Flux Linkage in Phase Winding
For Fundamental Harmonic
in Double Layer Lap Winding
Cross Section Diagram
b axis
θd
q axis
d axis
θd = θa −θm
θm
θa
a axis
c axis
Stator Winding
Fractional Pitch
3γ m
2
(exaggerated end turns)
γm
2
θa
a axis
ρm γ
m
θa
a axis
q=4
γ =
P
γm
2
q=2
q coils per group
Geometry of One Turn
θa
a axis
a axis
ia
0
rθ a
ρm γ
m
ζm
B
l
ia
z
θa =
ρm
2
+
ζ mz
r
+
q −1
γ m − υγ m
2
θa = −
ρm
2
+
ζ mz
r
+
q −1
γ m − υγ m
2
υ = 0,1,2  ( q − 1)
ζ mz
r
l
r
ζm
α sm
q=2
r
Flux of One Turn (1)
Assume the magnetic field for the fundamental harmonic in airgap is:
P
B = B pk cos( θ a + β )
2
ρm
ζ mz
q −1
θa =
+
+
γ m − υγ m
r
2
2
θa = −
a axis
ζ mz
r
+
q −1
γ m − υγ m
2
υ = 0,1,2  ( q − 1)
ζm
l
0
rθ a
2
+
ζm
B
ia
ρm
ia
α sm
l
r
q −1
γ m −υγ m
2
q −1
− l / 2 − ρ m / 2 +ζ m z / r +
γ m −υγ m
2
Φ=∫
l/2
ρ m / 2 +ζ m z / r +
∫
z
Brdθ a dz
q −1
γ m −υγ m
2
q −1
− l / 2 − ρ m / 2 +ζ m z / r +
γ m −υγ m
2
= B pk r ∫
l/2
∫
ρ m / 2 +ζ m z / r +
P
cos( θ a + β )dθ a dz
2
γ =
P
γm
2
−
ρ m / 2 +ζ m z / r +
γ m −υγ m
P
2
=
+
dz
sin(
θ
β
)
|
a
q −1
γ m −υγ m
− ρ m / 2 +ζ m z / r +
P / 2 ∫−l / 2
2
2
B pk r
l/2
q 1
ζ m zP
ζ m zP
q −1
q −1
ρ
ρ 

+
+
−
+
−
+
+
−
−
sin(
β
γ
υγ
)
sin(
β
γ
υγ
) dz
P ∫−l / 2 
2r
2
2
2r
2
2 
2 B pk D  ρ  l / 2
ζ zP
q −1
sin   ∫ cos( m + β +
γ − υγ )dz
=
P
2r
2
 2  −l / 2
=
2 B pk r
l/2
ρ=
P
ρm
2
Flux of One Turn (2)
q −1
ς =β+
γ m − υγ m
2
lζ m
r
P
α s = α sm
2
α sm =
l
ζm
α sm
r
2 B pk D
sin
αs
ρ
2 cos(ς )
sin  
P
 2  αs
2
2 B pk Dl
α
q −1
ρ
γ m − υγ m )
sin  sinc( s ) cos( β +
=
P
2
2
2
 
=
2 B pk Dl
ρ
k s = sinc(
k p = sin ( )
2
q −1
= N C k p k s Φ pk cos( β +
γ m − υγ m )
2
pitch factor
λcoil
ζ zP
 ρ  l/2
sin   ∫ cos( m + ς )dz
P
2r
 2  −l / 2
2 B pk D  ρ  1
ζ zP
sin  
sin( m + ς ) l−/l2/ 2
=
P
2r
 2  ζ mP
2r
2 B pk D  ρ  l
ζ lP
ζ lP
sin  
[sin(ς + m ) − sin(ς − m )]
=
P
4r
4r
 2  lζ m P
r2
2 B pk D  ρ  l
α
sin  
2 sin s cos(ς )
=
P
2
 2  αs
Φ=
αs
2
)
skew factor
Φ pk =
2 B pk Dl
P
Flux Linkage of A Group (1)
Consider the phase delay, the total flux of a group:
q −1
λgroup = ∑ λcoil,υ
υ =0
q −1
q −1
γ − υγ )
= ∑ N c k p k s Φ pk cos( β +
2
υ =0
q −1
q −1
γ − υγ )
= N c k p k s Φ pk ∑ cos( β +
2
υ =0
Let
q −1
ξ =β+
γ
2
Flux Linkage of A Group (2)
− jqγ
q −1




