!"##$%&'()* ,)-*%! &./01.23 4567648 439: ;#
!"#$#!%
'())*+,-./0 1/20+3
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
'())*+,-./0 1/20+3
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
!"#$#!%
'())*+,-./0 1/20+3
<=>? 01 > @A>BC?DE %IK A>J?KH1 I=0J= 2.?.HL0F. ?=. UHKU.H 1.C.J^KF KA J0HJB0? SH.>P.H1 >H. ?=. JBHH.F? MKI0FR 0LL.20>?.CG >\.H ?=. A>BC? KJJBH1 >F2 ?=. JBHH.F? I=0J= ?=. SH.>P.H LB1? 0F?.HHBU?N 'F A>BC? >F>CG101 />CB.1 KA ?=.1. JBHH.F?1 >H. >H . J>CJBC>?.2 AKH 20].H.F? ?GU.1 KA A>BC?1 >? 20].H.F? CKJ>^KF1 0F ?=. 1G1?.LN %=. 0FAKHL>^KF KS?>0F.2 AKHL ?=.1. J>CJBC>^KF1 >H. B1.2 ?K 2.?.HL0F. ?=. H.C>G 1._FR ?=>? JKF?HKC ?=. J0HJB0? SH.>P.H1N
!"#$#!%
'())*+,-./0 1/20+3
%H>F10.F?1 0F !.H0.1 &* (0HJB0?1 %=. 1.C.J^KF KA > J0HJB0? SH.>P.H 2.U.F21 KF3 • FKHL>C KU.H>^FR JBHH.F? • ?=. JBHH.F? 0? =>1 ?K 0F?.HHBU? • ?=. L>W0LBL JBHH.F? 0? L>G =>/. ?K LKL.F?>H0CG J>HHG `KI 2K.1 >F K/.HVJBHH.F? KJJBHE %K >UUHK>J= ?=. UHKSC.L KA J>CJBC>^FR ?=. 0F0^>C JBHH.F? I=.F > 1G1?.L 01 1=KH?VJ0HJB0?.2T JKF102.H I=>? =>UU.F1 I=.F >F >J /KC?>R. 01 >UUC0.2 ?K >F &* J0HJB0?N
!"#$#!%
'())*+,-./0 1/20+3
#K^/>^KF a %H>F10.F?1 0F !.H0.1 &* (0HJB0?1 t = 0
V max sin
(
! t + "
()
i t
=
V max Z
Z =
!"#$#!%
R
()
)
i t
b0].H.F^>C .ZB>^KF3 !KCB^KF3
L
V max sin
(
! t + "
%&sin (! t + " # # $ ) # e#
)
= Ri + L
Rt L
2
2
2
R + ! L
, " = tan
#1
(
di dt
)('
sin " # # $
(! L R )
'())*+,-./0 1/20+3
#K^/>^KF a %H>F10.F?1 0F !.H0.1 &* (0HJB0?1
()
i t
=
V max Z
%&sin (! t + " # # $ ) # e#
Rt L
>JVJKLUKF.F? JKF1?>F? >LUC0?B2. ! " " #
! " " #
=
0
"
2
<=>? >SKB? I=.F > H.>C 1GFJ=HKFKB1 R.F.H>?KH 01 1=KH?.2E !"#$#!%
'())*+,-./0 1/20+3
)('
2JVJKLUKF.F? .WUKF.F^>CCG 2.J>G0FR
$ =
(
sin " # # $
&./0.I KA !GFJ=HKFKB1 c.F.H>?KH1N
!"#$#!%
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH 9:* 3(;.:,<;<23 =*;*,/+<, -3 >,-?*; @( / +2,@-;* +< .<;?*,+ )*.:/;-./0 *;*,=( -;+< *0*.+,-./0 *;*,=(A 9:* B-;>-;=3
2.D?*0( .<2E0*> *0*.+,-. .-,.2-+3F 3<)* 2.+/;.*3 /,* ?/,-/@0*A 9:* )<>*03 >*?*0 C<, +:* +:* ?/,-<23 B-;>-;=3 /,* /EE0-./@0* +< @<+: 3+*/>(G3+/+* /;> +,/;3-*;+ /;/0(3-3A H;0( 0-;*/, )/=;*D. .-,.2-+3 /,* .<;3->*,*>A .<;3->*,*>A
!"#$#!%
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH 9:* +B< E,-;.-E/0 E/,+3 *,KF ./00*> +:* /,)/+2,* J3** ;*L+ 30->*KF :/3 0<;=-+2>-;/0 30<+3 -; B:-.: +:*,* /,* .<-03 -;=3A 9:*3* ./,,( +:* .2,,*;+ +< /; *0*.+,-./0 0A 9:* ,<+<, -3 +:* E/,+ <; +:* 3:/M /;> ,<+/+*3 -;3 ->* +:* :<00-;= <; +:* ,<+<,F ./00*> +:* N*0> B-;>-;=F -3 32EE0-*> B-+: >. .2,,*;+A .2,,*;+A 9:* O2L /.,<33 +:* /-, =/E @*+B**; +:* /,)/+2,* /;> ,<+<, =*;*,/+*3 ?<0+/=*3 -; +:* .<-03 -;=3A
!"#$#!%
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH a %IK %GU.1 Armature
Armature
Gap Gap
Rotor S
Rotor
N X
X
The cylindrical rotor is called pole machine. a non-salient pole !"#$#!%
The non-cylindrical rotor is called a salient pole machine.
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH 1<, +:* +B C<, */.: ,*?<02D<; -; */.: /,)/+2,* .<-0 E*, ,*?<02D<;A '-;.* +:* ;2)@*, ?<0+/=* -3R P N P f f m Hertz Hertz =
2 60
=
2
B:*,* C -3 +:* *0*.+,-./0 C,*Q2*;.( -; S*,+TF U -3 +:* ;2)@*, -; WUXF /;> C ) Y V#7" -3 +:* )*.:/;-./0 C,*Q2*;.( -; ,E)A Z* 3** +:/+ / +B
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH a (GC0F2H0J>C &K?KHT dKFV!>C0.F? ;KC. #>J=0F.
