5. Electromagnetics · 2020-02-04 · Electromagnetics. Electric and magnetic circuits. Lund University / LTH / IEA / Avo Reinap / EIEN20 / 2020-02-04 2. L5: Electromagnetics •Previous

04-FEB-2020EIEN20 Design of Electrical Machines

5. Electromagnetics Electric and magnetic circuits

Lund University / LTH / IEA / Avo Reinap / EIEN20 / 2020-02-04 2

L5: Electromagnetics• Previous lectures• Transformer example (A1)• Geometric optimisation of a transformer• Electromagnetic energy conversion• Equivalent circuits• Home assignment A2 and A3


Previous lectures• Basic formulation for

electromagnetic devices• Heat transfer is

described by heat equation

• Electromagnetism by Maxwell’s equations

• Electro-mechanism by electromagnetic stress tensor or virtual work

Gauss’s Law, Heat transfer

Faraday’s law

Ampere’s circuital law

Gauss’s Law, Electricity

Gauss’s Law, Electricity

Magnetic stress per unit of area

Change of system energy

S m dstF

tBE

JH

D

q

0 B

mWF


Previous lectures• Model – description of system and structure• Modelling – study of the behaviour of the model• Numerical modelling – handle the complexity of PDE

Method Finite element method (FEM)

Finite difference method (FDM)

Boundary element method (BEM)

Equivalent circuit method (ECM)

Point mirroring method (PMM)

Principle of discretisation

m1

m2

q

q* Geometry

approximation Extremely flexible Inflexible Extremely flexible Specific geometries

Simple geometries

Non-linearity Possible Possible Troublesome Possible By constant factors

Computational cost High High High Very low Low


FEM from user point of view• Design and development or Exploration of

component details for system performance and reliability improvement

• Analysis – from series of static analysis, towards quasistatic, transient, multiphysics, …

• Geometric input – programmable geometric modeller, CAD reader/reparation, livelink

• Material library• Analysis loops, sensitivity study, optimization• Comparing FEMM, Ansys and Comsol multiphysics• Closer look at electromagnetic design Ansys vs

Comsol


Short overview

Function Control & CAD

Attractive feature

Femm 2D XY RZ, Static & quasistatic

*.lua, *.m. *.py

Simplicity & accessibility

Ansoft ALL *.js, *.py Integrated competence

Comsol ALL With and without *.m multiphysics

• List of Finite Element Software packages

• ..

W

https://en.wikipedia.org/wiki/List_of_finite_element_software_packages


• User defined: Workbench -> design modeller -> Maxwell

• Library based: RMxprt, 2D @ 3D

Ansys Maxwell


Comsol• All-in-One• Features in

multiphysics overweight the detailed electromagnetic features for machines – self define instead of built in

– Torque calculation

– Core loss modelling


Energy conversion• Energy is the capacity of a system to do work• Energy cannot be created or destroyed, but only

converted from one form into another• Coupling between the different fields obeys to the

principle of energy conversion


Transformer example (A)• In essence the common

purpose of the transformer is

– Power transfer– Power conditioning – Galvanic separation

• The model of a transformer can be established either by starting from power transfer or power conditioning requirement

Intercoupled circuits: Electrical, magnetic and thermal


Transformer example (B)

• The geometry of a single-phase shell and core type of transformer at the point of the geometrical optimum

• The largest available cross-section area of electric and the magnetic ‘conductor’


Transformer example (C)• A single phase shell type

of transformer• Height, htr, undefined i.e.

calculate transformer per meter

• Transferred power is ideally P=UI=0.5ωBm Am Jm Ae

• Losses Ploss =qe Ve +qm Vm


Transformer example (D)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

1000

2000

3000

4000

5000

6000

7000

8000

slotting factor, Ks [-]po

wer

per

met

re, P

/m [W

/m]

fem =0.2W/mKecm =0.2W/mKfem =0.05W/mKecm =0.05W/mK

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5x 107

slotting factor, Ks [-]

curre

nt d

ensi

ty, J

m [A

/m2 ]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

50

100

150

200

250

300

350

400

450


loss

es, Q

/m [W

/m]


