Overview Electrical Machines and Drives - TU Delft OCW · Challenge the future 1 Overview...

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


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)


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⋅ =∫∫


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 = = =



circuits










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


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?


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


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


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



circuits










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!


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⋅ = − = −∫


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⋅ = − ⋅∫ ∫∫


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

=


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 ===



circuits










Magnetization curves

2T: heavily saturated silicon steel

limitation


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


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

≈ ≈

∫∫∫ ∫∫∫



circuits










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


Magnetizing magnets


Calculating magnetic fields with

magnets

d dm mC S

H s J n Aτ⋅ = ⋅∫ ∫∫

0=+ ggmm lHlH

permeability iron assumed infinite


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⋅ =∫∫


Rare earth magnets

rmrmm BHB += µµ0


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


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


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



Drives








• 2-10 5.13-6.3: IM drives, SM

• 5-10 6.4-6.13: SM, PMACM



• 9-11: exam


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)


Electromagnetic energy conversion

(4.1)


Induced voltage

BlvE =


Lorentz force

BliF =

FvBlviEiP ===Power balance holds:

)( BvEqF

×+=


Lorentz force



Drives








• 2-10 5.13-6.3: IM drives, SM

• 5-10 6.4-6.13: SM, PMACM



• 9-11: exam

Overview Electrical Machines and Drives - TU Delft OCW · Challenge the future 1 Overview...

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Transcript of Overview Electrical Machines and Drives - TU Delft OCW · Challenge the future 1 Overview...