Industrial Electrical Engineering and Automation Lund University, Sweden Torque generation with Electrical Machines
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Industrial Electrical Engineering and Automation
Lund University, Sweden
Torque generation
with
Electrical Machines
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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 ==
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
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