11 self and mutual inductances

8/3/2019 11 Self and Mutual Inductances

1/28

Self and Mutual Inductances for

uSynchronous Machine with

Round Rotor

Double Layer Lap Winding


2/28

Cross Section Diagram

baxisd

qaxis daxis

mad

m

a

aaxis

caxis


3/28

Stator WindingFractional Pitch

m

(exaggerated end turns)2

3m

a axis

a2

m

m

a

m

P

2

a axis

q=2q=4

q coils per group


4/28

Self and Mutual Inductances (1)

ib(t)mad d

ia(t)

am

aax s

ic(t)


5/28

Self and Mutual Inductances (2)

Alsccbbaa LLLLLLinear ModelBalanced Winding

)cos( mesfaf

scabcab

LL

)3

2

cos(

mesfbf LL

)2

cos(

mesfcf LL

mme

P

2

lsL is leakage inductance of armature phase A winding which is about10% of the maximum self inductance.


6/28

Flux Linkage (1)

)(

fafcbsaaa

fafcacbabaaaa

iLiiMiL

iLiLiLiL

)(

fbfcasbaa

fbfcbcbbbabab

iLiiMiLiLiLiLiL

)(

fcfbascaa

fcfcccbcbacac

iLiiMiL

iLiLiLiL, f .

cos[

measfff

ccfbbfaaffff

iLiL

iLiLiLiL

)]3

2cos()3

2cos( mecmeb ii

LLL

lf

L is leakage inductance of field winding.


7/28

Flux Linkage (2)

a

mesfsaas

mesfssaa

a iLMLM

LMML

)2

cos(

cos

f

c

b

mesfaass

f

c

b

i

iLLMM

22

)3

2cos(

fmesfmesfmesf

33


8/28

Flux Linkage (3)

iLiL

0 cba iiiY connected without neutral return or balanced connected :

bbsaab

mefsfas

iLiML

iLiL

)(

)cos(

mefsfbs iLiL )

3

2cos(

mefsfcs

fcfcsaac

iLiL

)3

2cos(

measffff

ii

iLiL

2cos

2cos

cos[

sAlssaas MLLMLL

33


9/28

Flux Linkage (4)

0 cba iiiWhen

a

mesfs

mesfs

a iLLLL

)2

cos(00cos00

f

c

b

mesfs

f

c

b

i

iLL

22

)3

2cos(00

fmesfmesfmesf

33


10/28

Self Inductance of Stator Winding

If we apply current in Phase A winding, then the magnetic field for fundamentalharmonic is:

aPN 4 0 This equation is true no matter how those P groups

aa

eff

aPg 2

Now, we can calculate flux in Phase A winding from its own current.

. a .Na is effective number of turns connected in series per phase.

pkaapkaaa

NN,,

)0cos( P

DlB pka

pka

,

,

2 a

a

eff

pka i

P

N

g

B

4 0,

aa

aa

aa iP

NDli

P

N

P

DlN

2

00842

where

2

08

NDl

L aaA

effa

Alaccbbaa LLLLL


11/28

Mutual Inductance betweenStator Windings

If we apply current in Phase B winding, then the magnetic field is:

24 0 a PN

32

ab

eff

b

PgNow, we can calculate flux linkage in Phase A winding from Phase B current.

)

3

2cos(| ,windingBPhasefrom

pkbaa N

P

pkb

pkb

,

, ba

eff

pkb iPg

B

0,

where

ba

eff

ba

eff

aa iP

N

g

DliP

N

gP

DlN 00 44221

2

42

0 Aa

effb

as

L

P

N

g

Dl

iM


12/28

Mutual Inductance between Stator andRotor Field Winding

If we apply current in rotor field winding, then when rotor is moving,the magnetic field in airgap from rotor field winding is:

4 PN mad mme

P

2

Now, we can calculate flux in Phase A winding from field current.

)

2

cos( dfeff

f i

Pg

B

Define: fwff NkN Effective numberof turns on fieldwinding.

mepkfaa N cos| ,windingfieldfrom

DlB2 N 4

me

P

p

pkf

,

,

fafi

NNDli

NDl

cos

842cos 00

f

eff

pkfPg

,

w ere

me

effeff

meaa

PgPgP 2

mesfme

faaaf L

NNDlL

coscos

82

0

efff

2

0

8PNN

gDlL fa

eff

sfwhere


13/28

Self Inductance of Rotor Field Winding

For the magnetic field from rotor field winding is:

cos4 0 f PiN

B

Now, we can calculate flux in field winding by integrating on .