−
e
1
− jυγ
j (ξ −υγ )
jξ
jξ
cos(ξ − υγ ) = ∑ Re[e
] = Re e ∑ e
∑
 = Re e 1 − e − jγ 
υ =0
υ =0
 υ =0



qγ 

sin(
)
q −1
 j ( β + 2 γ ) e − jqγ / 2 (e jqγ / 2 − e − jqγ / 2 ) 
 jβ
2
e
= Re e
Re
=



− jγ / 2
− jγ / 2
jγ / 2
γ
e
e
−
e
(
)



sin( ) 
2 

qγ
sin( )
2 = qk cos β
= q cos β
q −1
q −1
γ
d
n sin( )
2
The distribution factor is defined ask d =
sin(
qγ
)
2
γ
q sin( )
2
Define the winding factor:
kw = k p kd ks
⇒ λgroup = qN c k wΦ pk cos β
≤1
≈1
Flux Linkage in Phase A Winding
There are a total of P groups. These groups may be connected in series, or parallel,
or partly series and partly parallel.
P = Ps C
Assume:
Then:
Note:
λa = Ps λgroup = Ps qN c k wΦ pk cos β
N a = Ps qN c
C
Nˆ a = N a k w
⇒ λa = Nˆ a Φ pk cos β
number of series turns per phase per circuit
number of parallel circuits
effective number of series turns per phase
per circuit on armature winding
P
B = B pk cos( θ a + β )
2
2 B pk Dl
Φ pk =
P
For hth Harmonic
in Double Layer Lap Winding
Flux of hth Harmonic in One Turn (1)
Bh = B pk ,h cos(h
Assume for the hth harmonic:
ρm
ζ mz
q −1
θa =
+
+
γ m − υγ m
r
2
2
θa = −
a axis
ia
ρm
2
+
ζ mz
r
+
q −1
γ m − υγ m
2
υ = 0,1,2  ( q − 1)
ζm
B
0
rθ a
P
θa + βh )
2
ia
l
ζm
α sm
l
r
z
q −1
γ m −υγ m
2
q −1
− l / 2 − ρ m / 2 +ζ m z / r +
γ m −υγ m
2
Φh = ∫
l/2
∫
ρ m / 2 +ζ m z / r +
Bh rdθ a dz
q −1
γ m −υγ m
2
q −1
− l / 2 − ρ m / 2 +ζ m z / r +
γ m −υγ m
2
= B pk , h r ∫
l/2
B pk ,h r
∫
ρ m / 2 +ζ m z / r +
cos(h
P
θ a + β h )dθ a dz
2
q −1
ρ m / 2 +ζ m z / r +
γ m −υγ m
P
2
=
+
sin(
θ
β
)
|
h
dz
1
a
h
q
−
γ m −υγ m
− ρ m / 2 +ζ m z / r +
2
hP / 2 ∫−l / 2
2
l/2
P
γ = γm
2
ρ=
P
ρm
2
ζ m zPh
ζ m zPh
q −1
hρ
q −1
hρ 

+
+
−
+
−
+
+
−
−
sin(
β
γ
υ
γ
)
sin(
β
γ
υ
γ
) dz
h
h
h
h
h
h
−l / 2 
2r
2
2
2r
2
2 
hP

2 B pk ,h D
ζ zPh
q −1
 hρ  l / 2
sin 
+ βh +
hγ − υhγ )dz
=
 ∫−l / 2 cos( m
2r
2
hP
 2 
=
2 B pk ,h r
∫
l/2
Flux of hth Harmonic in One Turn (2)
n −1
hγ − υhγ
ς = βh +
2
lζ m
r
P
α s = α sm
2
α sm =
l
ζ zPh
 hρ  l / 2
+ ς )dz
sin 
 ∫−l / 2 cos( m
hP
r
2
2


2 B pk ,h D  hρ  1
ζ zPh
=
+ ς ) l−/l2/ 2
sin 
sin( m

hP
2r
 2  ζ m Ph
2r
2 B pk ,h D  hρ  l
ζ lPh
ζ lPh
=
sin 
[sin(ς + m
) − sin(ς − m
)]

hP
4r
4r
 2  lζ m Ph
r2
2 B pk ,h D  hρ  l
hα s
=
sin 
2 sin
cos(ς )

