previous lectures today’s goal - iea - lund university · previous lectures • geometry + medium...

Industrial Electrical Engineering and AutomationLund University, Sweden

L8: Electric circuits

The MMF and field distribution of an AC winding

eien20vt18

Avo R Design of Electrical Machines 2

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Previous lectures

• Geometry + medium = a circuit

– Soft magnetic core– Permanent magnets

• Energy conversion = a balance

– Energy vs torque• Intermediate magnetic field

– Facilitate flux linkage– Prevent magnetisation

losses A

snA

s rdABBrdAtT0

1

MV

HC dVBdHWT

0

irBlNiT


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Today’s goal

• Scope on AC 3-phase machines

• Winding configuration for the stator

• Winding factor• Insulation system


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Electromagnet at generator operation

• Excitation -> establish H-field

• Core -> guide the B-field

• Coil -> link by inducted E-field

• Winding -> add e

I H

dAtBEdl


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Placement of coils A

• Distribution• Coil width(pitch)

• Skewing• …

ψa

Ψ=N·Φ=L·I

N·I=Aw·J·Kf

coil

Ψm

ia

θmech

ω

x

magnet


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Placement of coils B

• Concentrated (non overlapping coils)• Distributed (overlapping coils)• TARGET: n-phase symmetric winding


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3-phase system

• Excitation flux Ψm– Rotates and links (projects)

• Spatial distribution of coils– Phase-A ej0·⅔π = a0

– Phase-B ej1·⅔π = a1

– Phase-C ej2·⅔π = a2

• Coordinate frames αβ, dq, xy• Space vector

3

43

2

j

c

j

ba eekj

ψmψa

ψb

ψc


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Sinusoidally-fed PM motor

• N-phase system: Nph=3• P-pole excitation: Np=2 (4,6..)• S-slot stator: Ns={3,6,9,12, …}• Sinusoidal distribution

tPiDNk

tPrPK

dd

rKdrK

iPNktP

eis

s

espis

s

sp

ississsp

sspespsp

2cosˆ6

2cos

2)(

)(1)()()(

ˆ423

2sin)(

1

1

2

0

111

s

s

F

FF

FFF


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Number of coils vs poles

15 20 25 30 35

15

20

25

30

35

number of poles, Np []

num

ber o

f tee

th, N

t []

0.25

0.25

0.25

0.5

0.5

0.5

0.6

0.6

0.6

0.6

0.7

0.7

0.7

0.7

0.7

0.7

0.8

0.8

0.8

0.8

0.8

0.8

0.8

0.85

0.85

0.85

0.85

0.85

0.85

0.85

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.9

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.95

0.99

0.99

0.95

0.99

0.2

0.2

0.2

0.3

0.3

0.3

0.3

0.4

0.4

0.4

0.5

0.5

0.5

0.6

0.6

0.7

0.7

0.8

0.9

• Distribution of concentrated coils – distributedconcentrated winding

1

6

11

oo

xxooxx

ooxxoo

xxoo x x

ooxx

123456789101112q=1.00 Kw =1.00

Ns=12 Np=12

1

6

11

oxoo

xoxxoxoxoo

xoxx

ox ox oo xoxx

ox1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

q=1.25 Kw =0.95

Ns=15 Np=12

1

6

11

16

oxox

oxoxoxoxox

oxoxoxox

ox ox ox ox oxox

ox1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

q=1.50 Kw =0.87

Ns=18 Np=12

1

6

11

ox

ooxoxx

oxooxo

xxo x o o

xox

x

1

2

3

45

6

7

8

9

1011

12

q=0.86 Kw=0.93

Ns=12 Np=14

1

6

11

ooxx

oxooxxooxx

oxoo

xx oo xx oxoo

x

x

12

3

45 6

7 8

9101112

13

1415q=1.07 Kw=0.95

Ns=15 Np=14

1

6

11

16

oxoo

xoxoxxoxoxoo

xoxoxx

ox ox oo xoxo

xxox

1

2

3

4

5

6

7

8

9

10

1112

13

14

15

16

17

18

q=1.29 Kw=0.90

Ns=18 Np=14

1

6

11

ox

oxoxox

oxoxox

oxo x o x

oxo

x

1

2

3

4

5

6

7

8

9

10

11

12

q=0.75 Kw =0.87

Ns=12 Np=16

1

6

11

ooxx

oxooxxooxx

oxoo

xx oo xx oxoo

x

x

12

3

45 6

7 8

9101112

13

1415

q=0.94 Kw =0.95

Ns=15 Np=16

1

6

11

16

ooxo

xxooxoxxoo

xoxxooxo

xx oo xo xx ooxoxx

1

2

34

5

67

8

9

10

11

1213

14

1516

17

18

q=1.13 Kw =0.95

Ns=18 Np=16


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Winding specification

• Number of turns per phase: Nt=U1/Ψ1ω1

– Voltage and flux at nominal operation point

• Number of turns per coil: Ntc=Nt/integer

• Number of strands– Wire diameter

according to frequency


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AC machine

• Sinusoidally-fed PM motor

• Electromagnetic torque

• Sinusoidal distribution of stator MMF

BlNirNT qpem 5.0


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MMF of a distributed winding

