Optimal Design and Analysis of Linear Permanent Magnet Machines

8/17/2019 Optimal Design and Analysis of Linear Permanent Magnet Machines

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Optimal Design and Analysis ofLinear Permanent Magnet Machines


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Optimal Design and nalysis o Linear Permanent~ a g n e t ~ a c h i n e s

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1 0 2 0 6 2


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e-Theses & Dissertations (101 ) i

12

I wish to express my deepest gratitude to Mr. Philippe Wendling,vice president of Magsoft and Dr. Tan Pham, technical director at

Magsoft for their help and support. Flux and Got-IT are such great and

powerful tools. Without your guidance, I will not able to use them very

well and carry out my dissertation.

Last but not least, a grateful acknowledgement to my girlfriend Ms.

Daisy Ko.


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e-Theses & Dissertations (101 ) ii

8 9

:


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e-Theses & Dissertations (101 ) iii

Abstract

In this dissertation, two kinds of linear motors, the permanent magnet

linear motor (PMLM) and the hybrid excited linear flux switching

permanent magnet motor (HLFSPMM) are designed for use in machine

tools. In the initial design of the PMLM, the winding layout procedure is

presented. The number of slots per pole per phase and period of cogging

force are used to select the pole/slot combination of the PMLM. The

8-pole/9-slot configuration that generates a lower cogging force is

selected. Design techniques for reducing cogging force in slot regions andend regions which are equipped with auxiliary poles (APs) and aluminum

alloys (ALs) using the Taguchi’s parameter method coupled with finite

element analysis (FEA) are employed.

In the design of the HLFSPMM, a ferrite magnet in place of a rare

earth magnet is adopted and use a DC excited winding is used to

modulate the excitation of permanent magnets. After the operating principle and the winding layout procedure are presented, a suitable

mover slot and stator pole number combination is selected in the initial

design to achieve the desired performance capability. Structural

refinements using design sensitivity analysis (DSA) based on the finite

element method (FEM) are performed to enhance the machine

performance first. Then, the genetic algorithm (GA) coupled with FEA is

used to the design optimization for force ripple and copper losses

minimization.

Keywords : Linear permanent magnet motor, hybrid excited linear flux

switching permanent magnet motor, finite element analysis,

Taguchi method, genetic algorithm.


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e-Theses & Dissertations (101 ) iv

················································································ i ················································································ ii

Abstract ··········································································· iii

··············································································· iv ··········································································· vii

··········································································· xii

··························································· xiv ······································································ 1

1.1 ······················································ 11.2 ······························································· 3

1.2.1 ············································· 3

1.2.2 ············································· 61.3 ····························································· 10

1.4 ····························································· 161.5 ····························································· 17

··············································· 192.1 ····················································· 19

2.1.1 ················································· 192.1.2 ················································· 202.1.3 ··········································· 202.1.4 ··········································· 212.1.5 ··········································· 23

2.2 ························································ 24


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e-Theses & Dissertations (101 ) v

2.3 ·················································· 262.3.1 ······························· 27

2.3.1.1 ············································· 282.3.1.2 ················································ 302.3.1.3 ······································· 30

2.3.1.4 ······································· 33

2.3.2 ······························· 362.3.2.1 ························ 38

2.3.2.2 ······································· 40

························································ 453.1 ······························································ 453.2 ············································ 45

3.2.1 ···················································· 453.2.2 ······················································· 48

3.3 ································ 513.3.1 ······················································· 513.3.2 ···················································· 583.3.3 ···················································· 66

··················································· 744.1 ···································································· 744.2 ································································· 744.3 ···················· 774.4 ··························································· 86

4.4.1 ························· 86

4.4.2 ·································· 90


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4.5 GA ····· 944.5.1 ········································ 944.5.2 ······················································· 98

4.5.3 ················································ 1004.5.4 ················································ 106

·························································· 112 A ·················································· 115 B ··························································· 119

C ·················································· 123 D ························································ 125

······································································· 127

············································································· 137 ······································································· 138


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1.1 2010 2011 ···································· 1 1.2

(a) ·················································· 3(b) ············································ 3

1.3 ·········································· 6 1.4 ···························································· 7 1.5 ······························ 7 1.6 ··················································· 8 1.7 U ······················································ 9 1.8 ··························································· 9 1.9 ··················································· 10

1.10 50 ·········· 11 1.11 50 ················ 11 1.12 ················································ 14 2.1 ··················································· 19 2.2 ················································ 20 2.3 ··················································· 21 2.4

(a) ················································ 24(b) ····························· 24

2.5 ············································· 26 2.6 ························································· 29

3.1


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(a)4 6 ····························································· 48(b)6 9 ···························································· 48(c)7 6 ····························································· 48(d)8 9 ···························································· 48

3.2 ········································ 49 3.3 8 9 ·························· 50 3.4

(a) ·········································· 51

(b) ···································· 51 3.5

(a) a ································································ 52(b) b ······························································· 52(c) c ································································ 52(d) d ······························································· 52

3.6 ·································· 53 3.7 ······························· 54 3.8

(a) 2D ····························································· 56(b) 3D ····························································· 56

3.9 (a) ···························································· 57(b) ·································································· 57

3.10 ································ 58 3.11 ··································· 59

3.12


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e-Theses & Dissertations (101 ) ix

(a)5 6 ····························································· 60(b)7 6 ···························································· 60(c)8 6 ····························································· 61(d)10 6 ··························································· 61(e)11 6 ··························································· 61(f)13 6 ··························································· 61

