has one distributed winding and an essentially uniform airgap.

2- Low speed machines with salient pole rotorsSalient pole rotors have concentrated windings on the poles and a non-

uniform airgap. Salient pole generators have a large number of poles (sometimes as many as 50) and operates at lower speeds. These generators are rated for ten or hundred mega watts and are driven by water turbines. The rotors are shorter but have larger diameter.

Dr. Amr AbdAllah 3

Synchronous Machine Modeling

Nature of Inductance:1- Mutual inductances and self inductance of stator windings

depends on rotor position (r) and are periodic every 180° (that is equivalent to be dependent on 2r

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Synchronous Machine Modeling

r

as axis

cs axis

bs axis

fd axis

fq axisr

f

Kq

Kq‘

r

as axis

cs axis

bs axis

fd axis

fq axisr

fKq

Kq‘

POSITION 1 POSITION 2

Nature of Inductance:2- Mutual inductances between rotor and stator windings

depends on rotor position (r) and are periodic every 360° (because the flux linkage change direction at the 180°)

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Synchronous Machine Modeling

r

as axis

cs axis

bs axis

fd axis

fq axisr

f

Kq

Kq‘

r

as axis

cs axis

bs axis

fd axis

fq axisr

fKq

Kq‘

POSITION 1 POSITION 2

Value is only repeated after 360°

Value is repeated each 180°

Nature of Inductance:3- Mutual and self inductances between rotor windings are

independent on rotor position (r).

A general equation that describes the mutual coupling between any two windings of a salient pole machine can then be given as:

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Synchronous Machine Modeling

POSITION 1 POSITION 2

fKd

Kq‘

f ‘

Kd‘

Kq

fKd

f ‘Kd‘

Kq

Kq‘

)cos()-cos(20 yxyx LLxyL

Notes on the salient pole machine inductances:

All the salient pole machine inductance can be computed from the relation:

Definitions

Dr. Amr AbdAllah 7

Synchronous Machine Modeling

)cos()-cos(20 yxyx LLxyL

22

0)

2()

2(2

lrL yNxN

10)

2()

2(0 lrL yNxN

)11

(maxmin2

11 gg

)11

(maxmin2

12 gg

102)

2( lrL s

A

N

22

02)

2(

lrL s

B

N

)22

1(0

)2

()2

(

lrL fssfd

NN )22

1(0

2)2

(

lrL fmfd

N

Machine inductances:

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Synchronous Machine Modeling

Machine stator inductances:

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Synchronous Machine Modeling

)2402cos(

)1202cos(

)2cos(

csc

rBLALlsLLrBLALlsLLrBLALlsLL

s

bsbs

asas

0;90

120

;240;

21 kqkqkdfd

rcs

rbsras

)1202cos(

2

1

)2cos(2

1

)1202cos(2

1

rBLALL

rBLALL

rBLALL

csas

bscs

asbs

Machine rotor self inductances:

Similarly,

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Synchronous Machine Modeling

mfdLlLLfN

LLlL

LLlLL

fd

fd

fd

fdfd

fdfd

yNxN

with

)2(

)9090cos(2)0cos(

0

0

22

11

22

11

mkqlkq

mkqlkq

mkdlkd

LLL

LLLLLL

kqkq

kqkq

kdkd

)2

( 210

2dd )

2(

lrL k

mk

N

)2

( 210

21q1q )

2(

lrL k

mk

N)

2( 2

1022q

2q )2

(

lrL kmk

N

0;90

120

;240;

21 kqkqkdfd

rcs

rbsras

Machine rotor mutual inductances:

Similarly,

Note

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Synchronous Machine Modeling

mkdkd

fmfd

f

kd

kdyfx

LN

NL

N

NL

NNNN

LL

LLL

fdkd

fdkd

; with

)(

)9090cos()0cos(

2

2

0

0

22

11

1

2

21

2

2

21

21

; with

)(

)00cos()0cos(

0

0

mkqkq

kqmkq

kq

kq

kqykqx

LN

NL

N

NL

NNNN

LL

LLL

kqkq

kqkq

02121 kdkqkdkqkfkqkfkq LLLL

0;90

120

;240;

21 kqkqkdfd

rcs

rbsras

Machine stator-rotor mutual inductances:

Similarly,

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Synchronous Machine Modeling

)120sin(

)120sin(

)sin(

; equations above theof allin )120sin()(

)90120cos()90120cos()120sin()(

)90240cos()90240cos()sin()(

)90cos()90cos(

thus

2

2

2

2

2

2

0

0

0

0

0

0

rLLrLLrLL

NN

rLLrLrLL

rLLrLrLL

rLLrLrLL

sfd

sfd

sfd

csfd

bsfd

asfd

acfd

bsfd

asfd

fyNsxN

0;90

120

;240;

21 kqkqkdfd

rcs

rbsras

)22

1(0

)2

()2

(

lrL fssfd

NN

Machine stator-rotor mutual inductances:

where

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Synchronous Machine Modeling

)120sin()120sin(

)sin(

rLLrLLrLL

skd

skd

skd

acfd

bsfd

askd

)22

1(

)22

1(

)22

1(

0

0

0

)2

()2

(

)2

()2

(

)2

()2

(

22

11

lrL

lrL

lrL

kqsskq

kqsskq

kdsskd

NN

NN

NN

)120cos(

)120cos(

)cos(

)120cos(

)120cos(

)cos(

2

2

2

1

1

1

2

2

2

1

1

1

rLLrLLrLLrLLrLLrLL

skq

skq

skq

skq

skq

skq

cskq

bskq

askq

cskq

bskq

askq

0;90

120

;240;

21 kqkqkdfd

rcs

rbsras

Machine voltage equations in machine variables:

Note that the negative sign in the stator voltage equations is to indicate the currents are assumed to be positive out of the terminals. This assumption is considered since the synchronous machine is generally operated as a generator.