Gap
Armature
Rotor S
2V>W01
N b0H.J? )W01 X
!"#$#!%
X
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH a dKFVJGC0F2H0J>C &K?KHT !>C0.F? ;KC. #>J=0F. Armature <$e** (f#$ g)(h %f %`'! !`f&%*"N
S
Gap
2V>W01
N
Rotor
N
S
ZV>W01 !"#$#!%
iB>2H>?BH. )W01
'())*+,-./0 1/20+3
b0H.J? )W01
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C >V>W01
ia
a
+
R Laa
!?>^KF>HG )HL>?BH.
,
2V>W01
v a
! 2V>W01
R L ff ,
i f +
R Lbb ,
!"#$#!%
b
v f f
!
"
R Lcc ,
vb
+
SV>W01
HK?>^KF
ib
!
! '())*+,-./0 1/20+3
v c
+ c
i
JV>W01
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C >V>W01
ia
R Laa ,
2V>W01
a
+ v a
Laf
!?>^KF>HG )HL>?BH.
! R L ff ,
i f +
R Lbb ,
HK?>^KF
v f f
!
Lcf
" R Lcc
Lbf
!"#$#!%
vb
+
SV>W01
b
ib
,
!
! '())*+,-./0 1/20+3
v c
+ c
i
JV>W01
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C >V>W01
ia
R Laa ,
2V>W01
a
v a
Laf Lab
=
Lba
=
! M s
Laa
+
=
Lbb
=
Lcc
=
L s
!?>^KF>HG )HL>?BH.
! Lac
R L ff
=
Lca
=
! M
s
,
i f +
R Lbb ,
HK?>^KF
v f f
!
Lcf
" R Lcc
Lbf
!"#$#!%
vb
+
SV>W01
b
ib
,
! Lbc
=
Lcb
=
! M s
'())*+,-./0 1/20+3
!
v c
+ c
i
JV>W01
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C >V>W01
ia
R Laa
! d
,
2V>W01
a
v a
Laf Lab
=
Lba
=
! M s
Laa
+
=
Lbb
=
Lcc
=
L s
!?>^KF>HG )HL>?BH.
! Lac
R L ff
=
Lca
=
! M
s
,
i f +
R Lbb ,
HK?>^KF
v f f
!
Lcf
" R Lcc
Lbf
!"#$#!%
vb
+
SV>W01
b
ib
,
! Lbc
=
Lcb
=
! M s
'())*+,-./0 1/20+3
!
v c
+ c
i
JV>W01
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C
!"#$#!%
Laf
=
M f cos! d
Lbf
=
M f cos (! d " 120° )
Lcf
=
M f cos (! d " 240° )
'())*+,-./0 1/20+3
)HL>?BH. ,CBW *0FP>R.1 Laa ia + Labib + Lac ic + Laf i f = L s ia " M s ( ib + ic ) + Laf i f
! a
=
! b
=
Lba ia + Lbbib + Lbc ic + Lbf i f = L s ib " M s ( ia + ic ) + Lbf i f
! c
=
Lca ia + Lcbib + Lcc ic + Lcf i f = L s ic " M s ( ia + ib ) + Lcf i f
)HL>?BH. ,CBW *0FP>R.1 ! f
=
Laf ia + Lbf ib + Lcf ic + L ff i f
g>C>FJ.2 %=H..V;=>1. !G1?.L ia
!"#$#!%
+
ib
+
ic
=
0
'())*+,-./0 1/20+3
)CR.SH>j ! a
=
! b
=
! c
=
(L (L (L
M s ) ia + Laf i f
s
+
s
+
M s ) ib + Lbf i f
s
+
M s ) ic + Lcf i f
!0FJ. ?=. k.C2 JBHH.F? 01 2J >F2 ?=. k.C2 HK?>?.1 I0?= JKF1?>F? >FRBC>H /.CKJ0?GT d ! ! d dt
( = (L = (L
=
! d = " t + ! o
"
,
) + M ) i + M ) i
,
i f = I f
( ) 120° + # ) + M I cos (" t $ 120° 240° + # ) t $ 240° + M I cos (" t
! a = L s + M s ia + M f I f cos " t t + # o ! b ! c
!"#$#!%
s
s
s
b
s
c
f f
o
f f
o
'())*+,-./0 1/20+3
(0HJB0? $ZB>^KF1 # a d #
dia
va = ! Ra ia ! M f I f sin ($ t + % o ) ! Ra ia ! ( L s + M s ) $ M = ! R + dt !## # "### ### $ " " dt !#"# $ ea& = 2 E i sin ($ t t + % o ) generator gener ator
'F?.HF>C .LA3
I=.H.3 E i
ea
!
=
(
2 E i sin " t + # o
) lKH 1GFJ=HKFKB1 .LAm
M f I f M
! =
2
,KH JKF/.F0.FJ. l0?e1 >HS0?H>HG >FGI>Gm 1.? <. =>/. ?=. J0HJB0? LK2.Cj
!"#$#!%
'())*+,-./0 1/20+3
90° ! = " + 90° o
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C
va = ! Ra ia ! ( L s + M s )
ia
R
dia dt
L s
M M f I f sin (" t + # o )
+ "
~ +
_
+ M
!
v
s
ea
eb
a
!
a
(
2 E i cos " t + #
=
+
)
_ !
!
e
c!
v c
L s R
!"#$#!%
+ M
s
L s
+ M
s
R
'())*+,-./0 1/20+3
vb
ic + c
ib
+
b
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C )C1K3 ia = ib = ic =
( ) 2 I cos (! t 120° ) t + " # $ # 120° 2 I cos (! t 240° ) t + " # $ # 240° 2 I a cos ! t + " # $ a a
a
a
a
I=.H. n> 01 ?=. U=>1. >FRC. KA ?=. C>R KA ?=. JBHH.F? 0 > I0?= H.1U.J? ?K .>eN
!"#$#!%
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C
L s
+ M
ia
R
s
(
2 I a cos ! t + " # $ a
=
)
+
±
ea
!