• Focus on stator (Hpm=0,μpm=1)• Amperes circuital law (PQRS)

• H-field uniform in gap (μfe=∞)

• MMF distribution

θ2 θ1

θ

P

Q R

S g

P

S

S

R

R

Q

Q

P

HdlHdlHdlHdlHdlNi

12 rr

S

R

Q

P

gHgHHdlHdlNi

0 gHgHNiF Avo R Design of Electrical Machines 14

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Flux & MMF waveforms

• Electromagnetic force

• The air-gap flux density

• Sinusoidal current density

θ

0 π 2π rotor

stator F(θ), B(θ)

0.5NI B

egmg BB

sinsin4)(

tPirNkK

ddF

rKdrKF

tiNNkF

eis

ss

sp

ississsp

eep

ssp

2cosˆ

26)(

)(1)()()(

sinˆ423)(

1

2

0

1

s

s

ssgmesgmeis

eissgm

iNkBlKBlD

dlrKBF

ˆ32

)()(

1111

2

0


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Full pitch winding

• MMF changes abruptly• A single-phase full-pitch

winding – one slot per pole• Lengthy end-turns• Open for space harmonics

...5sin513sin

31sin

24

sin12

4

INF

oddhhh

INF

tp

tph

θ

0 π 2π rotor

stator F(θ) NtI

½NtI


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Fractional pitch winding

• Coil pitch/Pole pitch is a fractional number

• Ordinarily two layer lap wound type

• MMF waveform becomes more sinusoidal

• Prevent harmonic content

θ

0 π 2π rotor

stator F(θ) ½NtI

½NtI

γ

,5,3,1sin

odd2cos

24

hphp

tph

hFF

hh

INF

h


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Pitch factor

• Coil pitch w

• Pole pitch p

• Pitch factor kph

• w/p ≈0.83 suppress 5th and 7th harmonic

hoddhh

hk

p

w

ph

2sin

2sin

...2

cos


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Distributed windings

• Removal of undesirable harmonics

• If q=2 is not possible, rather than selectingq=1, q can be selectedas a fraction between 1 and 2

θ

0 π 2π rotor

stator F(θ) 1/(2q)NtI

1/(2q)NtI

γ

q=2

,5,3,1sin

odd2

cos

2sin

2sin

24

hphp

tph

hFF

hhqhq

hq

hINF


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Distribution factor

• Number of slots per phase and per pole q

• Harmonic distribution factors for a winding with 600 phase belt

• Slot hamonics hk = 6kq +/- 1 = 2kS/P +/- 1

oddh

qhZq

hZ

hq

hq

k

p

pdh

2sin

2sin

2sin

2sin


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Slot openings

• Slot opening has filtering effect on the MMF

• It is assumed that the MMF varies linearly across the slot

θ

0 π 2π rotor

stator F(θ) NtI

½NtI

χ

,5,3,1sin

odd

2

2sin

12

4

hphp

tph

FF

hh

INF

h

h


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Skewing

• Reduces slot harmonics• Reduces flux variation

that is due to the fringing• Skew angle

θ

0 π 2π rotor

stator F(θ) NtI

½NtI

α

,5,3,1sin

odd

2

2sin

12

4

hphp

tph

FF

hh

INF

h

h


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Slot opening factor

• Skew angle α• Slot width angle χ• Filtering effect β

odd

2

2sin

2

2sin

hZ

Z

k

p

ph

h

h

h

h


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Winding factor

• Distribution factor kdh

• Pitch factor kph

• Slot opening factor koh

• Skewing factor ksh

• Winding factor is ratio of actual winding MMF to the full-pitched winding MMF

• A machine with a high polenumber AND a high numberof slots per pole and phaseis impractical to build

• The number of slots MUST be a multiple of the numberof phases – symmetry.

odd2

4 hkkkkhINF shhphdh

tph


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Specify 3-phase winding


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nFractional slot winding example : I

• 10-pole machine• Select number of slots

– 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...