2eff Pg

d

pkfff N ,

DlB pkf ,2f

k iNB

4 0where

Pp,

f

f

f

f

fff iNDl

iNDl

Nk

2

00842

effg

effeff2

08

P

N

g

Dl

iL

f

efff

f

mf

mflff LLL


14/28

Self and Mutual Inductances for

Fundamental Harmonic in

Rotor

Double Layer Lap Winding

on Stator


15/28

Cross Section Diagram

baxisd

mad

daxisqaxis

m

a

aaxis

caxis


16/28

Self Inductance of Stator Winding (1)

If we apply current in Phase A winding, then the magnetic field is:

cos

'

4 0aa

aa

Pi

NB

)(' avdeff P

gg

)

22

2cos(

2)

2cos(

4 0ma

g

aaa

deff

PPPi

P

N

2dg

Now, we can calculate flux in Phase A winding from its own current using the formuladerived in Notes Flux Linkage in Phase Winding.

NDl 42 P

ga

meaa

av

aa

iNDl

PgP

2cos18

cos2

2

0

mme 2De ine:

2cosa LLL

ame

av Pg 2

NDlNDl 8822

ai

A

av

B

av

APgPg 22

,


17/28

Self Inductance of Stator Winding (2)

)2cos( meBAlsaa LLLL

For balanced 3 phase windings

)3

22cos()3

2(2cos

meBAlsmeBAlsbb LLLLLLL

)3

22cos()

3

2(2cos

meBAlsmeBAlscc LLLLLLL

lsL is stator leakage inductance.

gaga NDlNDl88

2

0

2

0

Aav

B

av

A

PgPg 22

,

)1(3

)(3 g

ABAmd LLLL

)21(2

3)(2

3 gABAmq LLLL


18/28

Mutual Inductance of Stator Winding

If we apply current in Phase B winding, then the magnetic field for fundamentalharmonic is:

2

cos

'

4 0

aba

b

Pi

NB

)3

2

22

2cos(

2)

3

2

2cos(4 0

ma

g

aba

av

e

PPPiP

N

g

Now, we can calculate flux in Phase A winding from its own current using the formuladerived in Notes Flux Linkage in Phase Winding.

2242 0 gaNDl P

)2

2cos(18

323

2

0

me

g

ba

meb

av

aa

iNDl

PgPmme

2

av

g

)3

22cos(

2

1

meBA

b

aab LL

iL

)3

2

2cos(2

1

meBAac LLL )2cos(2

1meBAbc LLL


19/28

Mutual Inductance betweenStator and Rotor Field Winding (1)

If we apply current in rotor field winding, then when rotor is moving,the magnetic field from rotor field winding is:

cos4 0 f PiN

B

P

)2

22

cos(2

)2

cos(4

2'

0mema

g

meaf

f

av

deff

PPPi

P

N

g

Pg

2


)2

cos(2

14

0 meag

f

f

av

Pi

P

N

g

a

me

g

f

f

av

aa

NDl

iPgP

N

8

cos)2

1( 0

fmeav Pg cos2 2

mesfme

gfaaaf L

NNDlL

coscos)1(

82

0

avf

)2

1(

82

0 gfa

av

sfPNN

gDlL

where


20/28


)cos( mesfaf LL

)3cos(

mesfbf LL

)

3

2cos(

mesfcf LL

where

)2

1(82

0 gfa

av

sfP

NN

g

DlL


21/28

Self Inductance of Rotor Field Winding

For the magnetic field from rotor field winding is:

)cos(4 0 f PiN

B

)1)(cos(4

0gf

deff

Pi

N

Pg

Now, we can calculate flux in field winding by integrating on .

av Pg

d

f

gf

av

g

f

f

av

ff iP

N

g

Dli

P

N

gP

DlN )

21(

8)

21(

42

2

00

)2

1(8

2

0 gf

avf

f

mfP

N

g

Dl

iL

mflff LLL

lfL is leakage inductance of field winding.