=
(
2 E i cos " t + #
)
va
=
2 V a cos ! t
"
%=01 01 > 1?.>2GV1?>?. 1?.>2GV1?>?. LK2.C LK2.C ?=>? L011.1 2.?>0C1 KA ?=. k.C2 JBHH.F? F..2.2 AKH ?H>F10.F? >F>CG101N
!"#$#!%
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH V (GC0F2H0J>C &K?KH #K2.C %K A>J0C0?>?. ?H>F10.F? >F>CG101 l>1 I0?= > A>BC?m 0? 01 F.J.11>HG ?K H.J>1? ?=01 LK2.C 0F?K > 20].H.F? AKHLN &.J>CC3
! f
=
Laf ia + Lbf ib + Lcf ic + L ff i f
>F23
!BS1^?B^FR3
Laf
=
M f cos! d
Lbf
=
M f cos (! d " 120° )
Lcf
=
M f cos (! d " 240° )
$ ' ! f = M f & ia cos" d + ib cos (" d # 120° ) + ic cos (" d # 240° ) ) + L ff i f # # $ & !" ) % i cos! = 2 I cos! cos (" t + # $ ! ) ( a
d
a
= !"#$#!%
d
(
a
)
(
2 I a cos " t + # + 90° 90° cos " t + # $ ! a '())*+,-./0 1/20+3
)
%K A>J0C0?>?. > 1GLL.?H0J A>BC? 0? =.CU1 ?K H.J>1? ?=01 LK2.C > S0?N
ia cos! d = =
(
) 2 I cos (" t + # + 90° ) cos (" t + # $ ! ) 2 I a cos ! d cos cos " t + # $ ! a a
a
2cos ! cos " = cos
=
=
!"#$#!%
I a 2 I a 2
# " ) (! #
%&cos (! a + 90° ) + cos( 2" t + 2# $ ! a + 90° ) '( %& $ sin! a $ sin( 2" t + 2# $ ! a ) '(
'())*+,-./0 1/20+3
+
cos
(!
+
" )
!0L0C>HCG AKH ?=. K?=.H ?IK ?.HL13
(
)
ib cos ! d " 120° =
(
)
ic cos ! d " 240° =
I a 2 I a 2
%& " sin! a " sin( 2# t t + 2$ " ! a " 120° ) ' ( %& " sin! a " sin( 2# t + 2$ " ! a " 240° ) '(
!BS1^?B^FR3
(
)
(
ia cos! d + ib cos ! d " 120° + ic cos ! d " 240°
)
%3sin ! a ( I a ' * = " 2# t + 2$ " ! a ) + sin( 2# t + 2$ " ! a " 120° ) + sin( 2# t + 2$ " " ! a " 240°) * ( ' + sin #"############## $ 2 ' !############# * & ) =0 = "3
I a 2
sin! a
!"#$#!%
9:-3 -3 / @/0/;.*> 3*.<;> :/,)<;-. +*,)3 +:/+ 32)3 +< T*,< /+ *?*,( E<-;+ -; D)*A
'())*+,-./0 1/20+3
!BS1^?B^FR3 ! f =
M f $% ia cos" d + ib cos (" d # 120° ) + ic cos (" d # 240° ) '& + L ff i f
= L ff i f
I=.H.
KH
!"#$#!%
id = !
id =
M f I a
#3
2
sin" a = L ff i f +
3
M f id
2
3 I a sin" sin" a
2
#$ ia cos" cos" d + ib cos (" d ! 120° ) + ic cos (" d ! 240° ) % & 3
'())*+,-./0 1/20+3
fS1.H/>^KF13 ! f
=
L ff i f +
3
M f id ,
2
id = " 3 I a sin# a
%=. MBW C0FP>R.1 I0?= ?=. k.C2 I0F20FR ?=>? >H. 2B. ?K > JKLS0F>^KF KA ?=. ?=. ?=H.. C0F. JBHH.F?1 2K FK? />HG I0?= ^L.N %=.G J>F ?=B1 S. H.R>H2.2 >1 JKL0FR AHKL > 1?.>2G 2J JBHH.F? 0 2 0F > kJ^^KB1 kJ^^KB1 2J 2J J0HJB0? JK0FJ02.F? I0?= ?=. 2V>W01 >F2 ?=B1 1?>^KF>HG I0?= H.1U.J? ?K ?=. k.C2 J0HJB0?N %=. ?IK J0HJB0?1 HK?>?. ?KR.?=.H 0F 1GFJ=HKF01L >F2 =>/. LB?B>C 0F2BJ?>FJ. l86om46o#A S.?I..F ?=.LN (0HJB0? LK2.Cj !"#$#!%
'())*+,-./0 1/20+3
)C?.HF>?. (0HJB0? #K2.C !B0?>SC. AKH %H>F10.F? )F>CG101 >V>W01 ! d
2V>W01 3
M f
2
id )HL>?BH. .ZB0/>C.F? I0F20FR HK?>^FR I0?= HK?KH
i f
+
R L ff ,
v f f
,0.C2 I0F20FR HK?>^FR I0?= HK?KH
!