3.13 8 11 (a)8 ·························································· 62

(b)11 ························································ 62 3.14 ························ 64 3.15 11

(a) ··················································· 65

(b) ················································ 65

(c) ················································ 65(d) ················································ 65

3.16 SD2 ························ 67 3.17 ·················································· 69 3.18 ··························································· 70 3.19 ····················································· 70

3.20 ············································ 71 3.21 ························································· 72 3.22 ····················································· 73 4.1 ···························································· 78 4.2

(a) ······························································· 79


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(b) ······························································· 79

4.3 ······································· 81 4.4 ······································· 82 4.5 ······························ 84 4.6 ························································· 85 4.7 ············································· 85 4.8 ·········································· 88 4.9 ··················································· 89

4.10 ·················································· 89 4.11 GA ··········································· 89 4.12 Got-It ······················································· 97 4.13 Flux Got-It ···································· 97 4.14 ····························· 98 4.15 ····························· 99 4.16 ····································· 100 4.17 ·································· 101 4.18 ····································· 101 4.19 ····································· 102 4.20 ········································ 104

4.21 ······································ 104 4.22 ······································ 105 4.23 ·································· 106 4.24 ········································ 107 4.25 ····································· 107

4.26 ·································· 108


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4.27 ····································· 108 4.28 ······················································· 110 4.29

(a) ·························································· 110(b) ······················································ 111

B.1 ························································ 119 B.2 ········································· 120 B.3 ············································ 122

D.1 ··································· 126


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1.1 ·········································· 122.1 ··············· 222.2 ······································· 28

2.3 7 6 (a) 0 ··· 34(b) 270 º ································ 34(c) ± 90º ································· 34

2.4 7 6 ···································· 352.5 ······································· 37

2.6 7 6 (a) 0 ··· 40

(b) ·········································· 41

(c) 270 º ································ 41(d) ± 90º ································· 41

2.7 7 6 ···································· 422.8 8 6

(a) 0 ··· 42(b) ·········································· 43

(c) 270 º ································ 43(d) ± 90º ································· 43

2.9 8 6 ···································· 443.1 ···················································· 453.2 ···································· 46


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e-Theses & Dissertations (101 ) xiii

3.3 k cog ······················ 473.4 ································· 493.5 ···················································· 503.6 ························································· 583.7 ·················· 633.8 SD2 ························ 68

3.9 ················································ 723.10 ··············································· 72

4.1 L 8(27) ·························································· 76

4.2 ··················································· 774.3 L 9(3

4) ······················································· 794.4 ························································· 804.5 L 9(3

4) ···················································· 804.6 ················································ 804.7 ········································ 814.8 ········································ 824.9 ························································· 834.10 ··············································· 854.11 ································ 914.12 ··················································· 96

4.13 ····································· 1034.14 ·············································· 1054.15 ··········································· 109

4.16 ······································ 109


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e-Theses & Dissertations (101 ) xiv

Wb/m Am A/ mm 2 f Hz

A F max N F min N F avg N F rip F cog N g mm GCD

hM mm I phase AIa A

IAC A K 0 k pv k w

k d

k s

k c

k cog l m mm


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e-Theses & Dissertations (101 ) xv

m MW mm

N r N p N 1 N s N cph

p

R DC Ω

R phase Ω

Rmi Rmt SY mm

SD1 mm

SD2 mmSD3 mmTW1 mm

TW2 mmTW3 mmY1 mm

Y2 mmY3 mmθ s

τ p mm

μrec


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1.1

Gardner Publications, Inc. 2011

938 2010 34% 2011

1.1[1] 2011

1.1 2010 2011 [1]


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X-Y

[2]-[4]


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1.2 1.2.1

(Pneumatic)

(Ballscrew)

1.2

(a) (b)

1.2


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1.2.2 1.3

(Primary) (Secondary)

( ) ( )

1.4

1.3


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(a)

(b)

1.4

1.5(a)

[5]

1.5(b)

(a) (b) 1.5 [5]


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1.4 (Single

sided)

(Double

sided) 1.6

1.6

(Flat)

U (U-shaped or U-channel) (Tubular)

U 1.7

(Ironless)

1.7


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1.7 U [6]

1.8 [7]


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1.9

1.9

1.3

1845 Charles Wheatstone [5]

1.10 1.11

[8]-[10] 1.1

1.11


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1.10 50 [8] [9]

1.11 50 [8] [10]

[2]-[5]

[11]-[13]

1.1 [12]


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1.1

○ ╳ △ ○ ╳ ○ ○ ○

╳ ○ ○ ○ ○ ╳ △ ○

╳ ○ △ ○ ○ △ △ ○

△ △ ○ ○ ○ △ ╳ ○

○ △ △ ╳ ○ ╳ ╳ ╳

○ △ ╳

1.1

[5] Hellinger

[13] Chevailler

(Cogging force)

(a) Chevailler[13] Hwang [15]

(b) Lim [16] Jeans [17]


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(c) Gieras

[2] Jeans [17] Jung [18]

(d) Jeans [17]

(e) Inoue ( )

[19]

(f) [14] Zhu [20]-[22]

(Flux switching PM motor) 1950 Rauch

[23] 1.12

Zhu [24]

Chen [25]

Zhu Ilhan Li

Zhou [26]-[29] Wang Jin Krop

Min Cao Huang [30]-[37]


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Huang [37]

1.12

(Hybrid excited)

Hoang

[38] Hoang Sulaiman

[39] [40] Chen

[41] Cao

[35] Cao [35] Huang


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[37]

(Taguchi method) (Genetic algorithm)

(Off-line)

Roy Park [43] [44]

Chen Low Hwang [45]-[48]

Sim Bianchi

Chai Kurpati Hwang Chen

[49]-[55] Sim [49]

Chai [51]

Pollock

Hwang [53]


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Chen

[55]

Flux

2D/3D[56] Got-It[57]

1.4

1.