22/28

Flux Linkage (1)

a

mesfmeBAmeBAmeBAls

a i

LLLLLLLL cos)3

22cos(

2

1)

3

22cos(

2

12cos

f

c

b

mesfmeBAlsmeBAmeBA

mesfmeBAmeBAlsmeBA

f

c

b

i

i

i

LLLLL

LLLLLLLL

LLLLLLLL

2cos

2coscos

)3

2cos()3

22()2cos(21)

322cos(

21

)3

cos()2cos(2

)3

2cos()3

2cos(2

mmesmesmes33

Agag

Ba

A LNDlLNDlL

8,

8

2

0

2

0

avav gg

)1(82

0 gfa

sf

NNDlL

av

)2

1(8

2

0 gf

mfP

NDlL

av


23/28

Flux Linkage (2)

mesmeBmeBmeBAls LLLLLL cos)2

2cos()2

2cos(2cos3

0 cba iiiWhen

c

b

a

mesfmeBAlsmeBmeB

mesfmeBmeBAlsmeB

c

b

a

i

i

i

LLLLLL

LLLLLL

)3

2cos()

3

22(

2

3)2cos()

3

22cos(

)3

2

cos()2cos()3

2

2cos(2

3

)3

2

2cos(

f

mflfmesfmesfmesf

f

LLLLL )3

2cos()

3

2cos(cos

22cos

22cos2cos

))(2

3(

cbaAlscba

iiiL

iiiLL

)3

2cos()

3

2cos(cos

33

mememefsf iL

0


24/28

Terminal Voltage

aasa iRv 000

f

c

f

c

f

s

s

f

c dtii

RR

vv

000000

0 cba iiiWhen

)()( cbacbascbad

iiiRvvv

0


25/28

Self and Mutual Inductances Related toRotor d-axis Damper Winding

If the machine has a damper winding whose magnetic field is along d-axis, theanalysis is the same as field winding. Here we summarize the results:

LLL Self inductance

)1(8

2

0 gk

mk

NDlL d

d

ddd m

where

av

)cos(meskak dd

LLMutual inductances

2

)3

cos(

meskbk dd LL

3

meskck dd

where )2

1(82

0 gka

skP

NNDlL d

d

)21(

82

0 gkf

av

fkP

NN

g

DlL

d

d

Also


26/28

Mutual Inductance between Stator andRotor q-Axis Damper Winding (1)

If we apply current in q-axis damper winding, then when rotor is moving, themagnetic field from q-axis damper winding is:

)cos(

'

4 0

qkk

k

Pi

NB

q

P

))2

(2

22

cos(2

))2

(2

cos(4

0

mema

g

meak

k

av

deff

PPPi

P

N

g qq

Pmaq 2


))2

(2

cos(2

14

0

meag

k

k

av

Pi

P

N

g qq

q

q

ka

me

g

k

k

av

aa

NDl

iPgP

N

8

)2

cos()2

1( 0

q

q

kmeav iPg sin)21( 2

meskme

gkaaak q

q

qL

NNDlL

sinsin)1(

8

2

0

avkq

)2

1(

82

0 gka

av

skPNN

gDlL q

q

where


27/28


)sin( meskak qq LL

)3

2sin(

meskbk qq LL

)

3

2sin(

meskck qq LL

where

NN

)21(2

0 ga

av

skPgL

q

q


28/28

Self Inductance of Rotor q-Axis Winding

For the magnetic field from rotor q-axis damper winding is:

))2

(2

cos(2

1

4 0

mea

g

k

k

av

k

Pi

P

N

gB

q

q

q

)2

cos(2

1

4 0 qgkkav

PiP

N

g qq

, - .

q

q

q

q

qq k

gk

av

g

k

k

avkk iP

N

g

Dl

iP

N

gP

Dl

N )21(

8

)0cos()21(

42

2

00

q

)2

1(

8

2

0 gk

avk

k

mkP

N

g

Dl

iL

q

q

q

q

qqq mklkkLLL

qlk - .

Besides, we can calculate flux in field winding by integrating on .d

42 0

gkqNDl

22

k

av

ff qPgP

0q

q

k

f

fki

L

0qdkkLAlso

11 self and mutual inductances

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