Post on 28-Apr-2018
1Challenge the future
Overview Electrical Machines and
Drives
• 7-9 1: Introduction, Maxwell’s equations, magnetic circuits
• 11-9 1.2-3: Magnetic circuits, Principles
• 14-9 3-4.2: Principles, DC machines
• 18-9 4.3-4.7: DC machines and drives
• 21-9 5.2-5.6: IM introduction, IM principles
• 25-9 Guest lecture Emile Brink
• 28-9 5.8-5.10: IM equivalent circuits and characteristics
• 2-10 5.13-6.3: IM drives, SM
• 5-10 6.4-6.13: SM, PMACM
• 12-10 6.14-8.3: PMACM, other machines
• 19-10: rest, questions
• 9-11: exam
2Challenge the future
Maxwell’s equations / magnetic
circuits
• Introduction of Maxwell’s equations (for quasi-static fields)
• Ampere’s law used to calculate flux densities (1.1)
• Around a wire in air
• In magnetic circuit
• In a magnetic circuit with an gap
• Second of Maxwell’s equations used to calculate voltages (1.1)
• Soft magnetic materials: hysteresis and eddy currents (1.2)
• Hard magnetic materials: permanent magnets (1.4)
3Challenge the future
Maxwell’s equations for quasi-
static fields: Know by heart!
H : magnetic field intensity
J : current density
E : electric field intensity
B : magnetic flux density
n, τ: unit vectors
d dm mC S
H s J n Aτ⋅ = ⋅∫ ∫∫
dd d
de eC S
E s B n At
τ⋅ = − ⋅∫ ∫∫
d 0S
B n A⋅ =∫∫
4Challenge the future
Magnetic field in a magnetic circuit
Contour follows magnetic path:
d dm mC S
H s J n Aτ⋅ = ⋅∫ ∫∫
l
NiHNiHl =⇒=
l
NiHB rr µµµµ 00 ==
0
corem
r core
Ni NiBA
lRAµ µ
Φ = = =
5Challenge the future
Maxwell’s equations / magnetic
circuits
• Introduction of Maxwell’s equations (for quasi-static fields)
• Ampere’s law used to calculate flux densities (1.1)
• Around a wire in air
• In magnetic circuit
• In a magnetic circuit with an gap
• Second of Maxwell’s equations used to calculate voltages (1.1)
• Soft magnetic materials: hysteresis and eddy currents (1.2)
• Hard magnetic materials: permanent magnets (1.4)
6Challenge the future
Magnetic circuit with air gap
d dm mC S
H s J n Aτ⋅ = ⋅∫ ∫∫
NilHlH ggcc =+
NilB
lB
gg
crc
c =+00 µµµ
d 0S
B n A⋅ =∫∫
ggcc ABAB =Two equations, two unknowns,
can be solved
7Challenge the future
Magnetic circuit with air gap
mcmg RR
Ni
+=Φ
cr
cmc A
lR
µµ0
=
g
gmg A
lR
σµ0
=
The factor σ
Which reluctance is dominating?
8Challenge the future
Example magnetic circuit with air
gap
• Given
• N = 50
• I = 1 A
• μry = 5000
• μ0 = 4π10-7 H/m
• lc = 0.2 m
• lg = 1 mm
• Ac = 4 cm2
• σ = 1
• Calculate
• Reluctances
• Flux
• Flux density
9Challenge the future
Example magnetic circuit with air
gap
Wb
kA6.79
0
==cr
cmc A
lR
µµ
Wb
MA99.1
0
==g
gmg A
lR
σµ
Wb24.2µ=+
=Φmcmg RR
Ni
mT61=Φ=cA
B
10Challenge the future
Comparison
of circuits
Circuit Water Electric Magnetic
Driving force Force F
Pressure
Voltage V
Electric field E
Mmf Ni
Magnetic field H
Produces Flow density
Water flow
Current density J
Current I
Flux density B
Flux Φ
Limited by Resistance Resistance R Reluctance Rm
Isolators Yes Yes No
Physical Water Electrons ?
