Rotating MMF Wave
from Three Phase Winding
Stator MMF of Three Phases (1)
θb
θa
θc
2π
θ be = θ ae −
3
2π
θ ce = θ ae +
3
Stator MMF of Three Phases (2)
Fa =
Fb =
Fc =
P
θa )
2
∑ Fah
Fah = Fh ia cos(hθ ae ) = Fh ia cos(h
∑ Fbh
 
 P
2π 
2π 
Fbh = Fh ib cos hθ ae −
F
i
h
θ
cos
=
−


h b
 2 a
3
3


 
 
∑ Fch
 
 P
2π 
2π 
Fch = Fh ic cos hθ ae +
 = Fh ic cos h θ a +

3
2
3


 
 
h =1, 3, 5...
h =1, 3, 5...
h =1, 3, 5...
Total MMF from stator winding:
Fstator = Fa + Fb + Fc =

P
2hπ 
F
i
i
i
h
(
)
cos
cos(
θa ) +
+
+


∑
h a
b
c
3 
2
h =1, 3, 5...

(ib − ic )sin 2hπ sin(h P θ a )
3
2

Stator MMF of Three Phases (3)
For h = 1, 5, 7, 11 …
cos
2 hπ
1
=−
3
2
2hπ
3
=
3
2
For h = 1, 7, 13 …
sin
For h = 5, 11, 17 …
2 hπ
3
sin
=−
3
2
For h = 3, 9, …
cos
2 hπ
=1
3
sin
2 hπ
=0
3
For Y connected windings without neutral return:
ia + ib + ic = 0
Stator MMF of Three Phases (4)
Fstator
P
P
3
3
(ib − ic )sin(h θ a ) +
= ∑ Fh  ia cos(h θ a ) +
2
2
2
2

h =1, 7 ,13...
P
P
3
3

(
)
F
i
h
i
i
h
cos(
)
sin(
)
θ
θ
−
−
∑
h
a
a
b
c
a 
2
2
2
2


h = 5 ,11,17...
Note that the third harmonic components have been eliminated
if the machine is Y connected without neutral return.
Consider balanced three phase sinusoidal currents:
=
ia i pk cos(ωet + θi )
2π
)
=
ib i pk cos(ωet + θi −
3
2π
)
=
ic i pk cos(ωet + θi +
3
⇒
3
3
=
i pk sin(ωet + θi )
(ib − ic )
2
2
Note: ipk may change with time. It’s
a constant Ipk at steady state.
Stator MMF of Three Phases (5)
Fstator
h
P
3
−
ω
θ a + θi ) +
F
i
t
h
cos(
∑
h pk
e
2
2
=
1,7,13...
h
wave rotating in forward
direction (counterclockwise)
P
3
+
ω
θ a + θi )
F
i
t
h
cos(
∑
h pk
e
2
5,11,17... 2
wave rotating in backward
direction (clockwise)
From:



P
θa
θa 
=


ωet  h θ a ω=
t
ω
t


e 
e 
ω
ω
hP
2
2
/
e
h 



2ω
⇒ ωh =e
hP
angular velocity of the hth harmonic
Physical Understanding of Wave (1)
z
f (t − )
vp
z
f (− )
vp
z
f (t − )
vp
= f (−
z − v pt
vp
)
0
z
z1 = v p t
z
f (t − ) : stands for a wave propagating in + z direction with velocity v p .
vp
“forward wave”
Likewise:
z
g (t + ) : stands for a wave propagating in − z direction with velocity v p .
vp
“backward wave”
Physical Understanding of Wave (2)
Plot u + ( z , t ) = A cos(ωt − kz )
A
0
-A
z
Double Layer Lap Winding
Scheme
Double Layer Lap Winding Scheme (1)
Step 1. Find the nominal coil span in slots
S
S=
−1
If S/P is an integer
c
P
S


If S/P is not an integer Sc = max  fix   ,1
 P 
Step 2. Calculate the relative electric angle of In slot of all coils
The relative electrical angle (expressed in the range of -180o to 180o )
of the kth slot is:



θ slot =
(k ) mod (k − 1)γ + 180 ,360  − 180
Of course, the relative electrical angle of In slot of the kth coil is also:
θ coil =
(k ) mod (k − 1)γ + 180 ,360  − 180
Step 3. Readjust In slot angles if their magnitude are greater than 90o
For coil angles having a magnitude greater than 90o , the coil
direction is reversed; thereby changing the angle by 180o.
Double Layer Lap Winding Scheme (2)
Step 4. Picking up (S/m) coils for Phase A
For the angles calculated in last step, pick up those (S/m) close to 0o
as phase A coils.
Step 5. Calculate the slot offset to wind other phases
From
360
mod  Soff γ ,360  =
m

⇒ Soff
P 360
γ=
2 S
S
= 2
(1 + mn ) = 2q(1 + mn )
mP
where n is a integer value that makes Soff also an integer.
Phase B will start from slot (mod(Soff ,S)+1).
Step 6. Check to find out whether the winding is valid.
The winding is valid if all slots contain two coil sides each. If it’s not
valid, go back to Step 4 and pick up coils for Phase A.
Example 1: 3 Phase, 4 Pole, 15 Slot Machine (1)
Step 1. Find the nominal coil span in slots
  15  
=
Sc max
=
 fix  4  ,1 3
   
=
q
15
= 1.25
3× 4
Step 2. Calculate the relative electric angle of In slot of all coils
Step 3. Readjust In slot angles if their magnitude are greater than 90o
Example 1: 3 Phase, 4 Pole, 15 Slot Machine (2)
Step 4. Picking up (S/m) coils for Phase A
Step 5. Calculate the slot offset and wind other phases
Soff = 2q(1 + mn ) = 2 × 1.25 × (1 + 3 × 1) = 10
Phase B will start from slot 11
Phase C will start from slot mod(11+10, 15) = 6.
Example 1: 3 Phase, 4 Pole, 15 Slot Machine (3)
Step 6. Check and find out the winding is valid.
Example 2: 3 Phase, 10 Pole, 12 Slot Machine (1)
Step 1. Find the nominal coil span in slots
  12  
=
Sc max
=
 fix  10  ,1 1
   
=
q
12
= 0.4
3 × 10
Step 2. Calculate the relative electric angle of In slot of all coils
Step 3. Readjust In slot angles if their magnitude are greater than 90o
Example 2: 3 Phase, 10 Pole, 12 Slot Machine (2)
Step 4. Picking up (S/m) coils for Phase A
Step 5. Calculate the slot offset and wind other phases
Soff = 2q(1 + mn ) = 2 × 0.4 × (1 + 3 × 3) = 8
Phase B will start from slot 9
Phase C will start from slot mod(9+8, 12) = 5
Example 2: 3 Phase, 10 Pole, 12 Slot Machine (3)
Phase C – Group 1 5
6
4Phase B – Group 2
3
5
4
6
3
7
7
Phase A – Group 2
2
2
8
Phase A – Group 1
1
8
9
1
10
9
Phase B – Group 1 10
12
11
11
12
Phase C – Group 2
Example 2: 3 Phase, 10 Pole, 12 Slot Machine (4)
Slot Number
Phase A
1
In
2
Out & Out
3
In
Phase B
Phase C
Out
Out
4
In & In
5
Out
6
In
Out & Out
7
Out
8
In & In
9
Out
In
In
10
Out & Out
11
In
12
Step 6. Check and find out the winding is valid.
Out
In & In
Slot Pole Combination for Three Phase Machine
S/P Ratios for Three Phase Machine
Winding Diagram
Example 1
3 Phase, 24 Slot, 2 Pole, 5/6 Pitch
Three Phase Windings
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
B2
A1
C2
B1
A2
C1
Phase A
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Phase B
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Phase C
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Example 2
3 Phase, 36 Slot, 4 Pole, 8/9 Pitch
Three Phase Windings
1
2
3
4
B2
5
6
A1
7
8
9
C2
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
B1
C1
A2
Phase A
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Phase B
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
Phase C
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36