"
SV>W01 JV>W01 !"#$#!%
'())*+,-./0 1/20+3
$W>LUC. a ) p5V`q ?=H..VU=>1. 1GFJ=HKFKB1 R.F.H>?KH I0?= F.RC0R0SC. >HL>?BH. H.101?>FJ. =>1 ?=. AKCCKI0FR 0F2BJ?>FJ. U>H>L.?.H13 Laa L s 2.7656 mH , =
=
L ff 433.6569 mH , =
Lab M f
=
=
M s
=
1.3828mH
31.6950 mH
%=. L>J=0F. 01 H>?.2 >? p8O #Q)T 5T: UKI.HVA>J?KH C>RR0FRT 8p55 HULT o9 PQN <=.F KU.H>^FR BF2.H H>?.2 JKF20^KF1T ?=. C0F.V?KV F.B?H>C ?.HL0F>C /KC?>R. >F2 ?=. C0F. JBHH.F? JBHH .F? KA U=>1. > >H. va = 19596cos! t V ,
ia = 21603cos
(
" 25.6569°)
! t
b.?.HL0F. ?=. L>RF0?B2. KA ?=. 1GFJ=HKFKB1 0F?.HF>C /KC?>R.T ?=. k.C2 JBHH.F? 'A T >F2 ?=. MBW C0FP>R.1 I0?= ?=. k.C2 I0F20FRN &.U.>? ?=.1. J>CJBC>^KF1 I=.F > CK>2 KA p8O #Q) 01 1.H/.2 >? H>?.2 /KC?>R. >F2 BF0?G UKI.H A>J?KHN <=>? 01 ?=. k.C2 JBHH.F? AKH H>?.2 >HL>?BH. /KC?>R. KF > KU.F J0HJB0?E !"#$#!%
'())*+,-./0 1/20+3
$W>LUC. va
24,000 2 3
=
max
ia
=
max
VA V
cos " = cos
1
#
=
2
V
= 19596
635, 635, 000
A
= 21603
3 ! 24 0.9 = 25.8419°, lagging
!GFJ=HKFKB1 'F?.HF>C QKC?>R.3 ea! =
(
2 E i cos " t + #
= Ra
=0
)
(
ia + va + L s + M s
dia
) dt = v + ( 2.7656 + 1.3828)10 a
(
)
= 19596cos " t + 21603 2.7656 + 1.3828 10
(
)
$3
= 19596cos " t $ " 21603 2.7656 + 1.3828 10
!"#$#!%
'())*+,-./0 1/20+3
d
dia
$3
dt
(
cos " t t $ 25.6569 ° dt $3
(
sin " t $ 25.6569°
)
)
!
$W>LUC.
=
120"
(
sin x # y
)
=
sin x cos y # cos x sin y
!GFJ=HKFKB1 'F?.HF>C QKC?>R.3 2 E i cos (! t + " )
( 2.7656 + 1.3828)10 sin (! t # 25.6569°) 3
= 19596cos! t # ! 21603
#
(
= 19596cos! t # 33875sin ! t # 25.6569° =
)
34323cos ! t # 30407sin! t
(
= 45855cos ! t + 41.5384° $
2 E i
= 45855,
,0.C2 (BHH.F?3 E i
!"#$#!%
)
" = 41.5384°
M f I f
E i
! =
2
I f
"
=
2
M f
!
'())*+,-./0 1/20+3
=
3838 A
$W>LUC. ,CBW C0FP>R.1 I0?= k.C2 I0F20FR13
! f
=
M f I a
L ff i f " 3
2
sin# a
n> 01 ?=. >FRC. KA C>R L.>1BH.2 IH? . >eN !0FJ. 0> C>R1 oON794:K S.=0F2 />T I=0J= C>R1 94NO879K S.=0F2 .>eT ?=.F n> r oON794:K s 94NO879K r pXN8758K I a sin ! a
%=B13 ! f
=
=
21603
sin67.3803° = 14100.6 A
2
M f I a
L ff i f " 3
2
sin# a
433.6569 $ 3838 $ 10
=
"3
"
3$
31.6950
$ 14100.6 $ 10
2 =
!"#$#!%
1664.38 " 948.06
=
716.31 Weber-tur Weber-turns ns '())*+,-./0 1/20+3
"3
$W>LUC. a &.U.>^FR AKH BF0?G UKI.H A>J?KH3
(
)
(
)
2 E i cos " t + # = va + 2.7656 + 1.3828 10
ea! =
$3
dia dt
(
= 19596cos " t t $ 33785sin " t = 39057cos " t t + 59.8854°
I f
=
E i
2
39057
M f
!
=
45855
"
3838
=
)
3269 A
%=. JBHH.F? 0> 01 0FVU=>1. I0?= / >T >F2 C>R1 .>e SG O:N77O9K sin! a I a sin!
! f =
!"#$#!%
=
= 15276sin59.8854° = 13214 A
433.6569 " 3269 " 10
1417.62 # 888.43
#3
#
3"
31.6950 2
=
529.19 Weber-turns Weber-turns
'())*+,-./0 1/20+3
" 13214 " 10
#3
$W>LUC. a Z* 3** +:/+ B:*; +:* E =<*3 C,<) "AI 0/==-;= +< !A" 2;>*, ,/+*> )*=/G?<0+/)E*,*3 0-;= /;> ?<0+/=* .<;>-D<;3F +:* N*0> .2,,*;+ -3 ,*>2.*> C,<) %$%$ +< %47I [ A [03 B-;>-;= 2.*> /0<;= B-+: +:* >*)/=;*DT-;= -;O2*;.* .2,,*;+ ,*Q2-,*> +< )/-;+/-; ,/+*> +*,)-;/0 ?<0+/=* -; +:* )/.:-;* 2;>*, -D<;3 J- / Y "K -3
I f
!"#$#!%
=
2
E i M M f
!