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Synchronous Machine Modeling

qdrqdrrqdr

abcsabcssabcs

pp

λirvλirv

kdfdkqkqT

qdr

csbsasT

abcs

fffffff )(f

)(f

21

Machine voltage equations in machine variables:The flux linkage equation can thus be written as:

Again the negative sign preceding the stator current is to describe the positive assumption of the current out of the machine terminals, which is in the direction of negative flux linkages.

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Synchronous Machine Modeling

qdr

abcs

rrs

srs

qdr

abcs

ii-

L LL L

λλ

Tsrrs L L

Machine voltage equations in machine variables:The inductance matrix for the stator circuits Ls is given by:

The mutual coupling matrix between stator and rotor windings Lsr is give as:

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Synchronous Machine Modeling

Machine voltage equations in machine variables:The rotor circuits inductance matrix Lr is given by:

It is convenient to define the following inductances for synchronous machines:

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Synchronous Machine Modeling

)(2

3

)(2

3

BAmd

BAmq

LLL

LLL

Notice that for cylinderical rotor (non-salient LB=0This means that Lmq=Lmd=3/2 LA where =1/g (uniform air-gap) with g

lrL sA

N 02)

2(

Machine inductances using Lmq and Lmd inductances:

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Synchronous Machine Modeling

mqLN

Nlr

NNL

mqLN

Nlr

NNL

s

kqkqsskq

s

kqkqsskq

)(3

2)

22

1()2

()2

(

)(3

2)

22

1()2

()2

(

222

111

0

0

mdLN

Nlr

NNL

mdLN

Nlr

NNL

s

kdkdsskd

s

ffssfd

)(3

2)

22

1()2

()2

(

)(3

2)

22

1()2

()2

(

0

0

Machine inductances using Lmq and Lmd inductances:

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Synchronous Machine Modeling

mds

kdkmk

mds

ffmfd

LN

Nlr

NL

LN

Nlr

NL

22dd

22

)(3

2)

2()

2(

)(3

2)

22

1()2

(

210

0

mqs

kkmk

mqs

kkmk

LN

Nlr

NL

LN

Nlr

NL

22q22q2q

21q21q1q

)(3

2)

2()

2(

)(3

2)

2()

2(

210

210

Referring rotor variables to stator:The following relations are needed to refer the rotor

variables to the stator:

Using these equations for referring the rotor variables to stator we get:

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Synchronous Machine Modeling

kdfdKqKqj or , , becan 21

qdr

abcs

rrs

srs

qdr

abcs

ii-

L LL L

λλ Tsrrs L

3

2L

Referring rotor variables to stator:L’sr and L’r are given as:

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Synchronous Machine Modeling

kdfdKqKqj or , , becan 21ljj

slj L

N

NL

2

2

3

Machine voltage equations with variables referred to stator:

The elements of the matrix r’r is given by:

The voltage equations expressed above can be adapted for the positive direction of the current being going to the machine terminals (that motor operation) just by removing the negative sign preceding the vector of stator currents.

Dr. Amr AbdAllah 22

Synchronous Machine Modeling

qdr

abcs

rrT

sr

srss

qdr

abcs

pp

pp

ii-

Lr )L(3

2

L Lr

vv

jj

sj r

N

Nr

2

2

3kdfdKqKqj or , , becan 21

Dr. Amr AbdAllah 23

NEXT LECTURE

CHAPTER III Cont’d1- Torque equation.2- Synchronous Machine Dynamic Modeling in

qd0 reference frames

Dr. Amr AbdAllah 24

Derivation

mdLN

NL

mdLN

NL

mqLN

NL

mqLN

NL

s

kdskd

s

fsfd

s

kqskq

s

kqskq

)(3

2

)(3

2

)(3

2

)(3

2

22

11

))cos()()(

)cos()()(

)cos()()(

)cos()()(

))1202cos(5.0())1202cos(5.0(

))2cos(((

2

3

3

22

3

3

22

3

3

22

3

3

2

22

22

11

11

rskdkd

skd

s

kd

rsfdf

sf

s

f

rskqkq

skq

s

kq

rskqkq

skq

s

kq

rBAcs

rBAbs

rBAlsasass

asassas

LN

Ni

N

N

LN

Ni

N

N

LN

Ni

N

N

LN

Ni

N

NLLiLLi

LLLipirpλirv

Dr. Amr AbdAllah 25

Derivation

mqLN

N

N

N

mkqLN

NL

mqLN

NL

mqLN

NL

mqLN

NL

s

kq

kq

kq

kq

kqkqkq

s

kqmkq

s

kqskq

s

kqskq

21

1

2

1

221

211

22

11

)()(3

2

1)(3

2

)(3

2

)(3

2

)(3

2

)())((2

3)(

3

2

)()()(2

3)(

3

2

)))(3

2)120cos()(

2

3(

))(3

2)120cos()(

2

3(

)(3

2)cos()(

2

3(()(

3

2)(

2

3

121

122

2

111

2

11

1

11

1

11

1

11

11

11

1

1111

s

kqkqkq

kq

s

kq

skq

s

kq

s

kqmkqlkq

kq

skq

s

kq

s

kqrskq

kq

scs

s

kqrskq

kq

sbs

s

kqrskq

kq

saskq

s

kqkq

kq

s

kqkqkqkq

N

NL

N

N

N

Ni

N

N

N

NLL

N

Ni

N

N

N

NL

N

Ni

N

NL

N

Ni

N

NL

N

Nipi

N

Nr

N

N

pλirv

Lmq

This is the reason that [Lrs‘]=2/3 [Lsr‘]T

Lmq

Lmq