2.

3.


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4.

5.

6.

1.5


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36/158


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2.1.2

2.2 [58]

2.2 [58]

2.1.3

(Transverse flux)

2.3 3D


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2.3 [59]

100

2.1.4

1.7


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(

) 1.4

2.1

[60]

2.1


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2.1.5

2.4(a) U

U

2.4(b)

U


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(a) [61]

(b) 2.4

2.2

[2] 2.5

y x z

(a)


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(b)

(c) xz y (d) b p τ p

(e) z

(f)

(g)

(g)

(h)

(a) (b)

(2.1) g' = k c g (2.1)

k c g’ g

-

(2.2) (2.3) (2.2)

(2.3)

m p N I phase k w1


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2.5

A

)cosh()tanh()sinh(

)tanh(

2sin

4

g h g

h A LB p F

M rec

M i

mr pdx

(2.4)

i p

pi

b

2.3

[62]-[64]


44/158


45/158


2.2

N s N p 3 6 9 12 15 18 …

1 ○ ○ ○ ○ ○ ○ …

2 ○ ○ ○ ○ ○ ○ …

3 ╳ ╳ ○ ╳ ╳ ○ …

4 ○ ○ ○ ○ ○ ○ …

5 ○ ○ ○ ○ ○ ○ …

6 ╳ ╳ ○ ╳ ╳ ○ …

7 ○ ○ ○ ○ ○ ○ …

8 ○ ○ ○ ○ ○ ○ …

9 ╳ ╳ ╳ ╳ ╳ ╳ …

… … … … … … … …

※○ ╳

2.3.1.1

(Coil span or coil pitch)

θ s (2.7)

s

p

s N

N 180

(2.7)

180°E

S * (2.8)

1,max1,

180max*

p

s

s N N

Int Int S

(2.8)


46/158


Int ( x, y) x/ y max[ x, y] x y

1

2cos

s pk (2.9)

ν 180°E k p1

2/3 ( 120 °E)

0

2.6

21 ee E

21

ee E

2.6


47/158


48/158


s pd w k k k k (2.11)

k d k p k s

2.7 (2.12)

cph N

k c

cphd k N

k 1

)(11

(2.12)

N cph

2.7

k s = 1 (2.9) (2.12) (2.11)

12

cos)(11

1

s N

k c

cphw

cph

k N

k (2.13)


49/158


1. (2.5)

2. (2.8) S *

1 (1+ S *)

2 (2+ S *)

(2.14)

s sc N j j j ,,3,2,1,)1()(

(2.14)

3.

±90 °E

270 °E

E j Mod j cc 360,ˆ (2.15)

Mod ( x,y) x/y

4. 360 90 °E < θ c <

270 °E (2.16) ±90 °E

2.8

E j Mod j cc 180,ˆ (2.16)


50/158


2.8

5. ±90 °E

A

0

6. A A K 0 2 K 0

B C

2.3.1.4

7 6


51/158


1.

213

67,63

6GCD

2. θ s = 210oE S * = 1

3. 360

180 2.3

2.3 7 6

(a) 0 1 2 3 4 5 6 0o 210 o 420 o 630 o 840 o 1050 o

In 1 2 3 4 5 6Out 2 3 4 5 6 1

(b) 270º 1 2 3 4 5 6 0o 210 o 60o -90 o 120 o -30 o

In 1 2 3 4 5 6Out 2 3 4 5 6 1

(c) ±90º 1 2 3 4 5 6 0o -30 o 60o -90 o -60 o -30 o

In 1 3 3 4 6 6Out 2 2 4 5 5 1

4. A A N cph = 6 / 3 = 2

1 2 1 6


52/158


1 6 A B C

5. K 0

210360120360120

0qq

K s q = 2 K

0 = 4

6. B C K 0 2 K 0 2.4

2.9

2.4 7 6 A B C

1 2 1 2 1 2In 1 3 5 1 3 5

Out 2 2 6 6 4 4

2.9 7 6


53/158


2.3.2

[25]

(2.17)

f N r s

60

(2.17)

N r

'6k N s

(2.18)

k ′

(2.5)

k

N N GCD N

r s

s

2,3 (2.19)

2.5 24 15

3n 3n (2.20)

3 18 9

54

k N

n s 2

3 1 (2.20)


54/158


k

2.5

N s N r

6 12 18 24 …

1 ○ ○ ○ ○ …

2 ○ ○ ○ ○ …

3 ╳ ╳ ○ ╳ …

4 ○ ○ ○ ○ …

5 ○ ○ ○ ○ …

6 ╳ ╳ ○ ╳ …

7 ○ ○ ○ ○ …

8 ○ ○ ○ ○ …

9 ╳ ╳ ╳ ╳ …

10 ○ ○ ○ ○ …

11 ○ ○ ○ ○ …

12 ╳ ╳ ○ ╳ …

13 ○ ○ ○ ○ …

14 ○ ○ ○ ○ …

15 ╳ ╳ ○ ╳ …… … … … … …

※○ ╳


55/158


2.3.2.1

(2.21) S *

(2.22)

s

r s N

N 360 (2.21)

1,

2max1,

180max*

r

s

s N

N Int Int S

(2.22)

K 0 (2.21) (2.23)

)3(13

360120 0 q N

N q K

r

s

s (2.23)

q 1 ( N r /2) – 1

1. (2.18)

2. (2.21) S *

1 (1+ S *)

2 (2+ S *)

(2.14)


56/158


3.