19596 " 10 =
3
120# " 31.695
'())*+,-./0 1/20+3
=
1640 A
%=. !GFJ=HKFKB1 c.F.H>?KH V !>C0.F? &K?KH #K2.C 9:* ,<2;>G,<+<, +:*<,( /0,*/>( ]23+ .<;3->*,*> =-?*3 =<<> ,*320+3 C<, +:* 3+*/>(G3+/+* E*,C<,)/;.* +< .<;3->*, / +B*0AA )<>*0 Z* ;2.* +:* +B*0 @( )*/;3 ,-?*; B-+: ,<2;>G ,<+<, .<;3+,2.D<;^ C<33-0GN,*> 2;-+3 :/?* +B< E<0*3 /;> ;2.0*/, 2;-+3 :/?* C<2, E<0* 3 C<, ,*/3<;3 *3-=; /;> *_.-*;.(A S(>,<*0*.+,-. =*;*,/+<,3 232/00( :/?* )<,* E<0*GE/-,3 /;> /,* 3 3< /3 +< /?<-> )*.:/;-./0 >/)/=* >2* +< .*;+,-C2=/0 C<,.*3A !"#$#!%
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH V !>C0.F? &K?KH #K2.C 9:* +:,**GE:/3* 3/0-*;+GE<0* )/.:-;*F 0-\* -+3 ,<2;>G,<+<, .<2;+*,E/,+F :/3 +:,** 3())*+,-./00( >-3+,-@2+*> /,)/+2,* B-;>-;=3 /F @F /;> .F /;> / N*0> B-;>-;= C <; +:* ,<+<, B:-.: E,<>2.*3 / 3-;23<->/0 O2L >-3+,-@2D<; /,<2;> +:* /-, =/EA P; @<+: +(E*3 `3**3a +:* 3/)* /-, =/E /;> )/=;*DT-;= E/+:3 -; +:* 3+/+<, ,*=/,>0*33 B-;>-;= :/3 .<;3+/;+ 3*0CG-;>2.+/;.* b c A X<,*/0 ?/,(-;= )2+2/0 -;>2.+/;.*3 /3 @*C<,*A 9:* >-c*,*;.* -3 +:/+ +:* 3*0CG -;>2.+/;.*3 b// F b@@F /;> b.. /;> +:* )2+2/0 -;>2.+/;.*3 b /@F [email protected] /;> b./ @*+B**; +:*) /,* ;< 0<;=*, .<;3+/;+ @2+ /03< ?/,( /3 / C2;.D<; -3E0/.*)*;+A !"#$#!%
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH V !>C0.F? &K?KH #K2.C ,CBW *0FP>R.13
! a
=
! b
=
Lba ia + Lbbib + Lbc ic + Lbf i f
! c
=
Lca ia + Lcbib + Lcc ic + Lcf i f
Laa ia + Labib + Lac ic + Laf i f
%=.1. CKKP ?=. 1>L. >1 1..F S.AKH. l!C02. 4:mT SB? BFC0P. ?=. J>1. KA ?=. HKBF2 HK?KH ?=. *e1 >H. > H. FK CKFR.H I.CCV>UUHKW0L>?.2 SG JKF1?>F?1N
X X
!"#$#!%
'())*+,-./0 1/20+3
%=. !GFJ=HKFKB1 c.F.H>?KH V !>C0.F? &K?KH #K2.C ,KH ?=. >HL>?BH. >HL>?BH.TT !.CAV0F2BJ?>FJ.13
L s
0
>
m
Laa
= L + L
cos2! d
Lbb
= L + L
Lcc
= L + L
( cos2 (!
s
m
s
m
s
#B?B>CV0F2BJ?>FJ.13
!"#$#!%
> L
m
M s
m
= L
ba
=
Lbc
= L
cb
=
Lca
= L
=
ac
d
> L
Lab
) 2# 3)
cos2 ! d " 2# 3
>
+
0
#K1?CG S>1.2 KF R.KL.?H0J>C JKF102.H>^KF1
! M s ! Lm cos2 (" d + #
! M s ! Lm cos2 (" d ! #
) 2)
! M s ! Lm cos2 (" d + 5# '())*+,-./0 1/20+3
6
6
)
%=. !GFJ=HKFKB1 c.F.H>?KH V !>C0.F? &K?KH #K2.C ,KH ?=. HK?KH HK?KHTT !.CAV0F2BJ?>FJ.13 L ff ,0.C2 I0F20FR3 bVb>LU.H I0F20FR3 L D iVb>LU.H I0F20FR3 LQ #B?B>CV0F2BJ?>FJ.13 ,0.C26bVI0F20FR3 M ,0.C26iVI0F20FR3 0 bVI0F20FR6iVI0F20FR3 0
r
)HL>?BH.6,0.C23 Laf L fa
=
M f cos2! d
Lbf L fb
=
M f cos2 (! d " 2#
Lcf L fc
=
=
=
=
!"#$#!%
#K1?CG S>1.2 KF R.KL.?H0J>C JKF102.H>^KF1
) M cos2 (! " 4# 3) f
d
'())*+,-./0 1/20+3
3
%=. !GFJ=HKFKB1 c.F.H>?KH V !>C0.F? &K?KH #K2.C )HL>?BH.V2>LU.H I0F20FR LB?B>CV0F2BJ?>FJ.1T LB?B>CV0F2BJ?>FJ.1 T )HL>?BH.6bVI0F20FR3
<=>? >H. ?=.1.E
LaD
=
L Da
=
M D cos2! d
LbD
=
L Db
=
M D cos2 ! d " 2# 3
LcD
=
L Dc
=
M D
( cos2 (!
d
) " 4# 3)
)HL>?BH.6iVI0F20FR3 LaQ LQa
=
M Q cos2! d
LbQ LQb
=
M Q cos2 (! d " 2#
LcQ LQc
=
=
=
=
!"#$#!%
) M cos2 (! " 4# 3) Q
d
'())*+,-./0 1/20+3
3
#K1?CG S>1.2 KF R.KL.?H0J>C JKF102.H>^KF1
%=. !GFJ=HKFKB1 c.F.H>?KH V !>C0.F? &K?KH #K2.C
Damper Windings or Amortisseur Windings Windings
Shorting Bar
Field Windings !"#$#!%
'())*+,-./0 1/20+3
%IKV)W01 #>J=0F. #K2.C d0*/,0( +:* *Q2/D<;3 C<, +:* O2L 0-;\/=*3 +:/; +:*-, ,<2;>G,<+<, .<2;+*,E/,+3A 1<,+2;/+*0(F +:* *Q2/D<;3 -; / 3-)E0* C<,) @( +,/;3C<,)-;= +:* / F @F /;> . ?/,-/@0*3 -;= 3*+3 +:* >-,*.+G/L-3 F Q2/>,/+2,*G/L-3 F /;> T*,-3D;=2-3:*> @( +:* 32@3.,-E+3 >F Q F /;> "R ,*3E*.D?*0(A 9:* +:,** 3+/+<, .2,,*;+3 -/F -@F /;> -. ./; @* +,/;3C<,)*> -;+< +:,** *Q2-?/0*;+ .2,,*;+3F ./00*> +:* >-,*.+G/L-3 .2,,*;+ - >F +:* Q2/>,/+2,*G /L-3 .2,,*;+ -Q /;> +:* T*,F1AKHL>^KFN -3 )/>* @( / )/+,-L ; ./00*> ;>HPe1 %H>F1AKHL>^KFN
!"#$#!%
'())*+,-./0 1/20+3
;>HPe1 %H>F1AKHL>^KF W*./00 C<, +:* ,<2;> ,<+<, )/.:-;* B* C<2;> J'0->* %"K - >F B:/+ B* B-00 ;-,*.+ .2,,*;+ /3R id =
2
#$ ia cos! d + ib cos (! d " 120° ) + ic cos (! d " 240° ) &% 3 ZV>W01 iB>2H>?BH. )W01 !
