7. torque production - iea•magnetomotive force, field intensity, flux •electromagnets and...
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7. Torque production Electromagnetic and magnetic forces Rotating magnetic field
10-FEB-2020EIEN25 Power Electronics
Devices, Converters, Control and Applications
Lund University / LTH / IEA / Avo Reinap / EIEN25 / 2020-02-10 2
L7: Torque production• Electromagnetic & magnetic forces• Magnetomotive force, field intensity, flux• Electromagnets and permanent magnets• Flux variation & electromotive force• Electromagnetic layouts for machines• Machine types & energy conversion• MMF waveforms and torque• Three-phase system
Lund University / LTH / IEA / Avo Reinap / EIEN25 / 2020-02-10 3
Electromagnetic & magnetic forces• Distance forces – air-gap• Conductor in magnetic
field – Force due to current –
Flemming’s left-hand rule– Induced voltage due to
motion – Flemming’s right-hand rule
• Parts: field and armature
• Arrangement of magnets– Action & Reaction: share,
attraction, repulsion
BI
F
B
Ev• Types of forces and torques
– Excitation, electrodynamical– Reluctance
• Energy=capacity for doing work=Forced Motion W=Fx
two magnetsmagnet and iron
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Forces on wire, slotting, leakage
• Force on current-carrying conductor in an uniform magnetic field Fx =By Iz z
• Slotted coils = mechanical support + reduced reluctance of the mutual flux path Fx =tx Axz =1/μ0 *Bx By xz
• Leakage flux, does not contribute gap (mutual) flux and torque generation
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Magnetomotive force, field intensity & flux
RR
RF
RFFW
Fe
Fe
Femag
22
22
21
21
21
21
2
• Amperes Law
• Constitutive relation
• Magnetic circuit
• Magnetic flux• Magnetic reluctance
l
t dlHINmmfF
HHB r 0
RRFFF
BlBHlHF
FeFe
Fe
FeFeFeFe
00
AlR
• Magnetic energyBA
AI
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Electro- & permanent magnets• Flux density in air-gap –
measure of magnetization• Electromagnet – variable flux
driven by magnemotive force
• Permanent magnet – constant flux driven by electron spins
DemagnetizingMagnetizing
000
Bh
BBpm
pm
rpm
NIB
0
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Flux variation & electromotive force• Faradays Law
• Lenz’s law – any current produced by the emf tends to oppose the flux change
• Flux variation due to magnet movement and current change
dtdN
dtdE
dAtBdlE
t
l
sss LKKLLLLdtdLi
dtdiLLi
dtd
dtde
Li
1
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Electromechanical energy converters
• Conversion of electric energy into mechanical energy or vice versa
– Reversible except for the energy losses – motoring and generating mode
– In the presence of magnetic field (energy density)– Mechanical motion – translational, rotational, …– Electrical EMF waveform – pulsating DC,
alternating AC
8.1 p 257-258
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Energy density• Medium ability to maintain
magnetic field or/and electric field
• The permittivity ε is for polarization whereas the permeability μ is for magnetization
• Flux density in the air-gap• Bg =1T → ~400000 J/m3 > PM• Break down field for air• Eb =3kV/mm → ~40 J/m3
• Compare energy density [MJ/Lit] and specific energy [MJ/kg]
8.1 p 258-259
VmAs
mF
cEED
AmVs
mHBHB
ED
HB
9
020
0
20
70
0
2
1036
1122
10422
Storage material MJ/L MJ/kg Liquid hydrogen 10 142Diesel 35.8 48Lithium metal battery 4.3 1.8Lithium-ion battery 2.6 0.8carbohydrates 43 17
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Machine layouts• Machine classification
– Saliency: none, single, double
– Supply: single or double fed
– Excitation: EM & PM– Magnetization: IM &
RM
8.2 p 259-260
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Generalized machine theory• Different types of machines –
unified mathematical theory• Doubly fed machine expressed as
two sets (rotor & stator) magnetically coupled / cross- coupled two-phase circuits (direct axis & quadrature axis)