hP
2
 2  hα s
Φh =
ζm
α sm
r
=
2 B pk ,h D
2 B pk ,h Dl
 hρ 
sin 

 2 
hα s
2 cos(ς )
hα s
2
2 B pk ,h Dl
hα
q −1
 hρ 
=
hγ − υhγ )
sin 
sinc( s ) cos( β h +
hP
2
2
 2 
hP
hα s skew factor for
hρ
sinc
(
) th
=
k
sh
k ph = sin ( )
2 h harmonic
2
2 B pk ,h Dl
q −1
Φ
=
pk , h
= N C k ph k sh Φ pk ,h cos( β h +
hγ − υhγ )
hP
pitch factor for
hth harmonic
λcoil,h
sin
2
Flux Linkage of hth Harmonic in A Group (1)
Consider the phase delay, the total flux of a group:
λgroup,h
q −1
q −1
h γ − υ hγ )
= ∑ N c k ph k sh Φ pk ,h cos( β h +
2
υ =0
q −1
q −1
h γ − υ hγ )
= N c k ph k sh Φ pk ,h ∑ cos( β h +
2
υ =0
Let
q −1
ξh = βh +
hγ
2
Flux Linkage of hth Harmonic in A Group (2)
− jqhγ
q −1




−
1
e
j (ξ h −υhγ )
jξ h
jξ h
− jυhγ
cos(ξ h − qhγ ) = ∑ Re[e
] = Re e ∑ e
∑
 = Re e 1 − e − jhγ 
υ =0
υ =0
υ =0




qhγ 

 j ( β h + q2−1 hγ ) e − jqhγ / 2 (e jqhγ / 2 − e − jqhγ / 2 ) 
 jβ h sin( 2 ) 
= Re e
 = Re e

jhγ / 2
− jhγ / 2
− jhγ / 2
γ
h
(
)
−
e
e
e



sin( ) 
2 

qhγ
sin(
)
2 = qk cos β
= q cos β h
dh
h
hγ
q sin( )
qhγ
sin(
)
2
2
=
k
dh
The distribution factor for hth harmonic is defined as:
hγ
q sin( )
2
λgroup,h = qN c k ph k sh k dh Φ pk ,h cos β h
q −1
q −1
Define Winding Factor:
k wh = k ph k sh k dh
⇒ λgroup,h = qN c k wh Φ pk ,h cos β h
Flux Linkage of hth Harmonic
in Phase A Winding
There are a total of P groups. These groups may be connected in series, or
parallel, or partly series and partly parallel.
P = Ps C
Assume:
Then:
Note:
λa ,h = Ps λgroup ,h = Ps qN c k wh Φ pk ,h cos β h
N a = Ps qN c
C
Nˆ a ,h = N a k wh
⇒ λa ,h = Nˆ a ,h Φ pk ,h cos β
number of series turns per phase per circuit
number of parallel circuits
effective number of series turns per phase
per circuit for hth harmonic
P
Bh = B pk ,h cos(h θ a + β h )
2
2 B pk ,h Dl
h
Φ pk ,h =
hP
Summary of Winding Factor
for hth Harmonic
Coil pitch in electrical angle:
S
P
ρ = ρm = c π
2
SP
Slot pitch in electrical angle:
γ=
for
α sm is the skewed mechanical angle
P
2
α sm =S skewγ m , S skewis the number of slots skewed
k ph
 hρ 
= sin  
 2 
k wh = k ph k dh k sh
hqγ
)
2
=
hγ
q sin( )
2
pitch coil
P
πP
γm =
2
S
Skewed angle in electrical radian: α s = α sm
For the hth harmonic
Sc
SP
l
ζm
α sm
r
sin(
k dh
q=
S
3P
hα s
k sh = sinc(
)
2
Self and Mutual Inductances for
Fundamental Harmonic in
Synchronous Machine with
Round Rotor
Double Layer Lap Winding
on Stator
Cross Section Diagram
b axis
θd
q axis
d axis
θd = θa −θm
θm
θa
a axis
c axis
Stator Winding
Fractional Pitch
3γ m
2
(exaggerated end turns)
γm
2
θa
a axis
ρm γ
m
θa
a axis
q=4
γ =
q=2
q coils per group
P
γm
2
Self and Mutual Inductances (1)
ib (t)
θd
θd = θa −θm
d axis
ia (t)
θm
θa
a axis
ic (t)
Self and Mutual Inductances (2)
Linear Model
Balanced Winding
θ me
P
= θm
2
Laa = Lbb = Lcc ≡ Lls + LA
L=
L=
Lca ≡ M s
ab
bc
L=
Llf + Lmf
f
Laf = Lsf cos(θ me )
2π
=
Lbf Lsf cos(θ me −
)
3
2π
=
Lcf Lsf cos(θ me +
)
3
Lls is leakage inductance of armature phase A winding which is about
10% of the maximum self inductance.
Llf
is leakage inductance of field winding.
Flux Linkage (1)
λa = Laaia + Labib + Lacic + Laf i f
= Laaia + M s (ib + ic ) + Laf i f
λb = Lbaia + Lbbib + Lbcic + Lbf i f
= Laaib + M s (ia + ic ) + Lbf i f
At steady state, i f is DC.
λc = Lcaia + Lcbib + Lccic + Lcf i f
= Laaic + M s (ia + ib ) + Lcf i f
λ f = L f i f + Laf ia + Lbf ib + Lcf ic
2π
2π
)]
= L f i f + Lsf [ia cos θ me + ib cos(θ me − ) + ic cos(θ me +
3
3
Flux Linkage (2)