180 °E

4. ±90 °E

270 °E (2.15)

5. 360 90 °E < θ c <

270 °E (2.16) ±90 °E

6. ±90 °E

A

0

7. A A K 0 2 K 0

B C


57/158


2.3.2.1

6 7 8

(A)7 6

1.

1236

14,636

GCD

2. θ s 1

E E

N E N

s

r s 4206

3607360

11,1max1,2max*

r

s

N N

Int S

3.

360 180 2.6

2.6 7 6

(a) 0 1 2 3 4 5 6 0o 420 o 840 o 1260 o 1680 o 2100 o

In 1 2 3 4 5 6Out 2 3 4 5 6 1


58/158


2.6 7 6 ( )

(b)

1 2 3 4 5 6 0o 420 o 840 o 1260 o 1680 o 2100 o In 1 3 3 5 5 1

Out 2 2 4 4 6 6

(c) 270º 1 2 3 4 5 6

0o

60 o

120o

180o

240 o

-60 o

In 1 3 3 5 5 1

Out 2 2 4 4 6 6

(d) ±90º 1 2 3 4 5 6 0o 60 o -60 o 0 o 60 o -60 o

In 1 3 4 4 6 6Out 2 2 3 5 5 1

4. A A N cph = 6 / 3 = 2

1 4

5. K 0

420360120360120

0qq

K s

q=2 K 0=2

6. B C K 0 2 K 0 2.7


59/158


2.7 7 6 A B C

1 2 1 2 1 2In 1 4 3 6 5 2

Out 2 5 4 1 6 3

(B)8 6

1.

1

236

16,636

GCD

2. θ s 1

E E

N E N

s

r s 4806

3608360

11,1max1,2

max*

r

s

N N

Int S

3.

360 180 2.8

2.8 8 6

(a) 0 1 2 3 4 5 6 0o 480 o 960 o 1440 o 1920 o 2400 o

In 1 2 3 4 5 6Out 2 3 4 5 6 1


60/158


2.8 8 6 ( )

(b)

1 2 3 4 5 6 0o 480 o 960 o 1440 o 1920 o 2400 o

In 1 3 3 5 5 1Out 2 2 4 4 6 6

(c) 270º 1 2 3 4 5 6 0o 120 o 240 o 0 120 o 240 o

In 1 3 3 5 5 1

Out 2 2 4 4 6 6

(d) ±90º

1 2 3 4 5 6 0o -60 o 60o 0 o -60 o 60 o In 1 2 4 5 6 6

Out 2 3 3 4 5 1

4. A A N cph = 6 / 3 = 2

1 4

5. K 0

480360120360120

0qq

K s

q=1 K 0=1

6. B C K 0 2 K 0 2.9


61/158


62/158


3.1

[65] 3.1

3.1 /

117 mm 90 mm 1.7 A

150 N < 20 N

N30H 35CS400

3.2 3.2.1

(2.26)

N spp

p

s spp N m

N N (3.1)


63/158


m N spp 0.33~0.5

[19] 3.2 N spp

3.2 N p \ N s 3 6 9 12 15 18 21

2 0.5 1 1.5 2 2.5 3 3.5

3 1 2

4 0.25 0.5 0.75 1 1.25 1.5 1.75

5 0.2 0.4 0.6 0.8 1 1.2 1.4

6 0.5 1

7 0.28571 0.42857 0.57143 0.71429 0.85714 1

8 0.25 0.375 0.5 0.625 0.75 0.875

9

10 0.2 0.3 0.4 0.5 0.6 0.7

11 0.18182 0.27273 0.36364 0.45455 0.54545 0.63636

12 0.25 0.5

13 0.23077 0.30769 0.38462 0.46154 0.53846

14 0.21429 0.28571 0.35714 0.42857 0.5

15 0.2 0.4

16 0.1875 0.25 0.3125 0.375 0.4375

[66] [19]

k cog

[13]


64/158


65/158


3.2.2

117 mm

12 4 6 6 9 7 6 8 9

[14] 3.1

3.2 3.1

0.7 mm 270 Flux 2D

3.4

(a)4 6 (b) 6 9

(c)7 6 (d) 8 9

3.1


66/158


3.2

3.4

N spp k cog F cog (N) F avg (N) (%)4P-6S 0.5 3 146.07 159.52 192.58 20.72

6P-9S0.5

3 117.02 160.84 190.4 18.387P-6S 0.2857 6 42.66 124.27 154.3 24.178P-9S 0.375 9 38.51 167.32 200.62 19.9

3.4 20

7


67/158


6 8 9 7 6

8 9 3.5 3.3

3.5

[mm] 14.625 [mm] 12 [mm] 13 [mm] 90 [mm] 6.5 [mm] 6.5 [mm] 26 [mm] 1.5 [mm] 117 [mm] 3.25 [mm] 90

3.3 8 9

8 9


68/158


3.3

3.3.1

3.4(a)

(

) ( ) 3.4(b)

