torque generation with electrical machines - iea.lth.se 14/torquegeneration2014_03_26.pdf ·...

Industrial Electrical Engineering and Automation

Lund University, Sweden

Torque generation

with

Electrical Machines

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© Mats Alaküla L9-torque generation

Torque generating phenomena

1 Conductor in magnetic field

2 Iron shape in magnetic field

3 Electrostatic

4 Piezostriction

Magnetostriction

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Linear / rotating

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Energy density

30

2

30

2

2:densityenergyElectric

2:densityenergyMagnetic

m

JE

m

JB

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Geometry

IM SM

DCM RM

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# of poles

N

S

N

S

N

S

e = d

dt

e = d

dt+

-

+

-

el = p

2 mec

Tmech = p

2 Tel

p(t) = el Tel = p

2 mec

2p Tmec = mec Tmec

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Sinusoidal mmf & flux

) ( q r F

) ( q s F

r F

s F

t o t a l t o t a l o r F

q

q

1. Superposition

2. Vectors

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MMF, Flux & Reluctance

radius stator r

length stator l where

=

= d r l B s s =

p

q q

0

) (

R = Ftotal,a verage

total

=

2p Ftotal

total

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Summary so far ...

• Magnetic fields for energy transfer

• Salient poles on one side (stator or rotor)

• 2 poles, but can be scaled later

• Sinusoidal mmf-distribution

• Flux=integral of flux density

• Reluctance = MMF/Flux

• Now, we create a machine with these properties ...

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The PM / PM –machine

RotorRotor

Stator

Stator

N S N

mF

mF

0 2p

sF

sF

• Salient poles in the rotor

• Rotor reference frame (x/y)

• Rotor mmf and stator mmf (sinusiodal)

• Reluctances Rx and Ry

• Ideal iron (no magnetic saturation effects)

mF

sF

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Air gap magnetic energy

Rotor

Stator

m F

s F

yxsm FFFFF ==

sysy

msxmsx

FFF

FFFFF

ˆ)sin(ˆ

ˆˆˆ)cos(ˆ

==

==

==

y

s

x

mmss

y

y

x

xmagn

R

F

R

FFFF

R

F

R

FW

2222222 sinˆ

2

1ˆcosˆˆ2cosˆ

2

1ˆ

2

1ˆ

2

1

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Torque : Derivative of magnetic energy

Td

dWmec =

Constant, i.e. no energy supplied = mec magn W W

===

yxsysx

x

msymagn

RRFF

R

FF

d

dW-T

11ˆˆˆˆ

...

Compare to linear movement

(F=dW/dx or W=F*x)

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Torque : Components

N on-m agnetized

M agnetizedN on-

magnetizedM agnetized M agnetized

N o torque

Tor

que

definedby equation XX

===

yxsysx

x

msymagn

RRFF

R

FF

d

dW-T

11ˆˆˆˆ

...

External magnetic field generated by Fs

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Electrically magnetized stator

N s number of

winding turns

) ( q s F

) ( q s F ) ( q s F

s F s F

s F

s F

,...7,5,32

2

4ˆ

2

2

4ˆ1

=

=

=

=

=

nn

iN

n

iNF

iNiN

F

sssssn

ssss

s

pp

ppfundamental

harmonics

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Sinusoidally distributed winding

) ( q s F ) ( q s F

s F

,... 7 , 5 , 3 1 2 2 ˆ

2 2 ˆ

1

1 1 1

=

=

=

= =

n i n

k N k n

i N F

i k N k i N F

s rn s r s s

sn

s r s r s s s

p p

p p

s F

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Torque : expressed in flux and mmf

s

ss

s

sxysyx

sxsyy

sysxmx

F

FF

F

FF

FFR

FFFR

T

=

===

==

==

==

p

p

qp

qqp

p

2

sinˆ

2sinˆ

2

cossinsincosˆ

2

ˆˆ

2

ˆˆ1ˆˆˆ1

sFF

mF

y

y

x

x

R

F

jR

F

pp

ˆ2ˆ2

=

yF

xF

q

Remember: Flux=MMF/Reluctance

Important conclusion

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Torque : expressed in flux linkage and

mmf

1, rseffs kNN =

sysxeffsseffssysxs jiiNiNFjFF === ,,22ˆˆpp

Effective number of turns,

(to create the fundamental mmf wave)

s

sxysyxsxeffsysyeffsx

sxeffsysyeffsxsxysyx

i

iiiNiN

iNiNFFT

=

==

=

==

p

p

p

p

,,

,,

22

2ˆˆ

2

Important conclusion

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Flux vectors, and inductances ...