– Electric equations U=Rs i + dΨ/dt– Magnetic equations Ψ= Ψm + Ls i– Mechanic equations T=Ψm x i
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Number of poles8.3 p 260-261
N
S
N
S
N
S
e = ddt
e = ddt+
-
+
-
p(t) = el Tel = p2 mec 2
p Tmec = mec Tmec
mecel Tp
T 2
mecelp 2
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Basic, single phase torque
• Magnetically coupled rotary and stationary coil– s=d/dt and M=f(θ)
2
1
2221
1211
2
1
ii
sLRsMsMsLR
uu
)cos( rlIBTr
γN S
B
F
F
ddLii
ddLi
ddLiT 12
2122
212
1 21
21
8.4 p 262
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Sinusoidally distributed windings
)(sF )(sF
sF sF
8.5 p 262-263
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The PM and RM machine
RotorRotor
Stator
Stator
N S NmF
mF
0 2
sF
sF
• Salient poles in the rotor
• Rotor reference frame (x/y) (or (d/q))
• Rotor mmf and stator mmf (sinusiodal)
• Reluctances Rx and Ry
• Ideal iron (no magnetic saturation effects)
mF
sF
8.6 p 264-265
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Air-gap magnetic energy
Rotor
Stator
mF
sF
yxsm FFFFF
sysy
msxmsx
FFF
FFFFFˆ)sin(ˆ
ˆˆˆ)cos(ˆ
y
s
x
mmss
y
y
x
xmagn R
FR
FFFFR
F
RF
W 2222222 sinˆ
21ˆcosˆˆ2cosˆ
21ˆ
21ˆ
21
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Torque: derivative of magnetic energy I
Td
dWmec
Constant, i.e. no energy supplied mecmagn WW
Compare to linear movement(F=dW/dx or W=F*x)
0 d
dWd
dW mecmagn
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Torque: derivative of magnetic energy II
yxsysx
x
msymagn
RRFF
R
FF
d
dW-T 11ˆˆ
ˆˆ...
yxsysx
x
msysxsy
ymsysysx
x
ssy
msssx
y
s
x
mss
magn
sy
mmssx
magn
RRFF
RFF
FFR
FFFFR
FFR
FFFFR
RF
RFFF
dWd
FR
FFFFR
W
11ˆˆˆˆ
ˆˆ1ˆˆˆˆ1
cosˆsinˆ1sinˆˆsinˆcosˆ1
2cossin2ˆ
2sinˆˆ2sincos2ˆ
sinˆ2
1ˆcosˆˆ2cosˆ2
1
22
22222
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Torque components
Non-magnetized
Magnetized Non-magnetized Magnetized Magnetized
No torque
Torq
ue
definedby equation XX
yxsysx
x
msymagn
RRFF
R
FF
d
dW-T 11ˆˆ
ˆˆ...
External magnetic field generated by Fs
8.5 p 266-267
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Electrically magnetized stator
N s number ofwinding turns
)(sF
)(sF )(sF
sF sF
sF
sF
sss iNF 2ˆ
1Fundamental mmf
8.7 p 267-271
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Torque expressed in flux and mmf
s
ss
s
sxysyx
sxsyy
sysxmx
F
FF
F
FF
FFR
FFFR
T
2
sinˆ2
sinˆ2
cossinsincosˆ2
ˆˆ2
ˆˆ1ˆˆˆ1
sF F
mF
y
y
x
x
R
Fj
R
F
ˆ2ˆ2
yF
xF
Remember: Flux=MMF/Reluctance
Important conclusion
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Torque expressed in flux linkage 1, rseffs kNN
sysxeffsseffssysxs jiiNiNFjFF ,,22ˆˆ
Effective number of turns,(to create the fundamental mmf wave)
s
sxysyxsxeffsysyeffsx
sxeffsysyeffsxsxysyx
i
iiiNiN
iNiNFFT
,,
,,22
2ˆˆ
2
Important conclusion
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MMF waves & torque• Maximum reaction
torque @ armature field is perpendicular to excitation field
• Maximum torque control
– Defined by hardware – commutator in DC motors
– Defined software wise – power electronic field oriented control in AC motors
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Rotation and multi-phase winding
ssc
ssb
sa
iii
iii
ii
23
21
32
23
21
32
32
tjitijiijiieieii
eijiii
rsrsss
ssj
sjxy
ss
jssysx
xys
rr
sincos
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Three-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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Phasors and vectors• Phasor=rotating vector,• Vector=Magnitude+angle• abc – positive sequence• acb – negative sequence• Phase voltage UL1N =√2Um
• Line voltage UL12 =√3UL1N
• Travelling MMF wave• Single-phase wave
consists of forward and backward components
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Exercises on MMF distribution (4)• PE ExercisesWithSolutions2019b vers 190206• Draw cross-section: EMSM (4.1) & DCM (4.2)• Sine wave MMF: Single (4.3) Dual (4.9) wave• Torque expression (4.10)• Flux vector (4.11) (4.12) + armature current (4.13)• Rotation problem (4.14)• Voltage equation (4.15)