2V>W01 b0H.J? )W01
d
!
d
d
HK?>^KF
X X
!
V<+* :G/L-3 @( f I"G>*=,**3A !"#$#!%
'())*+,-./0 1/20+3
2V>W01
;>HPe1 %H>F1AKHL>^KF V<+* :G/L-3 @( f I"G>*=,**3A id = iq
=
2
#$ ia cos! d + ib cos (! d " 120° ) + ic cos (! d " 240° ) &% 3
2
#$ ia sin! d + ib sin (! d " 120° ) + ic sin (! d " 240° ) &% 3
P =
# 2% % 3% %$
cos! cos! d sin! sin! d ?
( sin (!
) " 120° )
cos ! d " 120° d
?
( sin (!
) " 240° )
cos ! d " 240° d
?
& ( ( ( ('
g*N;* U /3 / 2;-+/,( )/+,-LR P !1 P T 9:* 2;-+/,( E, @( UA =
!"#$#!%
'())*+,-./0 1/20+3
;>HPe1 %H>F1AKHL>^KF
P =
# % 2% % 3% % %$
cos! cos! d sin! sin! d 1 2
( sin (!
) " 120°)
cos ! d " 120° d
1 2
( sin (!
) " 240°)
cos ! d " 240° d
1 2
& ( ( ( ( ( ('
U/,\h3 +,/;3C<,)/D<; =-?*3 B:/+ -3 \;*0
!"#$#!%
'())*+,-./0 1/20+3
;>HPe1 %H>F1AKHL>^KF
! i $ ! i $ d # & # a & # iq & P # ib & # & # & #" i &% #" ic &% =
0
! v # d # vq # #" v 0
$ ! v $ & # a & & P # vb & & # & &% #" vc &% =
" $ $ $ $#
! d ! q ! 0
% " ' $ ' P $ ' $ '& $# =
! a ! b ! c
% ' ' ' '&
9:* UG+,/;3C<,)/D<; >*N;*3 / 3*+ O2L 0-;\/=*3 C<, +:,** N.DD<23 .<-03F <;* G.<-0 /;> +:* QG.<-0F QG.<-0F B:-.: B:-.: ,<+/+* -; 3(;.:,<;-3) B-+: +:* ,<+<,A 9:* >G /;> QG.<-03 :/?* .<;3+/;+ O2L 0-;\/=*3 B-+: +:* N*0> /;> /;( <+:*, B-;>-;=3 B:-.: )/( *L-3+ <; +:* ,<+<,A 9:* ,*320D;= >F QF /;> " O2LG0-;\/=* *Q2/D<;3 /,*
!"#$#!%
'())*+,-./0 1/20+3
;>HPe1 %H>F1AKHL>^KF 1<, *L/)E0*F +:* ,*320D;= >F QF /;> " O2LG0-;\/=* *Q2/D<;3 /,* C,<) +:* 3+/+<, O2LG0-;\/=*3 /3 C<00
' ! * ' L Lab $ ) a , ) aa $ ! b = Lba ia + Lbbib + Lbc ic + Lbf i f # & ) ! , = ) L Lbb b ab , ) $ ) ! c , ) Lac Lbc ! c = Lca ia + Lcbib + Lcc ic + Lcf i f ) $% ( + ( ' ! * ' ! * ' ! * ' ! * ) d , ) d , ) a , ) a , ) ! q , = P ) ! b , & ) ! b , = P ) ! q , ) , ) , ) , ) , ! ! ! ! )( ,+ )( ,+ )( c ,+ )( c ,+ ' ! * ' L ' ! * ' Laf * Lab Lac * d aa , ) , ) , - ) d , ) P ) ! q , = ) Lab Lbb Lbc , P ) ! q , + ) Lbf , i f , ) , ) ) , ) , L L L ! ! L ) , )( ,+ )( ac )( ,+ bc cc , + ( cf + ! a
=
1
0
0
1
1
0
' ) ) ) )( !"#$#!%
! d ! q ! 0
* ' L , ) aa , = P ) Lab , ) ,+ )( Lac
0
' , - ) , P ) ) , )( ,+
Lab
Lac *
Lbb
Lbc
Lbc
Lcc
1
! d ! q ! 0
' L * ) af , , + P ) Lbf ) , ) Lcf ,+ (
* , ,i , f , +
'())*+,-./0 1/20+3
' L af ,) , ) , ) ib , + ) Lbf ,) , ) i ,+ )( c ,+ ) Lcf (
Lac * ' ia * Lbc Lcc
* , , i f , , +
;>HPe1 %H>F1AKHL>^KF '2@3D+2D;= C<, +:* bh3 C,<) +:* E,*?-<23 +/@0*R ! # # # #"
Laa
Lab
Lac
Lab
Lbb
Lbc
Lac
Lbc
Lcc
$ & & & &%
! ( M s ( Lm cos2 (' d + ) 6) ( M s ( Lm cos2 (' d + 5) 6) $ L s + Lm cos2' d # & # = ( M s ( Lm cos2 (' d + ) 6) L s + Lm cos2 (' d ( 2) 3) ( M s ( Lm cos2 (' d ( ) 2) & # & # ( M s ( Lm cos2 (' d + 5) 6) ( M s ( Lm cos2 (' d ( ) 2) L s + Lm cos2 (' d + 2) 3) & " % ! ! L ( cos2' d cos2 (' d + ) 6) cos2 (' d + 5) 6) $ ( M s ( M s $ s # & # & = # ( M L s ( M s & ( Lm # cos2 (' d + ) 6) ( cos2 (' d ( 2) 3) cos2 (' d ( ) 2) & s # & # & # M M L ( ( cos2 (' d + 5) 6 ) cos2 (' d ( ) 2 ) ( cos2 (' d + 2) 3) & #" &% s s s " % ! ( cos2' d cos2 (' d + ) 6) cos2 (' d + 5) 6) ! 1 0 0 $ ! 1 1 1 $ # # & # & # cos2 (' + ) 6) ( cos2 (' ( 2) 3) cos2 (' ( ) 2) = ( L + M ) 1 0 & ( M # 1 1 1 & ( Lm s s # 0 d d d # #" 0 0 1 &% #" 1 1 1 &% # cos2 (' d + 5) 6) cos2 (' d ( ) 2) ( cos2 (' d + 2) 3) "
