Industrial Electrical Engineering and Automation
L8 – Electric Circuits
L8: Electric circuits
The MMF and field distribution of an AC
winding
Industrial Electrical Engineering and Automation
Lund University, Sweden
Avo R
• Geometry + medium =
a circuit
– Soft magnetic core
– Permanent magnets
• Energy conversion =
a balance
– Energy vs torque
• Intermediate magnetic field
T  rBlNi   i
T   t s rdA  
A
A
1
0
Bn B s rdA
2
Today’s goal
– Facilitate flux linkage
– Prevent magnetisation
losses
Industrial Electrical Engineering and Automation
Industrial Electrical Engineering and Automation
Previous lectures
Design of Electrical Machines
• Scope on AC 3-phase
machines
• Winding configuration
for the stator
• Winding factor
• Insulation system

WC
BdH dV

 VM 0

H
T
Avo R
Design of Electrical Machines
3
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Avo R
Design of Electrical Machines
4
1
L8 – Electric Circuits
Placement of coils A
• Excitation ->
establish H-field
I
H
• Core ->
guide the B-field
• Coil ->
link by inducted E-field
• Winding ->
add e
B
 Edl    t dA
Avo R
Design of Electrical Machines
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Electromagnet at generator operation
5
N·I=Aw·J·Kf
ω
• Skewing
•…
magnet
Ψm
ψa
x
ia
Avo R
Design of Electrical Machines
ψb
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θmech
6
3-phase system
• Concentrated (non overlapping coils)
• Distributed (overlapping coils)
• TARGET: n-phase symmetric winding
Design of Electrical Machines
coil
• Coil width
(pitch)
Placement of coils B
Avo R
Ψ=N·Φ=L·I
• Distribution
– Rotates and links (projects)
• Spatial distribution of coils
ψm
EIEN20 Design of Electrical Machines, IEA, 2016
ψa
– Phase-A ej0·⅔π = a0
– Phase-B ej1·⅔π = a1
– Phase-C ej2·⅔π = a2
• Coordinate frames αβ, dq, xy
• Space vector

ψc
7
• Excitation flux Ψm
Avo R

     j   k  a   b e

Design of Electrical Machines
j
2
3
 c e
j
4
3




8
2
L8 – Electric Circuits
Fsp1 
3 4 k1 N s ˆ
 
 is
2  P
1 dFsp ( )
Fsp ( )   K s ( )  ris d  K s ( ) 
ris d
0
2
 P 

 P
 Fsp1 cos
 K s ( )  
 et  

 2
 2ris 
6 kN

 P
  1 s  iˆs  cos
 e t 
 Dis

 2
Avo R
Design of Electrical Machines
9
7
0.
0.
0.85
6
0.95
0.7
0.
0.8
5
0.9
0.8
0.5
0.9
0. 3
0.9
0.8
0.7
6
0.
5
0.8
20
0.95
0.
4
0.9
0.5
5
0.8 .8
0
5
0.9
0.2
0.6
0.7
0.9
0. 3
15
50.3
0.9
0.9
0.85
0.8
0.6
0.7
0.2
0.25
0.5
0.9
15
5
0.2
0.5
0.99
0.85
0.8
0.7
0.6
20
0.25
0.2
25
number of poles, N []
xo x x ox Np=12
xo 6 1383
oo
o
o 1272
x
o
x q=1.25 K w =0.95
11
11
6
o
x
o
10
5
xx 15
xx
ox 11 1494
o
ox oo x
Ns=15
Np=12
x x oo
xx
oo
x 6
o
x
o
11
91
78
56
34
12
10
21
q=1.00 K w =1.00
o
x
o
x
11
xx
oo
oo x x
Ns=12
Ns=15
0.95
0.9
9
30
Ns=12
xo xx ox o Np=14
3
xo 167 612
x
oo
oo 112
5
14
x
ox q=1.29 K =0.90
o
110
1
w
xo 11
x
615 o
18
xx 9 13
xx
4
16
ox 178
o
x
ox oo xo
Ns=18
xx o x oo x Np=16
oo 134 1456
x
ox
ox 123
18
9
oo
xx q=1.13 K =0.95
10
1 1
w
oo 11
x
11
2 x
xo 178 167
xo
16
xx 156
o
o
oo xo xx
x x oo xo Np=14
oo 5 66 7 8
xx
x
x 4
14
o
15
o
o q=1.07 K w =0.95
11
x
2
3
x
oo 13
1211
o
xx 11 10 9
xo
x
oo x x o
Ns=15
Np=14
xx ox
4
oo
xo 115 10
o 6
x
o
o
171
q=0.86 K w =0.93
x 2
6
12 x
o 8
11
x x 39
xo
xo x o o
Ns=12
x x o o xo Np=16
oo1261110 9
xx
x
x 13
3
o
2
o
o q=0.94 K w =0.95
1 1
x
15
14
x
oo 4
5 6
o
xx 11 7 8
xo
x
oo x x o
Np=16
xo x o
xo
xo 12963
o 6
x
x
o
10
1 1
4
7
q=0.75 K w =0.87
o
x
o
11
ox 11852
ox
xo x o x
35
p
• Distribution of concentrated coils – distributed
concentrated winding
Avo R
– Voltage and flux at
nominal operation
point
• Number of turns
per coil:
Ntc=Nt/integer
• Number of strands
– Wire diameter
according to
frequency
11
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0. 3
4
5
0.8
0.7
Ns=18
Design of Electrical Machines
10
AC machine
• Number of turns per
phase: Nt=U1/Ψ1ω1
Design of Electrical Machines
9
0.6
0. 7
25
Winding specification
Avo R
5
0. 9
0. 8
5
0.9
0.
0.95