Power Fv VI 0
11Challenge the future
Maxwell’s equations / magnetic
circuits
• Introduction of Maxwell’s equations (for quasi-static fields)
• Ampere’s law used to calculate flux densities (1.1)
• Around a wire in air
• In magnetic circuit
• In a magnetic circuit with an gap
• Second of Maxwell’s equations used to calculate voltages (1.1)
• Soft magnetic materials: hysteresis and eddy currents (1.2)
• Hard magnetic materials: permanent magnets (1.4)
12Challenge the future
Second of Maxwell’s equations
dd d
de eC S
E s B n At
τ⋅ = − ⋅∫ ∫∫
Contour is chosen in the electric path, in the wire!
13Challenge the future
Calculation of voltage
d d de
term term
C term term
E s E s E sτ τ τ− +
+ −
⋅ = ⋅ + ⋅∫ ∫ ∫
d de
term
Cu Cu Cu
C term
E s J s u l J uτ ρ τ ρ−
+
⋅ = ⋅ − = −∫ ∫
CuA
iJ =
de
Cu Cu
CuC
lE s i u Ri u
A
ρτ⋅ = − = −∫
14Challenge the future
Calculation of voltage
Φ≈⋅= ∫∫ NAnBeS
dλ
With the flux linkage
tNRi
tRiu
d
d
d
d Φ+≈+= λThis can be worked out to
dd d
de eC S
E s B n At
τ⋅ = − ⋅∫ ∫∫
15Challenge the future
For coils
Induced voltage:t
Nt
ed
d
d
d Φ≈= λ
For a coil: LiiR
NN
m
==Φ=2
λ
t
iL
tN
te
d
d
d
d
d
d =Φ== λmR
NL
2
=
16Challenge the future
Special cases
• for DC:
flux and flux linkage are determined by current
• for sinusoidal AC if resistance is negligible:
flux and flux linkage are determined by voltage
tRiu
d
dλ+=
Riu =
tu
d
dλ= λλπλω ˆ44.4ˆ2
2ˆ2
1ffU ===
17Challenge the future
Maxwell’s equations / magnetic
circuits
• Introduction of Maxwell’s equations (for quasi-static fields)
• Ampere’s law used to calculate flux densities (1.1)
• Around a wire in air
• In magnetic circuit
• In a magnetic circuit with an gap
• Second of Maxwell’s equations used to calculate voltages (1.1)
• Soft magnetic materials: hysteresis and eddy currents (1.2)
• Hard magnetic materials: permanent magnets (1.4)
18Challenge the future
Magnetization curves
2T: heavily saturated silicon steel
limitation
19Challenge the future
Iron: Hysteresis losses
∫∫ +== tt
iRituiW dd
dd 2 λ
∫∫ == λλdd
d
dit
tiWh
With NBA=λN
Hli =
∫= BHAlWh d
VBfkP Shh dˆ∫∫∫= 3.25.1 << S
20Challenge the future
Iron: eddy current losses
22 2 2
,0 0
d d12Fe
Fe e eFeV V
b B BP V k V
B
ω ωρ ω
≈ ≈
∫∫∫ ∫∫∫
21Challenge the future
Maxwell’s equations / magnetic
circuits
• Introduction of Maxwell’s equations (for quasi-static fields)
• Ampere’s law used to calculate flux densities (1.1)
• Around a wire in air
• In magnetic circuit
• In a magnetic circuit with an gap
• Second of Maxwell’s equations used to calculate voltages (1.1)
• Soft magnetic materials: hysteresis and eddy currents (1.2)
• Hard magnetic materials: permanent magnets (1.4)
22Challenge the future
Magnetic circuits with permanent
magnets
• Why increase of permanent magnet use?