!"#$#!%
'())*+,-./0 1/20+3
$ & & & & %
;>HPe1 %H>F1AKHL>^KF V-<23 E/,+ f 32@3D+2+* /00 +:-3i ! L # aa # Lab # #" Lac ! # # # #"
* d * q * 0
= P =
! ' cos2( d cos2 (( d + ) 6) cos2 (( d + 5) 6 ) ! $ ! $ # 1 0 0 1 1 1 & & = ( L s + M s ) ## 0 1 0 && ' M ## 1 1 1 && ' Lm # cos2(( d + ) 6) ' cos2(( d ' 2) 3) cos2 (( d ' ) 2) # & # & # & 0 0 1 1 1 1 # cos2(( d + 5) 6 ) cos2 (( d ' ) 2) ' cos2(( d + 2) 3) " % " % &% "
Lab
Lac $
Lbb
Lbc
Lbc
Lcc
$ ! L & # aa & = P # Lab & # &% #" Lac
! # & & P '1 # # & #" &%
Lab
Lac $
* d
Lbb
Lbc
* q
Lbc
Lcc
* 0
! $ $ M f cos cos 2( d # & & # & + P M f cos cos 2 (( d ' 2) 3) & i f # & & # M f cos cos 2 (( d ' 4) 3) & &% " %
! cos( cos (( ' 120° ) cos (( ' 240° ) d d d # 2 # sin( d sin (( d ' 120° ) sin (( d ' 240° ) # 3# 1 1 1 # 2 2 2 #"
$ & & T '1 & , P = P & & &%
+< =*+i !"#$#!%
'())*+,-./0 1/20+3
$ & & & & %
;>HPe1 %H>F1AKHL>^KF +< =*+i
" $ $ $ $#
! d ! q ! 0
% " L %" i % " 0 0 ' $ d ' $ d ' $ ' = $ 0 Lq 0 ' $ iq ' + $ ' $ '$ ' $ '& $# 0 0 L '& $# i '& $ $# 0
0
% M f ' 2 'i f ' 0 ' 0 '& 3
B:*,*R 3 L = L + M + Lm g-,*.+G/L-3 -;>2.+/;.*R d s s 2
j2/>,/+2,*G/L-3 -;>2.+/;.*R Lq k*,2.+/;.*R L0
=
3
=
L s + M s ! Lm
L s ! 2 M s
>CC JKF1?>F?1t
!"#$#!%
'())*+,-./0 1/20+3
2
;>HPe1 %H>F1AKHL>^KF 9:* *Q2/D<; C<, +:* O2L 0-;\/=*3 .-,.2-+ ,*)/-; 2;.:/;=*> J'0->* 4$KR
! f
!"#$#!%
=
L ff I f +
3
M f id
2
'())*+,-./0 1/20+3
;>HPe1 %H>F1AKHL>^KF
;
!"#$#!%
'())*+,-./0 1/20+3
P=;<,* >/)E*, B-;>-;= C<, +:* )<)*;+A
%IKV)W01 #>J=0F. #K2.C
>V>W01 ! d
ZV>W01
iq
2V>W01 3
M f
2
id
R Lq ,
R Ld ,
&K?>^KF
i f +
R L ff ,
v f f
!
"
SV>W01
[00 .<-03 ,<+/+* +<=*+:*,A JV>W01
!"#$#!%
'())*+,-./0 1/20+3
%IKV)W01 #>J=0F. #K2.C 9:* .<;3+/;+ -;>2.+/;.* .<*_.-*;+3 )/\* C<, Q2-+* 3-)E0* +< 23*A U:(3-./00(F +:*3* 3-)E0*, O2LG0-;\/=* *Q2/D<;3 3: -3 +:* 3*0CG-;>2.+/;.* G/L-3 /,)/+2,* B-;>-;= B:-.: ,<+/+*3 /+ +:* 3/)* 3E**> /3 +:* N*0> /;> B:-.: ./,,-*3 .2,,*;+ - > +< E,<>2.* +:* 3/)* ))C <; +:* >G/L-3 /3 >< +:* /.+2/0 3+/+<, .2,,*;+3 -/F -@F /;> -.A '-)-0/,0( C<, bQ /;> -Q /;> +:* QG/L-3A 9:* N.DD<23 >G/L-3 B-;>-;= /;> +:* N*0> B-;>-;= ,*E,*3*;D;= +:* E:(3-./0 N*0> ./; @* .<;3->*,*> +< /.+ 0-\* +B< .<2E0*> .<-03 B:-.: /,* 3+/D<;/,( B-+: ,*3E*.+ +< */.: <+:*, /3 +:*( ,<+/+* +<=*+:*, 3:/,-;= +:* )2+2/0 -;>2.+/;.* @*+B**; +:*)A
!"#$#!%
'())*+,-./0 1/20+3
%IKV)W01 #>J=0F. #K2.C 12,+:*,)<,*F +:* N*0> /;> +:* >G/L-3 .<-0 >< ;<+ .<2E0* )/=;*D./00( B-+: N.DD<23 Q B-;>-;= 3-;.* -+ 0/=3 +:* >G/L-3 -; 3E/.* @( I"eA 9:* T*,2.+/;.* b" -3 /33<.-/+*> B-+: / 3+/D<;/,( N.DD<23 /,)/+2,* .<-0 B-+: ;< .<2E0-;= +< /;( <+:*, .<-03A l;>*, @/0/;.*> .<;>-D<;3 +:-3 .<-0 ./,,-*3 ;< .2,,*;+F /;> +:*,*C<,* B* <)-+ -+ C,<) <2, >-3.233-<; A
!"#$#!%
'())*+,-./0 1/20+3
$W>LUC. a %K R.? > A..C0FR AKH ;>HPe1 %H>F1AKHL>^KF l;>*, 3+*/>(G3+/+* -D<;3 +:* /,)/+2,* /0 +:,**GE:/3* .2,,*;+3R ia
=
ib
=
ic
=
( 2 I sin (! 2 I sin (!