 P
Fsp ( )  Fsp1 sin 
 e t 

 2
0.95
0.95
xo 6xo xo x Np=12
xo 17
o
11
14
2
5
8
xo
xo
o
x
x q=1.50 K =0.87
o
10
13
16
11
4
7
w
ox 11
ox
ox
ox
12
15
18
3
6
9
16
ox
x
o
ox ox ox
Ns=18
0.95
0.9
0.95
• Sinusoidal distribution
4
0.9
30
number of teeth, Nt []
• S-slot stator: Ns={3,6,9,12, …}
0.
0.8
0.8
• P-pole excitation: Np=2 (4,6..)
35
0.5
• N-phase system: Nph=3
Industrial Electrical Engineering and Automation
Number of coils vs poles
0.9
Industrial Electrical Engineering and Automation
Sinusoidally-fed PM motor
• Sinusoidally-fed PM
motor
• Electromagnetic torque
Tem  0.5 N p rq BlNi
• Sinusoidal distribution of
stator MMF
Avo R
Design of Electrical Machines
12
3
L8 – Electric Circuits
MMF of a distributed winding
Flux & MMF waveforms
• Focus on stator (Hpm=0,μpm=1)
• Amperes circuital law (PQRS)
Q
R
P
S
θ2
Q
R
S
P
P
Q
R
S
 Ni   Hdl   Hdl   Hdl   Hdl   Hdl
θ
g
• H-field uniform in gap (μfe=∞)
θ1
Q
S
P
R
 Ni   Hdl   Hdl  gH    gH  
r
2
r
1
• MMF distribution
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F(θ), B(θ)
stator
0.5NI
Design of Electrical Machines
2
F   Bgm ( )  K s ( )  ris  le d 
0

2
Dis le Bgm1 K s1  3le Bgm1k1 N s iˆs
θ
0
Fph 
π
4 Nt I 1
 2 h
• A single-phase full-pitch
winding – one slot per pole
• Lengthy end-turns
2π
• Open for space harmonics
sin h  h odd
4 NI
Fp     t
 2
Avo R
rotor
1
1


 sin   sin 3  sin 5  ...
3
5


Design of Electrical Machines
B gm sin  sin  e 
3 4 k1 N s ˆ
 
 is sin  e  et 
2  Np
2
Fsp ( )   K s ( )  ris d  K s ( ) 
0
K s ( ) 
1 dFsp ( )
ris d
6 k1 N s ˆ
 P


 is  cos
 et 
2ris
 2


Design of Electrical Machines
14
Fractional pitch winding
• MMF changes abruptly
stator
4