(a) (b) 3.4


69/158


3.5

(a) a (b) b

(c) c (d) d 3.5

a 3

3

3 b


70/158


3

a d 3 4

3.6

3.6

7 6

-8 A 8 A 3.7

0 T 1.8 T 1.8 T


71/158


3.7

8A 7A

6A

3A4A

5A

2A 1A

0A -1A

-3A

-6A


72/158


3.7 2 A

( 1.8 T)3.6

3 A 6 A

6 A

7 A 8 A

3.8 3.8(a)

7 A 5 A

8 A

3.8(b) Z

XY 4 A


73/158


position (m)0.00 0.02 0.04 0.06

F l u x

d e n s

i t y ( T )

-1.5

-1.0

-0.5

0.0

0.5

1.0

1.5 8 A7 A

5 A

3 A

0 A-4 A

-8 A

(a) 2D

(b) 3D

3.8


74/158


75/158


3.3.2

3.1 115 mm 130 mm Y40

6 2.5 5 7 8 10 11

13 6 3.6 3.10

3.10 3.6

5, 7, 8, 10, 11, 13 (mm) 21.833 6 MW(mm) 3.5

Ferrite Y40 SD1 (mm) 3.5 35CS350 SD2 (mm) 24 (mm) 130 SD3 (mm) 15 (mm) 90 TW1 (mm) 4.727 (m/s) 4 TW2 (mm) 3.5

(N) 150 TW3 (mm) 5 (N) 20 Y1 (mm) 3.5 (A) 1.7 Y2 (mm) 4.5 (mm) 54 Y3 (mm) 3 (mm) 131/N r

(N r = 5, 7, 8, 10, 11, 13) (mm) 0.8


76/158


TW1 0.4

B

C (Favg) 150 N (F cog)

20 N

4 m/s 100 6 A

245 A

3.11 5

140 V

Electrical Degree

0 60 120 180 240 300 360

B E M F ( V )

-180-160-140-120-100

-80-60-40-20

020406080

100120140160180

5 poles7 poles8 poles

10 poles11 poles13 poles

3.11

3.12


77/158


78/158


Electrical Degree0 60 120 180 240 300 360

B E M F ( V )

-180-150-120

-90-60-30

0306090

120150180

phase A phase B phase C


B E M F ( V )

-180-150-120

-90-60-30

030

6090

120150180


(c) 8 6 (d) 10 6


B E M F ( V )

-180-150-120

-90-60-30

0306090120

150180



B E M F ( V )

-180-150-120

-90-60-30

0306090120

150180


(e) 11 6 (f) 13 6

3.12 ( )

3.12 8 11

3.13 8 11 8

B 30 V

11 12 V


79/158


Harmonic order 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

V o l

t a g e

( V )

0

10

2080

100

120

140

160

Phase APhase BPhase C

(a) 8

Harmonic order 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

V o l

t a g e

( V )

0

10

2080

100120140160180200

Phase APhase BPhase C

(b) 11

3.13 8 11

3.11 (THD)

A 3.14 5

15 V

5

8 10 7 11 13


80/158


81/158


8 10

11 6 3.15 11 (a) (d)

3.3.1

6 11

150 N

Number of Poles5P 7P 8P 10P 11P 13P

B E M F ( V )

0

20

40

60

80

100

120140

T H D ( % )

0

2

4

6

8

1012BEMF

THD

3.14


82/158


(a) (b)

(c) (d) 3.15 11


83/158


3.3.3

[9-10]

0.15

3.10

SD2 SD3 39 mm SD2 23 mm 27 mm 1

mm SD3 16 mm 12 mm

SD2

0 10 A

3.16 3.8 SD2

SD2 SD3

Ia Ia 6 A


84/158


SD2 27 mm

Ia 6 A 149.651 N SD1 = 3.5

mm MW = 3.5 mm TW1 = 4.727 mm TW2 = 3.5 mm TW3 = 5 mm

Y1 = 3.5 mm Y2 = 4.5 mm Y3 = 3 mm

020406080

100

120

140

160

02

4 6

8 10

2223

2425

2627

F a v g ( N

)

I a ( A )

S D 2 ( m m ) 3.16 SD2


85/158


3.8 SD2

SD2Ia

23 mm 24 mm 25 mm 26 mm 27 mm

0 A 60.085 N 63.901 N 67.681 N 71.415 N 75.15 N2 A 78.4 N 80.567 N 82.855 N 85.322 N 87.952 N4 A 126.99 N 125.155 N 122.819 N 120.837 N 119.244 N6 A 132.59 N 139.137 N 145.14 N 149.08 N 149.651 N8 A 127.013 N 131.011 N 135.343 N 140.488 N 147.214 N

10 A 128.046 N 128.046 N 131.45 N 135.11 N 139.283 N

MW TW2

(3.2)

mm72TWMW Rmt,TW2MW

(3.2)

3.17 Rmt 1.8

Rmt 1 MW TW2

MW = TW2 = 3.5 mm


86/158


Rmt

0.50 0.75 1.00 1.25 1.50 1.75 2.00

F a v g

( N )

120

125

130

135

140145

150

155

F r i p

( N )

14161820222426

2830

FavgFrip

3.17

SD2 SD3 Y2 Y3

(3.3)

mm 5.7Y3Y2Rmi,Y3Y2

(3.3)

3.18 Rmi 1 Rmi

1.14 Y3

Rmi 1.14 Y2 =

4 mm Y3 = 3.5 mm


87/158


Rmi0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

F a v g

( N )

