A Microscopic Investigation of Force Generation in a

1

A Microscopic Investigation of Force Generation in aPermanent Magnet Synchronous Machine

S. Pekarek, Purdue University(W. Zhu – UM-Rolla),

(B. Fahimi University of Texas-Arlington)

February 7, 2005

2

Outline

Torque and force characteristics

• Structure of permanent magnet (PM) synchronous machine

• Define macroscopic view of torque

• Define microscopic view of force

• Microscopic torque and force characteristics of PM synchronous machines

Tangential and radial force ripple minimization

• Field reconstruction (FR) method

• Optimization using FR method

3

Permanent Magnet Synchronous Machine

•• Has stator windings in stator slots• Uses permanent magnets as magnetic source on the rotor

4

Macroscopic Machine Model

abcs s abcs abcsd

dt= +v r i λ

'abcs s abcs abcpm= +λ L i λ

'

' 23

' 23

sin( )

sin( )

sin( )

m rapm

abcpm bpm m r

cpm m r

π

π

λ θλλ λ θλ λ θ

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥= = −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥+⎢ ⎥⎣ ⎦ ⎣ ⎦

'λ

( )2

aspm bspm cspm pmce as bs cs

r r r r r

WW PT i i i

λ λ λθ θ θ θ θ

∂ ∂ ∂ ∂∂= = + + +

∂ ∂ ∂ ∂ ∂

5

Machine Model in Rotor Frame of Reference

r r r rqs s qs r ds qsv r i pω λ λ= + +r r r rds s ds r qs dsv r i pω λ λ= − +r rqs ss qsL iλ =

'r rds ss ds mL iλ λ= +

'3( )

2 2r

e m qs cog rP

T i Tλ θ= +

• Average torque not function of d-axis current

6

Field-Based Solution of Magnetic Forces

0/t n tf B B µ= ⋅2 2

0( ) / 2n n tf B B µ= −Force Density:

Maxwell Stress Tensor (Neglecting z-component of flux density)

Material 1

B

f

f

t tF f dl= ⋅∫n nF f dl= ⋅∫

Overall Force:

l

e t stack contourT F L R= ⋅ ⋅

Material 2

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Microscopic View of Electric Machines

N S

nB

tB

1i 2i 3i

B

8

Cross-section of PM Machine Studied

3-phase, 4-pole, 12-slot, surface-mounted, 1 HP, 2000 rpm

9

Flux and Force Density Distribution by PMs

0, 5807t nF F= =

Distribution of Bt and Bn generated by permanent magnets

Average Forces: N/m

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Flux and Force Density Distribution −

Distribution of Bt and Bn at single rotor position when

Distribution of ft and fn at single rotor position when 0 , 4 .6r r

d s q si i= =

4 .6rq si =

0 , 4 .6r rd s q si i= =

1 2 1 9 , 6 2 3 9t nF F= =

A

A

Average Forces: N/m

A

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Flux and Force Density Distribution −

Distribution of Bt and Bn at single rotor position when

Distribution of ft and fn at single rotor position when4 .0 , 4 .6r r

d s q si i= = 4 .0 , 4 .6r rd s q si i= =

4 .0rd si =

1 2 1 8 , 9 1 3 2t nF F= =

A

Average Forces: N/m

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Effect of on Average Components of Force

d-axis current (A)

-9.0 -4.0 0 4.0 9.0

Average Ft (N/m)

1148 1154 1157 1160 1160

Average Fn (N/m)

1755 3852 6248 9278 13879

rd si

13

Effect of on Average Components of Force

5837

249

1

6531

1495

6

5996

748

3

70836249Average of Fn (N/m)

19851157Average of Ft (N/m)

84.6q-axis current (A)

rq si

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Detailed Force Density Expression

If effect of saturation is neglected, then:

Source of Magnetic Field:

• Phase currents

• Permanent Magnets

n npm ns

t tpm ts

B B B

B B B

= +

= +

2 2

0

1( ) ( )

