7. torque production - iea•magnetomotive force, field intensity, flux •electromagnets and...

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


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


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


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


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


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


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


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


Machine layouts• Machine classification

– Saliency: none, single, double

– Supply: single or double fed

– Excitation: EM & PM– Magnetization: IM &

RM

8.2 p 259-260


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


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


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


Sinusoidally distributed windings

)(sF )(sF

sF sF

8.5 p 262-263


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


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


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


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


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


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


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


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


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


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


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


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


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)

7. torque production - iea•magnetomotive force, field intensity, flux •electromagnets and...

Documents