• Sinusoidal current density
2π
Fsp ( ) 
Avo R
15
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½NtI
NtI
B g ( ) 
• Electromagnetic force
Full pitch winding
F(θ)
rotor
π

13
θ
B
0
F     Ni  gH    gH  0 
Avo R
• The air-gap flux density
F(θ)
½NtI
½NtI
• Coil pitch/Pole pitch is a
fractional number
stator
θ•
0
 
cos h 
4 NI
 2
Fph   t 
 2
h
Fp   
Avo R
rotor
π
γ

F
h 1, 3, 5,
ph
h odd
2π
Ordinarily two layer lap
wound type
• MMF waveform becomes
more sinusoidal
• Prevent harmonic content
sin h 
Design of Electrical Machines
16
4
L8 – Electric Circuits
Distributed windings
Industrial Electrical Engineering and Automation
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Pitch factor
• Coil pitch w
• Pole pitch p
• Pitch factor kph
h
 ...
2
 h w      h 
 sin 
 odd h
  sin 
  p  2   2 
k ph  cos
• w/p ≈0.83 suppress 5th
and 7th harmonic
Avo R
Design of Electrical Machines
17
1/(2q)NtI
1/(2q)NtI
F(θ)
q=2
θ
0
π
 hq 
sin 

hq
2 
 4  N I 
cos
Fph    t  
2
   2h  q sin  h 
 
 2 
γ
Fp 

F
h 1, 3, 5,
ph
Avo R
19
EIEN20 Design of Electrical Machines, IEA, 2016
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h odd
• Harmonic distribution
factors for a winding with
600 phase belt
• Slot hamonics hk = 6kq
+/- 1 = 2kS/P +/- 1
Design of Electrical Machines
2π
h odd
Design of Electrical Machines
18
Slot openings
• Number of slots per
phase and per pole q
Avo R
rotor
• If q=2 is not possible,
rather than selecting
q=1, q can be selected
as a fraction between 1
and 2
sin h 
Distribution factor
 hZ 
 hq 

sin 
sin 

 2 
2


 p 
k dh 
 h 


q sin   q sin  hZ 
  2q 
 2 
 p 
• Removal of undesirable
harmonics
stator
• Slot opening has filtering
effect on the MMF
½NtI
NtI
F(θ)
stator
χ
0
π
 
sin  h 
4 NI 1  2
Fph   t 
 2 h  
h 
 2
Fp   
Avo R

F
h 1, 3, 5,
ph
rotor
θ•
2π
It is assumed that the
MMF varies linearly
across the slot
h odd
sin  
Design of Electrical Machines
20
5
L8 – Electric Circuits
½NtI
NtI
F(θ)
• Reduces slot harmonics
stator
θ
0
Fph 
rotor
π
α
 
sin  h 
4 Nt I 1  2 


 2 h  
h 
 2
Fp   

F
h 1, 3, 5,
ph
Slot opening factor
Industrial Electrical Engineering and Automation
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Skewing
• Reduces flux variation
that is due to the fringing
• Skew angle
2π
h odd
sin  
Avo R
Design of Electrical Machines
21
• Skew angle α
• Slot width angle χ
• Filtering effect β
k h
Avo R
 4  N I 
Fph    t k dh k ph k h k sh h odd
   2h 
• Distribution factor kdh
• Pitch factor kph
• Slot opening factor koh
• Skewing factor ksh
Avo R
22
Specify 3-phase winding
• Winding factor is ratio of
actual winding MMF to the
full-pitched winding MMF
• A machine with a high pole
number AND a high number
of slots per pole and phase
is impractical to build
• The number of slots MUST
be a multiple of the number
of phases – symmetry.
Design of Electrical Machines
Design of Electrical Machines
23
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Winding factor
 Z 
 