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Fractional slot winding example : II• 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 factorkp1 = 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:


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Fractional slot winding example : III1

2

3

4

56

7

8

9 10

11

12

1314

15

16

1718

19

20

21

The phase coils forms an equivalent 60 degreephase belt, distributedover 7 slots:

956.0

790

2171sin7

9021

71sin

2sin

2sin

qhZq

hZ

k

p

pdh


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Fractional slot winding example : IV1 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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nFractional slot winding example : V


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End-turns

• Single-layer concentric and equal coils

• Double layer distributed winding with concentric coils and concentrated winding


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Double layer diamond winding

• Animation 1: Increasing coil span

• Animation 2: increasing q=Ns/(Np*Nph)


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Diamond windings in Ansys


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Winding configuration

• 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)– Turn to turn (3)


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Conductor fill factor

• Coating increases the diameter of copper strand by 5% to 7%

• Different layers of main insulation

• Round conductor in rectangular slot Kf=π/4

• Max range 60-70%

hins r

hs

hwht

bs1

hss

iscuis

isispispsf

sssinsinsinsis

sins

ins

wts

ddAdNdNNK

rbhhrbhhhrA

brh

hhhrb

rhhhh

/25.0

105.02225.05.0

22

22

222

211

1112

11

11

1


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Thermal limits

• Insulation lifetime is shortened radically if temperature exceeds the limit and that is due to accelerated oxidation process in the insulation material.

A E B F H0

20

40

60

80

100

120

140

160

180

insulation classes

tem

pera

ture

[

C]

40

60

5

40

75

5

40

80

10

40

105

10

40

125

15ambmaxallow edsafety

140 160 180 200 220 240 260 280102

103

104

105

temperature [C]

ther

mal

life

[h]

Class 200Class 155


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Modeling a winding

• 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

inscond

fcondfins

ins

f

cond

f

eff

kkLkLkLL

11

fcondfins

inscondeff kk

1


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Winding insulation and heat transfer

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

1

2

3

4

5

6

7

8

wire diameter dc, [mm]

rela

tive

thic

knes

s of

rhe

insu

latio

n d i, [

%]


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Winding design

• k1→Fsp()– Winding layout and

resulting MMF• →Ubr

– Dielectric properties and voltage limits

• →hs– Heat dissipation

and temperature limits

42

41

40

39

38

37

36

35

34

33

32

31

30

29

28

27

26

25

24

23

22 21

20

19

18

17

16

15

14

13

12

11

10

9

8

7

6

5

4

3

2

11

2

3

4

5

6

78

91011121314

15

16

17

18

19

20

21

22

23

24

25

26

27

2829

30 31 32 33 3435

36

37

38

39

40

41

42

B()MMF()T()

1 5 9 13 17 21 25 29 33 37 41

Ns=42 Np=10 q=1.40 span=4.0

kd=0.978 kp=0.997 ks=0.955 k1=0.931

Does not consider properly concentratedwindings where q<1

1 5 9 13 17 21 25 29 33 37 41

6

5

43

2

1

1

2

3

4

5

6 B()MMF()T()

1 5

Ns=6 Np=2 q=1.00 span=3.0

kd=1.000 kp=1.000 ks=0.955 k1=0.955

Does not consider properly concentratedwindings where q<1

1 5


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Assignment A4 part 1

-0.05 0 0.05-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.03

0.04

0.05

gap radiustooth ref nodesyoke ref nodes

0 30 60 90 120 150 180 210 240 270 300 330 360

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

angle , [deg]

Mag

netic

flux

dens

ity in

the

airg

ap B

( ),

[N/m

2 ]

BgnL(), [T]

BgtL(), [T]

Bgn0(), [T]

Bgt0(), [T]

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

harmonic order, [-]

Mag

netic

flux

den

sity

in th

e ai

rgap

Bg, [

T]

BgnL(), [T]

BgtL(), [T]

Bgn0(), [T]

Bgt0(), [T]

0 30 60 90 120 150 180 210 240 270 300 330 360

-3

-2

-1

0

1

2

3

4

x 105

angle , [deg]

Mag

netic

she

ar s

tress

in th

e ai

rgap

(

), [N

/m2 ]

gnL(), [N/m2]

gtL(), [N/m2]

gn0(), [N/m2]

gt0(), [N/m2]

• con.femm=1• θ=0 el.deg• I(θ),Ψ(θ),

T(θ), B(θ)• Bgm1?• EMF?• T calculation?

– Weighted– Line integral– Estimation

• Phasors?


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Assignment A4 part 2

0 50 100 150 200 250 300 350

-300

-200

-100

0

100

200

300

angle , [deg]

curre

nt I,

[A]

abc

0 50 100 150 200 250 300 350-10

-5

0

5x 10

-3

angle , [deg]

flux

linka

ge

, [V

s]

abc

0 50 100 150 200 250 300 350

3.5

4

4.5

5

5.5

6

angle , [deg]

torq

ue T

, [N

m]

0 50 100 150 200 250 300 350

0.2

0.4

0.6

0.8

1

1.2

angle , [deg]

flux

dens

ity in

the

stat

or c

ore

B, [

T]

Bst1Bst2

Bst3

Bsy1

Bsy2Bsy3

• con.femm=2• pos_end=120• pos_step=60

• I(θ),Ψ(θ), T(θ), B(θ)

• Isx=0, Isy>0 • T>0, ω>0

previous lectures today’s goal - iea - lund university · previous lectures • geometry + medium...

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