Download - 11 Self and Mutual Inductances
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Self and Mutual Inductances for
uSynchronous Machine with
Round Rotor
Double Layer Lap Winding
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Cross Section Diagram
baxisd
qaxis daxis
mad
m
a
aaxis
caxis
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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
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Self and Mutual Inductances (1)
ib(t)mad d
ia(t)
am
aax s
ic(t)
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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Self and Mutual Inductances for
Fundamental Harmonic in
Rotor
Double Layer Lap Winding
on Stator
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Cross Section Diagram
baxisd
mad
daxisqaxis
m
a
aaxis
caxis
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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
,
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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
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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
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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
Now, we can calculate flux in Phase A winding from field current.
)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
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Mutual Inductance betweenStator and Rotor Field Winding (2)
)cos( mesfaf LL
)3cos(
mesfbf LL
)
3
2cos(
mesfcf LL
where
)2
1(82
0 gfa
av
sfP
NN
g
DlL
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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.
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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
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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
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Terminal Voltage
aasa iRv 000
f
c
f
c
f
s
s
f
c dtii
RR
vv
000000
0 cba iiiWhen
)()( cbacbascbad
iiiRvvv
0
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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
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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
Now, we can calculate flux in Phase A winding from field current.
))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
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Mutual Inductance betweenStator and Rotor Field Winding (2)
)sin( meskak qq LL
)3
2sin(
meskbk qq LL
)
3
2sin(
meskck qq LL
where
NN
)21(2
0 ga
av
skPgL
q
q
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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