sin  h  sin  h
 p 2 
2


h odd


 
 Z 
h 
h



2


 p 2
Avo R
Design of Electrical Machines
24
6
L8 – Electric Circuits
• 10-pole machine
• Select number of slots
–
–
–
–
Fractional slot winding example : II
Industrial Electrical Engineering and Automation
Industrial Electrical Engineering and Automation
Fractional slot winding example : I
q=1 -> 10*3*1 = 30 slots
q=2 -> 10*3*2 = 60 slots
Pick 42 slots -> q = 42/(10*3) = 7/5
q=slots per pole per phase.
• q=7/5 implies 1 or 2 slots occupied per pole
• Coil arrangements: 2,1,1,2,1 OR 1,1,2,2,1 slots per
pole per phase for 5 poles.
• The other 5 poles are copies...
• Pick pitch – same for all coils
• Phase shift between coils =
360/42*p/2= 42 6/7 °
• Suggest coil span 4 slots ->
pitch angle = 180-4* 42 6/7 ° =
8 4/7 °
• Pitch factor
kp1 = cos(8 4/7°/2)=0.997
1
2
3
4
5
6
7
8
9 10
11 12 13 14 15
16
17
18 19
20
21 22 23
•Winding sketch:
Avo R
Design of Electrical Machines
25
Avo R
17
9
1 18
2
19
8
11
The phase coils forms
an equivalent 60 degree
phase belt, distributed
over 7 slots:
 hZ 
16

sin 
3
 p2 

 
20 k 
dh
7
12
15
6
Avo R
4
21
14
5
Fractional slot winding example : IV
1
10
13
Design of Electrical Machines
 hZ 

q sin
  p 2q 


 1 7

 90 
sin 
 21
  0.956

 1  7 90 
 
7 sin
 21 7 
27
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26
2
3
4
5
6
7
8
9 10
11 12 13 14 15
16
17
18 19
20
21 22 23
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Industrial Electrical Engineering and Automation
Fractional slot winding example : III
Design of Electrical Machines
Avo R
Design of Electrical Machines
28
7
L8 – Electric Circuits
End-turns
Industrial Electrical Engineering and Automation
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Fractional slot winding example : V
Avo R
Design of Electrical Machines
29
• Single-layer concentric
and equal coils
• Double layer distributed
winding with concentric
coils and concentrated
winding
Avo R
Design of Electrical Machines
Conductor fill factor
• Slot content
– Insulation system
– Electrical strands
• Insulation system
– Phase/overhang insulation (a)
– Ground insulation (b)
– Turn insulation (c)
• Insulation tests
– Phase to phase (1)
– Phase to ground (2)
Industrial Electrical Engineering and Automation
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Winding configuration
Design of Electrical Machines
• Coating increases the
diameter of copper
strand by 5% to 7%
hs
ht
hw
hins
r
• Different layers of main
insulation
bs1
• Round conductor in
rectangular slot Kf=π/4
hss
h1  hs  ht  hw  r
b1  2r  2hins 
h1  2hins
2r  bs1 
h1
2


K f  N s 0.25 N p1 d is1  N p 2 d is 2
31
EIEN20 Design of Electrical Machines, IEA, 2016
• Max range 60-70%
Ais  0.5 r  hins   0.5h1  2hins b1  2r  2hins   hss 0.5bs1  r  10 
– Turn to turn (3)
Avo R
30
Avo R
d is  d cu   is
2
2
/ A
is
Design of Electrical Machines
32
8
L8 – Electric Circuits
Thermal limits
5
10
4
125
5
100
105
80
60
60
75
80
10
3
10
40
40
20
40
40
40
40
E
B
F
H
2
0
A
10
140
Class 200
Class 155
160
180
200
220
240
260
280
temperature  [C]
insulation classes
• Insulation lifetime is shortened radically if temperature
exceeds the limit and that is due to accelerated oxidation
process in the insulation material.
Avo R
Design of Electrical Machines
Industrial Electrical Engineering and Automation
120
10
thermal life [h]
140
10
15
ambmax
allow ed
safety
160
temperature  [C]
Industrial Electrical Engineering and Automation
Modeling a winding
5
180
33
eff 
L
eff

cond  ins
ins  k f  cond  1  k f 
Lkf
cond

L  1  k f 
ins

L  ins  k f  cond  1  k f 
cond  ins
• Equivalent thermal conductivity of a winding is given
by the filling factor of the conductor strands (copper
in this example) and the thermal conductivity of the
medium between the conductor strands
Avo R
Design of Electrical Machines
34
Winding design
Winding insulation and heat transfer
Industrial Electrical Engineering and Automation
Industrial Electrical Engineering and Automation
8
relative thickness of rhe insulation di, [%]
7
6
5
4
3
2
1
0
0
0.5
1
1.5
2
2.5
3
3.5
wire diameter d , [mm]
4
4.5
5
c
5
2
2
1
6
EIEN20 Design of Electrical Machines, IEA, 2016
B()
MMF()
T()
4
6
Ns=6
Np=2
q=1.00
span=3.0
kd=1.000 kp=1.000 ks=0.955 k1=0.955
12
11
windings where q<1
10
41
16
20
33 37
12
9
14
8
15
7
16
1