s i

x

y s y m y s y

y

e f f s i L i

R

N =

2 ,

2 2

p

s x m x s x x

e f f s i L i

R

N =

2 ,

2 2

p

x

m

e f f s m R

F

N

ˆ 2

, p =

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2 Phases

m m

x x

yy

i

isi

sisi

jssysx

xys eijiii ==

qq

ssj

sjxy

ss jiieieii rr ===

)sin()cos( = tjitijii rsrsss

Stationary operation:

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3 Phases

=

=

=

ssc

ssb

sa

iii

iii

ii

2

3

2

1

3

2

2

3

2

1

3

2

3

2

m

x

y

ai

bi

ci

m

x

y

i

i

si si

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Leakage inductances

csceffsc

bsbeffsb

asaeffsa

iLN

iLN

iLN

==

==

==

,

,

,

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Stator voltage equations

csccsc

csc

bsbbsb

bsb

asaasa

asa

iLdt

diR

dt

diRu

iLdt

diR

dt

diRu

iLdt

diR

dt

diRu

==

==

==

sssss

sss iLdt

diR

dt

diRu

==

”+”

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Stator voltage in the

rotor reference frame

xyss

xyr

xyss

xyxyss

xys iLjiL

dt

diRu

=

m

x

y

a i

b i

c i

s i

sxsxmrsy

sysys

sxssxmxmrsyssymysyssy

sysyrsxsxmsxs

syssymyrsxssxmxmsxssx

iLdt

diLiR

iLiLiLiLdt

diRu

iLiLdt

diR

iLiLiLiLdt

diRu

=

==

=

==

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Control challenges

• Priority:

– Torque

– Stator flux

– Power factor

– Field weakening

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DC Machine control

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Why DC?

• Simple to control

• Cheap to produce – but only due to

effective production and large series.

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Mechanical design:I

m

0a

x

y

ai

fi

Arm ature winding

C om pensation widning

C om m utation pole

Field winding

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Mechanical design:II

ai ai

)(qAF

20

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Mathematical model:I

am

mra

aaaa

iT

dt

diLiRu

=

=

m

x

y

a i

b i

c i

s i

sxsxmrsy

sysys

sxssxmxmrsyssymysyssy

sysyrsxsxmsxs

syssymyrsxssxmxmsxssx

iLdt

diLiR

iLiLiLiLdt

diRu

iLiLdt

diR

iLiLiLiLdt

diRu

=

==

=

==

m

0 a

x

y

a i

f i

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Mathematical model:II

fmf

fmm

fffff

LLL

iL

dt

diLiRu

=

=

=

a u

f u

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Torque Control

mra

a

kn

n

aas

a

a

saa

a

s

aa

kke

keniniT

R

L

Tkiki

R

T

Lku

=

=

=

=

)()(

)()()(

2

)()(2

)(1

0

***

R Le

u

i

)()()(*

2

)()(*2

)(*1

0

keniniT

R

L

Tkiki

R

T

Lku

kn

ns

s

s

=

=

=

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Field weakening

=

=

nomrrr

nomrnomm

nomrrnomm

m

mra

if

if

e

,,

,

,,*

nom max

m

fi

*fi

*fi

PI

ae

max,ae

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Example

ref

act

u

if-controller

ref

act

emf

u

ia-controller

ref

act

if*

ea-controller1

Signal

Generator

ua

uf

Tl

ia

T

wr

ea

if

Psim

DC Machine

eamax

Udc

Udc

u*

sa

sb

um

4Q 2 level modulator

Udc

sa

sb

va

vb

a b

4Q 2 level inverter

Udc

u*

sa

um

2Q 2 level modulator

Udc

sa

va

2Q 2 level inverter

torque generation with electrical machines - iea.lth.se 14/torquegeneration2014_03_26.pdf ·...

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