2 I a sin ! d " ! a
)
a
d
" 120° " ! a )
a
d
" 240° " ! a )
1-;> +:* .<,,*3E<;>-;= >GQG" .2,,*;+3
!"#$#!%
'())*+,-./0 1/20+3
$W>LUC. a %K R.? > A..C0FR AKH ;>HPe1 %H>F1AKHL>^KF
! i $ # d & # iq & = # & #" i0 &%
! # 2# # 3# # #"
cos' cos' d sin' sin' d
( sin ('
) ( 120° )
cos ' d ( 120° d
1
1
2
2
( sin ('
) ( 240° )
cos ' d ( 240° d
1 2
X20DE0( +:,<2=:R id = iq i0 !"#$#!%
=
=
2
!"cos' cos' d ia + cos (' d ( 120° ) ib + cos (' d ( 240° ) ic $ % 3
2
!"sin' sin' d ia + sin (' d ( 120° ) ib + sin (' d ( 240° ) ic $ % 3
! # 3"
2
$ ( ia + ib + ic )& = 0 2 %
1
@/0/;.*> +:,**GE:/3*
'())*+,-./0 1/20+3
$ & ! ia $ & &# & # ib & & &# & #" ic &% &%
$W>LUC. a %K R.? > A..C0FR AKH ;>HPe1 %H>F1AKHL>^KF 2sin x cos y
'-)E0-C(-;=i
cos! d = ia cos!
(
ib cos ! d " 120°
) = =
(
(
(
) = =
I a
) (
)
#$sin( 2! 2! d " 240° " ! a ) " sin! sin! a % &
(
I a
) (
#$sin( 2! 2! d " 480° " ! a ) " sin! sin! a % &
'())*+,-./0 1/20+3
)
+
(
sin x ! y
#$sin( 2! 2! d " ! a ) " sin! sin! a % &
2 I a cos ! d " 240° sin ! d " 240° " ! d
2
!"#$#!%
2
(
sin x + y
2 I a cos ! d " 120° sin ! d " 120° " ! d
2 ic cos ! d " 240°
)
2 I a sin ! d " ! a cos! cos! d =
I a
=
)
)
$W>LUC. a %K R.? > A..C0FR AKH ;>HPe1 %H>F1AKHL>^KF '2@3D+2D;=i
id =
= "
2 3
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3 I a sin ! a
P+ 3:<20> .<)* /3 ;< 32,E,-3* +:/+
iq ! =
3 I a cos" a
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!"#$#!%
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va ! Ria ! " " dt !# "# $ =
generator generat or vb ! Rib ! ! Ri =
vc ! Ric ! ! Ri =
d # # b dt d # # c dt
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va vb vc vd vq v0
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V<+*RR V<+*
!"#$#!%
=
=
ia ib ic id iq i0
()
P P t =
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1
0
3-;.* 3-; .*
! d = " t + ! o
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=
ia ib ic
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( a ( b ( c
$ & & & &%
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! # # # #"
vd vq v0
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=
=
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0
1
1
0
0
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0
0
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!"#$#!%
!
=
d " " d dt
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$ & & ! ') ( ( q & # & + # ) ( ( d & # & #" 0 & &%
$ & & & &%
QKC?>R. $ZB>^KF1 !BLL>HG3 >G/L-3 vd = ! Rid ! " f
=
" d d "
L ff i f +
" d = Ld id +
QG/L-3 vq = ! Riq ! " q
=
dt
! # " " q
3
M f id
2 3
M f i f
2
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dt
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!
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d " " f dt
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(0HJB0? #K2.C
vd ! Rid ! Ld =
i f
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v f f
!
1-*0> B-;>-;=
3
M f
2
Ld
dt
! "# q id
R
L ff
did
+
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!
v f f
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d " " f
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q
dt
+
iq
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Lq
+
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! +
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vq ! Riq ! Lq =
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! "# d
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vd = ! Rid ! vq
= !
Riq !
d " " d dt " q d "
dt
! #"
q
,
+ #" , d
" f " q
=
=
=
L ff i f +
M f id ,
2
Lq iq , v f f
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$
=
R f i f +
" d = Ld id + " f d "
dt
3
M f i f
2
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vd = #
3
! d =
M f i f
2
! q
=
0
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=
0
"
vq
= #
d ! ! q dt v0
! d d !
dt
# $! = q
+ $! = d
Ri0
= #
9:23R
! v # d # vq # #" v 0
!"#$#!%
$
d ! ! 0 dt
0
3
M f i f
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0
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=
0
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! v $ ! v $ # d & # a & # vb & = P T # vq & = # & # & #" v0 &% #" vc &%
=
! # 3 # 2# # "
! # # # 2 # 3# # # #"
cos' cos' d
(
cos ' d ( 120°
(
( sin ('
) ( 240 )
sin ' d ( 120° d
sin' sin' d
)
cos ' d ( 240°
sin' sin' d
°
1
)
(
sin ' d ( 120°
(
2
)
sin ' d ( 240°
)
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v f f
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=
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id
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L ff
3
M f
2
t = 0
+
Ld >G/L-3
! vd !
! "# + = 0
iq
q
R
Lq
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t = 0
+
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=
M f id
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M f i f
2
3
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