5

9
13 17 21 25 29 

 41

33
37
24
3
7
28
32 15
11 36
6
17
5
18
26 5
4
19
9 30
3
34 13
20
2
21
1
22
42
23
5
9
13
17
21
25
29
33
37
41
39
26
38
27
37
25
8
31
10
6
27
40
25
42 21
14
35
41
24
Avo R
17 38
1
1 22
39
18
B()
MMF()
T()
Ns=42
4
29
23
40
2
19
Np=10
q=1.40
span=4.0
kd=0.978 kp=0.997 ks=0.955 k1=0.931
• →Ubr
3
– Dielectric
properties and
voltage limits
Does not consider properly concentrated
5
13
• →hs
– Heat dissipation
and temperature
limits
36
35
30
35
– Winding layout and
resulting MMF
5
1
4
29
Design of Electrical Machines
• k1→Fsp()
5
1
28
Avo R

1
3
31
32
33
34
Does not consider properly concentrated
windings where q<1
Design of Electrical Machines
36
9
L8 – Electric Circuits
Assignment A4 part 1
Assignment A4 part 2
-0.01
-0.02
-0.03
gap radius
tooth ref nodes
yoke ref nodes
-0.04
-0.05
-0.05
B
gnL
(), [T]
gtL
(), [T]
gn0
(), [T]
gt0
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
0.05
0
30
60
90
120
150 180 210
angle , [deg]
240
270
300
330
360
5
x 10
1
B
0.9
B
(), [T]
gnL
(), [T]
gtL
0.8
B gn0(), [T]
0.7
B
(), [T]
gt0
0.6
0.5
0.4
0.3
0.2
0.1
0
Avo R
B
0.4
0
2
4
6
8
10
12
harmonic order, [-]
14
16
18
20
Design of Electrical Machines
4
3
2
1
0
-1
 gnL(), [N/m2]
-2
 gtL(), [N/m2]
• con.femm=2
• pos_end=120
• pos_step=60
300
6
200
5.5
100
torque T, [Nm]
0
B
0.6
current I, [A]
0.01
0.8
0
-100
a
b
c
-300
0
 gt0(), [N/m2]
0
30
60
90
120
150 180 210
angle , [deg]
240
270
300
330
37
EIEN20 Design of Electrical Machines, IEA, 2016
150
200
angle , [deg]
250
300
350
0
50
100
150
200
angle , [deg]
250
300
100
150
200
angle , [deg]
250
300
350
1.2
0
-5
a
b
c
0
Avo R
100
-3
x 10
-10
360
50
3.5
5
 gn0(), [N/m2]
-3
5
4.5
4
-200
• I(θ),Ψ(θ),
T(θ), B(θ)
• Isx=0, Isy>0
• T>0, ω>0
flux linkage , [Vs]
Magneticflux density in the airgap B(), [N/m2]
0.02
(), [T]
flux density in the stator core B, [T]
• Phasors?
0.03
B
Industrial Electrical Engineering and Automation
– Weighted
– Line integral
– Estimation
0.04
1
Magnetic shear stress in the airgap (), [N/m2]
• con.femm=1
• θ=0 el.deg
• I(θ),Ψ(θ),
T(θ), B(θ)
• Bgm1?
• EMF?
• T calculation?
Magnetic flux density in the airgap Bg, [T]
Industrial Electrical Engineering and Automation
0.05
50
1
0.8
0.6
150
200
angle , [deg]
250
300
350
Design of Electrical Machines
B
B
0.2
100
B
st1
Bst2
0.4
0
st3
sy1
Bsy2
B
50
sy3
350
38
10