145

146

147

148149

150

151

F r i p

( N )

20

22

24

26

28

30

FavgFrip

3.18

TW1

TW1 α 3.19

α 0.25 0.3 156.562 N 158.696 N

23.937 N 27.541 N α

0.25 0.3 TW1 2.9545 3.5454 mm

α0.20 0.25 0.30 0.35 0.40 0.45 0.50

F a v g

( N )

135

140

145

150

155

160

165

F r i p

( N )

20

22

2426

28

30FavgFrip

3.19


88/158


SD1

(3.4) mm 7Y1SD1 ,

Y1SD1SD1

(3.4)

3.20 α 0.25 0.3

β β 0.5

150 N

β 0.4 β 0.4 α 0.25 β 0.5 α 0.3

β0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65

F a v g

( N )

146148150152154156158160

F r i p

( N )

12151821242730

333639

α - 0.3 F avgα - 0.25 F avgα - 0.3 F ripα - 0.25 F rip

3.20

3.9 3.20 β

0.5 3.21 3.10

139.137 N


89/158


14.05% 158.696N 17.44 N

11.36% 3.22

3.9

Ia 6 A TW1 2.9545, 3.5454 mm

SD1 1.75, 2.1 mm TW2 3.5 mm

SD2 27 mm Y1 5.25, 4.9 mmSD3 12 mm Y2 4 mmMW 3.5 mm Y3 3.5 mm

3.10 Favg (N) F rip (N) Ripple (%)

initial 139.137 20.5343 14.76

β =0.5, α=0.3 158.696 27.541 17.35 β =0.4, α=0.25 153.449 17.44 11.36


F a v g

( N )

120

130

140150

160

170

180 α = 0.3, β = 0.5α = 0.25, β 0.4Initial

3.21


90/158


3.22


91/158


92/158


(Orthogonal array , OA)

3 4 81

9

C. R. Rao 1947

(Factor) (Level)

(4.1) : La(b

c ) (4.1)

a b c

7 L8(27) 4.1 :


93/158


94/158


4.2

2 3 4 5

L4 4 3 3 - - - L8 8 7 7 - - - L9 9 4 - 4 - - L12 12 11 11 - - - L16 16 15 15 - - -* L16 16 5 - - 5 - L18 18 8 1 7 - - L25 25 6 - - - 6 L27 27 13 - 13 - -* L27 27 22 - 22 - - L32 32 31 31 - - -* L32 32 10 1 - 9 - L36 36 23 11 12 - -* L36 36 16 3 13 - - L50 50 12 1 - - 11 L54 54 26 1 25 - - L64 64 63 63 - - -* L64 64 21 - - 21 - L81 81 40 - 40 - -

4.3

(Auxiliary poles, AP)

(AL)


95/158


4.1

(A) (B)(C) (D)

(τ p) 4.2

L9 34 L9 34 4.3

4.4 4.3 4.5

4.1


96/158


(a)

(b)

4.2

4.3 L9 34

Ai Bi C i D i

1 1 1 1 1

2 1 2 2 23 1 3 3 34 2 1 2 35 2 2 3 16 2 3 1 27 3 1 3 28 3 2 1 39 3 3 2 1


97/158


4.4 Level 1 Level 2 Level 3

( A, τ p) 0 0.25 0.5 ( B, τ p) 1 0.5 0.25 (C , mm) 0 3 6 ( D, τ p) 1 0.5 0.25

4.5 L9 34

L9 Ai(τ p) Bi(τ p) C i(mm) D i(τ p)

1 0 1 0 12 0 0.25 3 0.253 0 0.5 6 0.54 0.25 1 3 0.55 0.25 0.25 6 16 0.25 0.5 0 0.257 0.5 1 6 0.258 0.5 0.25 0 0.59 0.5 0.5 3 1

4.5 9

4.6

4.6 L9 F cog (N) F avg (N)

1 98.74 139.822 21.17 143.013 40.18 142.024 40.41 137.535 22.24 139.016 63.45 137.417 22.89 128.058 7.85 136.69

9 9.46 129.48


98/158


4.5 (Analysis of means, ANOM)

))3()2()1((31

)(1 cog cog cog cog A F F F F m (4.2)

))6()5()4((31

)(2 cog cog cog cog A F F F F m (4.3)

))9()8()7((3

1)(

3 cog cog cog cog A F F F F m (4.4)

m A1( F cog ) A

1

4.7 4.3

4.7 Ai Bi C i D i i = 1 53.363 54.013 56.68 43.48i = 2 42.033 17.087 23.68 35.837i = 3 13.4 37.697 28.437 29.48

Setting of ParametersA1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

F c o g

( N )

10

20

30

40

50

60

4.3


99/158


4.3

A3 B2 C2 D3 6.24 N 129.74 N

(4.2) (4.3) (4.4) F cog F avg 3.13

4.8 4.4

A1 B2 C1 D3

144.02 N 50.42 N

4.8 Ai Bi C i D i

i = 1 141.616 135.132 137.972 136.100

i = 2 137.982 139.567 136.671 136.159i = 3 131.406 136.305 136.360 138.745

Setting of ParametersA1 A2 A3 B1 B2 B3 C1 C2 C3 D1 D2 D3

F a v g

( N )

120

125

130

135

140

145

4.4


100/158


(Analysis of variance, ANOVA) (4.5)

9

1)(

91

iall i F m (4.5)