2n npm ns ts tpmf B B B Bµ

⎡ ⎤= + − +⎣ ⎦

0

1t ts tpm npm nsf B B B B

µ⎡ ⎤ ⎡ ⎤= + +⎣ ⎦ ⎣ ⎦

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Components of Torque

ns tpmB B

tpm npmB B

ts npmB B

ns tsB B

(zero average)

(zero average)

(non-zero average)

(non-zero average)

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Flux Densities Generated by Current in Single Stator Slot

If set origin at center, then:• f1 is a even function.• f2 is a odd function.

Bn

BtΦs

Islot Iron1

Magnet

Airgap

Bn [T

]Bt

[T]

1

2

( ) ( )

( ) ( )tsk s slot s

nsk s slot s

B I f

B I f

φ φφ φ

= ⋅= ⋅

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Components of Tangential Force

1

cos( )npm npmk rk

B B kφ∞

=

=∑

1

sin( )tpm tpmk rk

B B kφ∞

=

=∑_ 1( ) ( )t as s as sB i Fφ φ=

1 11

cos( )k sk

F F kφ∞

=

=∑

_ 2( ) ( )n as s as sB i Fφ φ=

2 21

( ) sin( )s k sk

F F kφ φ∞

=

=∑

_ 1( ) ( 120)t bs s bs sB i Fφ φ= −

_ 1( ) ( 120)t cs s cs sB i Fφ φ= +_ 2( ) ( 120)n bs s bs sB i Fφ φ= −

_ 2( ) ( 120)n cs s cs sB i Fφ φ= +

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Average Tangential Force Due to Fundamental Harmonics

1 11 1 210

3( ) 2

2r

t npm tpm qsP

F B F B F R Iπ π

µ= ⋅ ⋅ + ⋅ ⋅ ⋅

2

21 20

1( ) sin( )s s sF F d

πφ φ φ

π= ⋅∫

2

11 10

1( ) cos( )s s sF F d

πφ φ φ

π= ⋅∫

• d-axis current doesn’t appear in average tangential force.

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Components and Average Radial Force

If consider only the fundamental components:

2

20

( )1

2 ( )

npm n as n bs n csn

t as t bs t cs tpm

B B B Bf

B B B Bµ− − −

− − −

⎡ ⎤+ + + −⎢ ⎥=⎢ ⎥+ + +⎣ ⎦

2 2 2 2 2 221 11 1 1

0

21 1 11 1

[9( )( )4

6( ) ]

r rn qs ds npm tpm

rnpm tpm ds

P RF F F i i B B

F B F B i

πµ

= − + + −

+ −

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Problems in PM Synchronous Machine Applications

Solutions for torque ripple mitigation:

• Improve machine design to obtain better back-emf waveform

• Employ excitation control methods to eliminate torque harmonics

Acoustic noise and vibration caused by:

• Torque ripple

• Harmonics in radial force

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Force Optimization using FEA

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Field Reconstruction

n npm ns

t tpm ts

B B B

B B B

= +

= +

1

L

ns nskk

B B=

=∑

1

L

ts tskk

B B=

=∑Slot# k k+1

Stator

Rotor Magnet

Air-gap

1

2

( ) ( )

( ) ( )tsk s slot s

nsk s slot s

B I f

B I f

φ φφ φ

= ⋅= ⋅

( , )

( , )

n n nsk npm

t t tsk tpm

B B B B

B B B B

=

=

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Field Reconstruction Method

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Field Reconstruction Results – Normal Component

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Field Reconstruction Results – Tangential Component

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Force Optimization Using Field Reconstruction

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Operation with Fixed Ft and Minimal Copper Loss

1000tF =

0as bs csi i i+ + =

max max, ,as bs csi i i i i− ≤ ≤Phase current waveform

Subject to:

0 50 100 150 200 250 300 350 400−5

−4

−3

−2

−1

0

1

2

3

4

5

Angle [Electrical degree]

ias,

ibs,

ics

[A]

iasibsics

2 2 2( )as bs csMin i i i+ +

N/m

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Waveform of Ft, Fn and Comparison with Sinusoidal Excitation

Peak-to-peak

204/6248145/1157Sinusoidal

991/61987.3/1101OptimalFn (N/m)Ft (N/m)

0 20 40 60 80 100 120 140 160 180100010501100115012001250130013501400


Ft [

N/m

]

optimizedsinusoidal currents

0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Ft [

N/m

]

Harmonic order


0 20 40 60 80 100 120 140 160 1805500

6000

6500

7000


Fn

[N/m

]


0 2 4 6 8 10 12 14 16 18 200

100

200

300

400

500

Fn

[N/m

]Harmonic order


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Operation with Fixed Fn and Ft

0as bs csi i i+ + =

max max, ,as bs csi i i i i− ≤ ≤

Phase current waveform

0 50 100 150 200 250 300 350 400−5

−4

−3

−2

−1

0

1

2

3

4

5


ias,

ibs,

ics

[A]

iasibsics

2

2

[( )

( ) ]

t t

n n

Min F F

F F

−

+ −

Subject to:

30


204/6248145/1157Sinusoidal


Peak-to-peak

0 20 40 60 80 100 120 140 160 1801000

1100

1200

1300

1400


Ft [

N/m

]


0 5 10 15 200

50

100

Ft [

N/m

]

Harmonic order


0 20 40 60 80 100 120 140 160 1806100

6200

6300

6400

6500


Fn

[N/m

]


0 5 10 15 200

50

100

Fn

[N/m

]Harmonic order


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Operation with Fixed Fn, Ft and Minimal Copper Loss

1000tF =6200nF =



Subject to:

0 50 100 150 200 250 300 350 400−5

−4

−3

−2

−1

0

1

2

3

4

5


ias,

ibs,

ics

[A]

iasibsics


N/m

N/m

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204/6248145/1157Sinusoidal


Peak-to-peak

0 20 40 60 80 100 120 140 160 180100010501100115012001250130013501400


Ft [

N/m

]


0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Ft [

N/m

]

Harmonic order


0 20 40 60 80 100 120 140 160 180610061506200625063006350640064506500


Fn

[N/m

]


0 2 4 6 8 10 12 14 16 18 200

20

40

60

80

100

Fn

[N/m

]Harmonic order


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Trapezoidal back-emf case: Fixed Fn, Ft and Minimal Copper Loss

1000tF =6200nF =



Subject to:


0 50 100 150 200 250 300 350 400−5

−4

−3

−2

−1

0

1

2

3

4

5


ias,

ibs,

ics

[A]

iasibsics

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Waveform of Ft, Fn and Comparison with Trapezoidal Excitation

2773/642797/1185Trapezoidal

31/628911/1100OptimalFn (N/m)Ft (N/m)

Peak-to-peak

0 20 40 60 80 100 120 140 160 1801000

1100

1200

1300


Ft [

N/m

]

optimizedtrapzoidal currents

0 5 10 15 200

10

20

30

40

50

Ft [

N/m

]

Harmonic order


0 20 40 60 80 100 120 140 160 1805000

5500

6000

6500

7000

7500

8000


Fn

[N/m

]


0 2 4 6 8 10 12 14 16 18 200

200

400

600

800

1000

1200

Fn

[N/m

]Harmonic order


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Conclusions

Microscopic investigation of forces leads to result that both d- and q-axis currents influence radial force (quadratic)

Area of tangential force density relatively small – leading to larger radial than tangential force

Opens the question - are alternative designs/excitation strategies possible to provide a more effective force profile?

Field reconstruction is a time-efficient tool to consider alternative methods of excitation for control force profile

Documents

A Microscopic Investigation of Force Generation in a