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.74

0.76

0.78

0.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94


effic

ienc

y,

[-]


Improved thermal design, by higher thermal conductivity of coil impregnation

transferred power increases Pout~I,

efficiency Ploss~I2

and proportions between coils and core remain nearly the same


Transformer example (E)• Power rating (hc=0.04m)

– Modeled 150-210 VA– Referred 125-150 VA

• Conductor losses (Pcu)– Modeled 4.5-10.5 W– Referred 12-13 W

• Magnetizing losses (Po)– Modeled 4.5 W– Referred 5 W


Shell type vs core type

• Parameterization of construction– Transformer volume Vtr=ltr*wtr*htr=ltr3*kw*kh

• Comparison of different layouts

Φ B I J

length

Volume

?tr

me

VAA


Performance equations• Apparent power

• Voltage

• Magnetic flux

• Current

• Ability to transfer power

• Losses

• Core losses are estimated from the specific loss curves

• Flux density is constant • Power density

mmmeeeloss qlAqlAP

memmmm AAJBIUS 21

21

mm IUS21

dt

tdNdt

tdtetUtu m cos

tABtN

Utt

mmm

m

sinsin

sin

tNAJ

tN

NItIti

em

mm

cos

coscos

ptrmmtr

klJBVS

21


Maximized cross-sections (A)

• geometrically optimal relation between the electric and the magnetic core

p

sTHH

p

sTWW

pHW

sslfttr

tr

meptr N

kkkN

kkkNkk

kkkkkl

VAAkl 11

2

0.25 0.5 1 2 4 8relative transformer width, kW [-]

core type

00

0

2

2

2

2

22

24

4

4

4

4 6

6

6

8

8

101012

14

0.25 0.5 1 2 4 80.25

0.5

1

2

4

8

relative transformer width, kW [-]

rela

tive

trans

form

er h

eigh

t, k H

[-]

shell type

0 0

22

22

2

2

2

44

4

4

4

6

6

68

8

10

10

1214

Height factor:hc =htr -kTH *lsWidth factor: ws =wtr -kTW *lc

Vm=ltr3*kH *kW


Minimal temperature rise

• According to the same loss density (W/m3) and thermal conductivity (W/mK)

0.25 0.5 1 2 4 8relative transformer width, kW [-]

core type

00

0

2

2

2

22

4

4

4

44

6

6

6

8

8

10

10

12

14

16

0.25 0.5 1 2 4 80.25

0.5

1

2

4

8

relative transformer width, kW [-]

rela

tive

trans

form

er h

eigh

t, k H

[-]

shell type

0 02 2

2

2

4 4

4

4

4

6

6

6

8

8 1010

12

14

16


electromagnetic coupling

• Magnetically coupled coils • Coupling facilitated by the magnetic core• FEM becomes useful when defining inductances: self,

mutual, leakage• Challenges: dynamic effects, saturation, hysteresis

2

11

221

121

22

11

III

LMML

nnnn

n

n

n i

ii

LMM

MLMMML

2

1

21

2221

1121

2

1


Electromagnetic energy converter

• Electromagnetic energy converter via intermediate magnetic field

• Equivalent circuit representation of the different fields

i1

e1

Rμ

N1i1

u1

RΩ1 Lσ1

i2

Rδ

e2

RΩ2Lσ2

u2

N2i2


Energy conversion principle

• The energy conservation principle says that the sum of electrical energy input to a device at each time instant has to be equal to the sum of accumulated electromagnetic energy and losses.

lossmagk

kk dWdWdtui

1


Electrical energy• Considering Kirchhoff’s voltage equations the

differential of the electric energy consists of resistive voltage term and electromotive voltage term

• According to Faradays law the opposing electromotive voltage can be either a transformed voltage or/and a motional voltage

kkconductorkkkkk didWdtedtRiidtui

dxdL

dtdxi

dtdiLiL

dtd

dtde


Magnetic energy• The magnet energy is distributed in all passive elements• The differential of flux is expressed through the

permeance and the mmf drop in the magnetic element• The permeances are: linear, parametric nonlinear and

inherently nonlinear ones• The differential of magnetic energy in non-parametric

elements are expressed in terms of currents and flux linkages

we N

kkk

N

kkkmag didFdW

11


Losses• Losses in an electromechanical energy converter can

be separated according the loss origin: electrical, magnetic (and mechanical)

• Electromagnetic losses shear often the same loss mechanism.