F (i) i

4.6 (4.5) 25.55 N

137.00 N

(Experimental error)

(Sum of squares, SS)

23

13

iall i-mmSS (4.6)

(Contribution)

4.6 4.8 (4.5) (4.6) 4.9 4.5

4.9 F cog F avg

(%) (%) A 2545.31 37.40 160.71 76.35 B 2054.58 30.19 31.68 15.05C 1909.31 28.06 4.39 2.08

D 294.83 4.35 13.69 6.52sum 6804.03 100 210.47 100


101/158


Setting of parameters A B C D

F a c t o r

C o n

t r i b u t

i o n ( % )

0

20

40

60

80

100 F cog F avg

4.5 4.5 A

A1 B

B2 C

C2 D

D3 A1 B2 C2 D3

4.10 2D

3D Flux 3D

4.10 3D 3D 2D 151.38 N

44.7 N 17.97 N

4.6 142.96 N

4.7


102/158


103/158


4.4 4.5

4.4

B

4.4.1

( GA ) Holland

1960

(Chromosome) GA


104/158


(Individual)

(Crossover)

GA 4.8 (1)

(2) (3) (Fitness) (4) (5)

(6)

GA GA

(Approximation function) (Response surfaces

method)

(Screening)


105/158


4.8

(Design of experiments)

Box-Behnken Box-Wilson

L12 L16 (Principle of interpolation )


106/158


4.9 Box-Behnken

(4.7) 4.10 GA

4.11

213

2

222

2

11222110

' )( x xa xa xa xa xaa x F (4.7)

4.9

4.10 4.11 GA


107/158


4.4.3

(Deterministicalgorithm) (Conjugate gradients)

(Sequential quadratic programming, SQP) BFGS (Broyden-fletcher-

goldfarb-shanno)

(Stochastic

algorithm) (Simulated annealing, SA)

(Particle swarm optimization, PSO) Niching

;

GA GA

Chun Mirzaeian Hsu GA

[68]-[70] Matlab Simulink


108/158


[55] [57] 4.11

(Sensitivity)

4.11

FEM

FEM

GA GA FEM

◎ △ △ ○ △ ○ △ ◎

△ ○ ◎ △ △ ○ ○ ◎

△ ○ ○ ◎ △ ○ ○ ◎

△ ○ ◎


109/158


GA

Got-It (Python)GA

GA tool box

GA Got-It

GA


110/158


GA

;

4 4

GA

( ) GA


111/158


4.5 GA

4.5.1

GA

(4.8)

fill ww

S DSW

INT DSD

INT Turns

(4.8)

INT ( x) x SD Dw SW

S fill

80% 75%

(4.8)

150 N

150 N (4.9)

150

150 avg F force (4.9)


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60 mm (4.10)

60603232

H limitSY Y Y SDSD

(4.10)

SD2 SD3 Y2

Y3 SY

(4.11)

phase AC DC acu R I R I P 22 3 (4.11)

R DC R phase I a

I AC

(4.12) (4.13) 4.12

]),0([MAX]),0([MAX limit penaltyrip H forcek F obj (4.12)

]),0([MAX]),0([MAX limit penaltycu H forcek P obj (4.13)

penaltyk


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4.12

(mm) SD2 27 [25, 28](mm) SD3 12 [10, 13]

Rmt 1 [0.75, 1.333](mm) TW3 5 [4, 7]

(mm) Y2 4 [3, 5](mm) Y3 3.5 [3, 4]

α = TW1/TP 0.3 [0.25, 0.4] β = SD1/SY 0.35 [0.3, 0.5]

(mm) SY 7 [5, 8](A) I a 2.5 [1.5, 3](A) I AC 6 [5, 7](mm) WD_AC 0.6 [0.5, 0.7](mm) WD_DC 0.9 [0.6, 0.9]

GA Cedrat Flux 2D

Got-It [56] [57] Got-It

4.12 Got-It

HLH

(HLHRBF) (Niching GA) (GA)

(GMGA) Sequential surrogate optimizer (SSO) Sequential quadratic

programming (SQP)

4.13 Flux F2G python

Got-It


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4.12 Got-It

4.13 Flux Got-It


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4.5.2

13 BOX 32 5-16

32

4.14

( I)

4.14

( )

13 9


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4.15

C

SD1_RATIO

4.14 4.18 MW

(Rmt) TW1_RATIO( )

4.15

4.15


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4.16

4.16

4.5.3 4.5.1

GA

3.10 β = 0.4 α = 0.25

4.17

4.18 4.19

MF


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4300 FMAX FMIN TURN_AC

TURN_DC 100

4.17

4.18


119/158


4.19

117 11.36 % 4.29 %

67.51 W 78.19 W

4.13 151.33 N


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121/158


4.20

4.21


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4.22

4.14

(mm) SD2 27 27.2(mm) SD3 12 12.69

Rmt 1 1.0526(mm) TW3 5 5.58

( ) (mm) Y2 4 3.615 ( ) (mm) Y3 3.5 4.445

α = TW1/TP 0.3 0.3β = SD1/SY 0.35 0.443

(mm) SY 7 7.93(A) Ia 2.5 2.462(A) IAC 6 5(mm) WD_AC 0.6 0.618(mm) WD_DC 0.9 0.9

(%) Frip 11.36 10.65 (W) Pcu 67.51 61.447


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4.5.4

4.4.1 (GMGA) 4.23

4.24 4.25

4.26

4.27

9800

4.23


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4.24

4.25


125/158


4.26

4.27

87 11.36 % 6.71 %

67.51 W 63.91 W 150.02 N 4.15

4.16


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4.28 4.29

4.15

(mm) 27 25.1 27.2 25.3(mm) 12 11.57 12.69 12.99 1 1.3294 1.0526 1.1116(mm) 5 4.625 5.58 4