• field without current: no losses, no windings, small volume
• new strong rare-earth magnets become cheaper
23Challenge the future
Magnetizing magnets
24Challenge the future
Calculating magnetic fields with
magnets
d dm mC S
H s J n Aτ⋅ = ⋅∫ ∫∫
0=+ ggmm lHlH
permeability iron assumed infinite
25Challenge the future
Calculating magnetic fields with
magnets
m
ggm l
lHH −=
gg HB 0µ=
mmgg ABAB =
mgm
mgm H
lA
lAB 0µ−=
Defines shear line (in Sen 1.47, a minus sign is missing)Operating point: intersection of shear line and BH characteristic
d 0S
B n A⋅ =∫∫
26Challenge the future
Rare earth magnets
rmrmm BHB += µµ0
27Challenge the future
Calculation with magnets
What is the effect of increasing the gap?
Combine shear line with BH curve:
Result: rmgrmgm
mmg B
AlAl
AlB
µ+=
rgrmm
mg B
ll
lB
µ+=
If gm AA =
mgm
mgm H
lA
lAB 0µ−=
rmrmm BHB += µµ0
28Challenge the future
Alternative: using magnetic circuit
theory
mr
mmm A
lR
µµ0
=gr
gmext A
lR
µµ0
=
mgmm
mc
RR
lH
+−=Φ
mrm
rmcm l
BlHF
µµ0
=−=Magnetomotive force of a magnet:
A magnet can be replaced by an air coilwith length lm and cross section Am and magnetomotive force Hclm
29Challenge the future
Permanent magnet characteristics
• Characteristics:
• Hc coercive force (Coërcitief veldsterkte) [A/m],
• Br remanent flux density (remanente inductie) [T]
• µrm relative permeability (1.05 ...1.15)
Br
(T)HcB
(kA/m)dBr/dT(%/K)
dHcB/dT(%/K)
ρ (µΩm)
Cost (€/kg)
Ferrite 0,4 -250 -0,2 +0,34 1012 2
Alnico 1,2 -130 -0,05 -0,25 0,5 20
SmCo 1,0 -750 -0,02 -0,03 0,5 100
NdFeB 1,4 -1000 -0,12 -0,55 1,4 75
30Challenge the future
Overview Electrical Machines and
Drives
• 7-9 1: Introduction, Maxwell’s equations, magnetic circuits
• 11-9 1.2-3: Magnetic circuits, Principles
• 14-9 3-4.2: Principles, DC machines
• 18-9 4.3-4.7: DC machines and drives
• 21-9 5.2-5.6: IM introduction, IM principles
• 25-9 Guest lecture Emile Brink
• 28-9 5.8-5.10: IM equivalent circuits and characteristics
• 2-10 5.13-6.3: IM drives, SM
• 5-10 6.4-6.13: SM, PMACM
• 12-10 6.14-8.3: PMACM, other machines
• 19-10: rest, questions
• 9-11: exam
31Challenge the future
Principles of electromechanics (3)
• Lorentz force, induced voltage (4.1)
• Energy or power balance (3.1)
• Energy and coenergy (3.2)
• Calculation of force from (co)energy (3.3)
• Application to actuators and rotating machines (3.4, 3.5)
32Challenge the future
Electromagnetic energy conversion
(4.1)
33Challenge the future
Induced voltage
BlvE =
34Challenge the future
Lorentz force
BliF =
FvBlviEiP ===Power balance holds:
)( BvEqF
×+=
35Challenge the future
Lorentz force
36Challenge the future
Overview Electrical Machines and
Drives
• 7-9 1: Introduction, Maxwell’s equations, magnetic circuits
• 11-9 1.2-3: Magnetic circuits, Principles
• 14-9 3-4.2: Principles, DC machines
• 18-9 4.3-4.7: DC machines and drives
• 21-9 5.2-5.6: IM introduction, IM principles
• 25-9 Guest lecture Emile Brink
• 28-9 5.8-5.10: IM equivalent circuits and characteristics
• 2-10 5.13-6.3: IM drives, SM
• 5-10 6.4-6.13: SM, PMACM
• 12-10 6.14-8.3: PMACM, other machines
• 19-10: rest, questions
• 9-11: exam