 λa  
λ  
 b =
 λc  
  
λ f   L
 sf

Laa
Ms
Ms
cos θ me
Lsf cos θ me 
2π   ia 
Laa
Ms
Lsf cos(θ me − )  
3  ib
2π   
Ms
Laa
Lsf cos(θ me + )  ic 
3  
 i f 
2π
2π
Lsf cos(θ me − ) Lsf cos(θ me + )
Lf

3
3

Ms
Ms
Flux Linkage (3)
Y connected without neutral return or balanced ∆ connected :
λa = ( Laa − M s )ia + Laf i f
= Ls ia + Lsf i f cos(θ me )
λb = ( Laa − M s )ib + Lbf i f
2π
= Ls ib + Lsf i f cos(θ me −
)
3
λc = ( Laa − M s )ic + Lcf i f
2π
)
3
λ f = L f i f + Lsf [ia cos θ me +
= Ls ic + Lsf i f cos(θ me +
2π
2π
ib cos(θ me −
) + ic cos(θ me +
)]
3
3
L s ≡ Laa − M s = Lls + LA − M s
ia + ib + ic = 0
Flux Linkage (4)
When
ia + ib + ic = 0

 λa  
λ  
 b =
 λc  
  
λ f   L
 sf

Ls
0
0
cos θ me
Lsf cos θ me 
2π   ia 
Ls
0
Lsf cos(θ me − )   
3  ib
2π   
Ls
Lsf cos(θ me + )  ic 
0
3  
 i f 
2π
2π
Lsf cos(θ me − ) Lsf cos(θ me + )
Lf

3
3

0
0
Flux Linkage in Phase A Winding
There are a total of P groups. These groups may be connected in series, or parallel,
or partly series and partly parallel.
P = Ps C
Assume:
Then:
Note:
λa = Ps λgroup = Ps qN c k wΦ pk cos β
N a = Ps qN c
C
Nˆ a = N a k w
⇒ λa = Nˆ a Φ pk cos β
number of series turns per phase per circuit
number of parallel circuits
effective number of series turns per phase
per circuit on armature winding
P
B = B pk cos( θ a + β )
2
2 B pk Dl
λ pk= Nˆ a Φ pk
Φ pk =
P
Self Inductance of Stator Winding
If we apply current in
harmonic is:
4 µ0
Ba =
π g eff
Phase A winding, then the magnetic field for fundamental
 Nˆ a 
P
equation is true no matter how those P groups

ia cos θ a  This
of
windings
are connected. Note ia is phase A terminal current.
 P 
2
 Na is effective number of turns connected in series per phase.



Now, we can calculate flux in Phase A winding from its own current.
Following the formula derived in Notes Flux Linkage in Phase Winding
λa = Nˆ a Φ a , pk cos(0) = Nˆ a Φ a , pk
where
Φ a , pk =
2 Dl 4 µ 0  Nˆ a
ˆ

λa = N a
P π g eff  P
2 Ba , pk Dl
P
Ba , pk
4 µ 0  Nˆ a 
 ia
=
π g eff  P 
2

8µ 0 Dl  Nˆ a 
ia =

 ia


π g eff  P 

λa 8µ 0 Dl  Nˆ a
⇒ LA =
=
π g eff  P
ia
Laa = Lbb = Lcc ≡ Lls + LA




2
β =0
Mutual Inductance between
Stator Windings
If we apply current in Phase B winding, then the magnetic field is:
4 µ0
Bb =
π g eff
 Nˆ a

 P


2π 
P
ib cos θ a −


3 
2

Now, we can calculate flux linkage in Phase A winding from Phase B current.
where
2π
ˆ
λa |from Phase B winding = N a Φ b , pk cos(− )
3
2 Bb , pk Dl B = 4 µ 0  Nˆ a i
b , pk
Φ b , pk =
 P b
π
g
eff 

P
− 1 ˆ 2 Dl 4 µ 0  Nˆ a

Na
λa =
P π g eff  P
2

− 4 µ 0 Dl  Nˆ a
ib =


 P
g
π
eff


λa − 4 µ 0 Dl  Nˆ a
⇒ Ms =
=
ib
π g eff  P
2

L
 =− A

2

2

 ib