( ) (mm) 4 3 3.615 4.135 ( ) (mm) 3.5 4.5 4.445 4.5

0.3 0.308 0.3 0.3360.35 0.368 0.443 0.4

(mm) 7 7.99 7.93 8(A) 2.5 2.142 2.462 2.406(A) 6 6.05 5 5.14(mm) 0.6 0.5715 0.618 0.62(mm) 0.9 0.799 0.9 0.9

(%) 11.36 4.29 10.65 6.71 (W) 67.51 78.19 61.447 63.91

4.16

Favg (N) 153.449 151.33 150.01 150.02Frip (%) 11.36 4.29 10.65 6.71Pcu (W) 67.51 78.19 61.447 63.91

Frip + P cu 78.87 82.48 72.097 70.62


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Electric Degree0 60 120 180 240 300 360

F a v g

( N )

140

145

150

155

160InitialOPT_rippleOPT_PcuOPT_Multi

4.28

(a)

4.29


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(b)

4.29 ( )

GA

GA


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7 6

3

mm 0.25

0.5 44.7 N 17.97 N

6

11 11

139.137 N 153.449 N

(Flux 2D) (Got-It)


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67.51 W 63.91 W 11.36% 6.71%

D

Got-It

Ferrite

NdFeB

200 NdFeB

NdFeB Ferrite


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(CFD)


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A

2.2 2.5 -

x (2.2)

xt k I N pm

dxt xd

t xa w phase p

cos2,

, 11F

(A.1)

x jt jm xt jm ee Ae At xa ReRe, (A.2)

p

τ p Am

p

phasewm p

I k N m A

112 (A.3)

x Ae A xa m x jm cosRe

(A.4)

(f) H c

A/m 0 rec

r M

B J

(A.5)

Br μrec


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2.5 I II

Laplace

022

2

2

z A

x A

(A.6)

curl

z z xt j y Be Aeet z x A ,, (A.7)

(2.5)

z z y Be Ae x z x A cos, (A.8)

(c) y

x

A B

z

A B y z

y x , (A.9)

I II

z z y x Be Ae x z z x A

z x B cos,

, (A.10)

z z y z Be Ae x z z x A

z x B sin,

, (A.11)

(a) z = 0

)(

0, xa

z x B

o

xI

(A.12)

(A.8)

mo I I A B A (A.13)


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Am (A.7)

z = g

,,

reco

xII

o

xI g z x B g z x B

g z x B g z x B zII zI ,,

g II g II g I g I rec e Be Ae Be A (A.14) g

II g

II g

I g

I e Be Ae Be A (A.15)

z = g+h M hM

0, M x h g z x B

0 M M h g II h g II e Be A (A.16)

A II B II mm h g

II h g

II eC BeC A ',' (A.17)

mreco

M M rec

A g h g h

C 2)cosh()sinh()sinh()cosh(

1'

A II B II (2.14)

]cosh[)sin('2, z h g xC z x B M zII (A.18)

Lorentz x = b p / 2

J M


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Ldz J z b x BdF M p zII dx ,5.0 (A.19)

L 2×2 p

dz z h g C pLJ F M h g g i M dx M ]cosh[2sin'8 (A.20)

)sinh()sinh(1]cosh[ M M h g g M hhdz z h g M

(A.21)

)cosh()tanh()sinh()tanh(

2sin

4 g h g

h A LB p F

M rec

M imr pdx

(A.22)

i p

pi

b


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B

(Objective function)

(Design variables) (B.1)

),,,,()( min 321 n x x x x f x f (B.1)

f ( x) x

B.1

(Local minimum)

(Global minimum)

B.1


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(B.2)B.2 (Feasible)

(Infeasible)

mi x g i ,...,2,1 ,0)( (B.2)

x g i (Inequality constraint)

B.2

(B.3)

maxmin j j j x x x (B.3)

min j x max j x


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)(,),(),(),()( min 321 n x f x f x f x f x F (B.4)

(Pareto optimal solutions)

(Pareto optimal

front) B.3

B.3

A ;

D A D


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B.3

A D A D

(Non-dominated solutions) 4.10

A D

B C P A

A B C P C P (Inferior

solutions) (Dominated solutions)


140/158


141/158


8)*Lin(TW1_RATIO,0.3,0.4))+6*(Lin(SD1_RATIO,0.3,0.5)*Lin(TW1 _RATIO,0.3,0.4)+Lin(MW,3,4)*Lin(TW3,4,7))+(-6)*Lin(SD2,25,28)+6*Lin(SY,5,8)*Lin(TW1_RATIO,0.3,0.4)+4.5*(Lin(MW,3,4)*Lin(TW1_ RATIO,0.3,0.4)+Lin(SD2,25,28)*Lin(Y2,3,5)+Lin(SD3,10,13)*Lin(Y3,3.5,4.5)+Lin(SD1_RATIO,0.3,0.5)*Lin(TW3,4,7))+4*Lin(TW3,4,7)

(C.3)


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D

D.1


143/158


(Motor-CAD)

FEM

/

D.1


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Optimal Design and Analysis of Linear Permanent Magnet Machines