• Common for the losses is a phenomenon of friction that opposes (current) flow, magnetisation, motion, etc and causes (irreversible) heat energy loss.

coreconductor

m

jcorejjloss dWdWdWdtRidW

1

2


Ideal transformer• Ideal loss-free

electromagnetic coupling• Electrical equations

• Magnetic equations

• The primary mmf balan- ces the secondary mmf

• The transformation coefficient n is in accordance with instantaneous power balance

• The transformer property is of changing impedances

dtdN

dtdu

dtdN

dtdu

22

2

11

1

22

11

iPNiPN

222

22

11

11 ZnI

UnI

UZi

2

1

1

2

1

2

ii

NN

uun


Real transformer• Real loss-leaky

electromagnetic coupling• Electrical equations

• Magnetic equations

• The imperfection of magnetic coupling

• substitution of the magnetic circuit into the electrical equations

dtdiM

dtdiLiRu

dtdiM

dtdiLiRu

122222

211111

2

22

1

2

2

221222

1

11

2

1

1

112111

NiL

NMi

NiL

NiL

NMi

NiL

MNNLL

MNNLL

1

222

2

111

212

2222

211

11111

niidtdM

dtdiLiRu

niidtdM

ndtdiLiRu


Equivalent circuit

• The resulting equivalent system according to equations• Corresponding components in phasor diagram

201 nIII

222111122222 jXRIjXRnInUjXRIEU


Electric circuit (A)

• Complete equivalent circuit of a transformer

11 NE m

m XNI 1

0

1

10 N

PRNI fe

cc


Electric circuit (B)

• The small voltage drop in R1 and X1 compared to U1 , and Small magnetizing current I0 in comparison with load current I1 allows the shunt terminals transfer to primary terminals,

• The secondary quantities R2 and X2 may be replaced on the primary side by using ideal transformer property R’2 =R2 /n2 and X’2 =X2 /n2 ,

• The new equivalent circuit can be used for small power transformer that simplifies the analysis and parameter identification from the experiments.


Magnetic circuit (A)• Transformer can be seen

as an electromagnetic inductor that is loaded with a secondary winding

• Loaded transformer– Source flux 1 ,– Linked flux 2 ,– Leakage flux σ ,

Gμ1 Gμσ Gμ2

φ2

φ1

i1N1

u1 u2

i2N2

NI

Pc1Pgap

Pc2

Pσ

c

σ


Magnetic circuit (B)• Short-circuited ideal

transformer – R2 =0Ω– Ψ2 =0Vs– Lσ = Ψ1 /i1

• Short-circuited real transformer

– R2 ≠0Ω– Ψ2 =- Ψ20

Gμ1 Gμσ Gμ2

φ2

φ1

i1N1

u1

i2N2


Thermal circuit (A)• Node points i, Qi [W], i [K]

1. Coil loss and temperature2. Tooth loss and temperature3. Yoke loss and temperature4. Ambience temperature

• Thermal conductivity elements Gij [W/K]– From coil to tooth G12

– From coil to yoke G12

– From tooth to yoke G23

– From yoke to ambience G34

G13

G12

G34

G23

4

3

2 1


Thermal circuit (B)• Topology matrix • Circuit formulation

ijji Gnodenodekelement:)T(k,

34

23

13

12

434323312211

GGGG

T

000

00

3

2

1

4

3

2

1

3434

343423132313

23231212

13121312

QQQ

GGGGGGGG

GGGGGGGG

QG

BBNNN

B giveninitially

GQG 1


Transient heat flow• Thermal model representing a

physical model• Many simplifications and

approximations• Heat is not internally

generated in the body• Losses are applied to specific

node-point

1

Rth 2

Cth P

2

1

12

121

av

ththth CRCP

dtd

t

ambmamb

thththththm

th

e

cVRRCRP

1


Electromagnetic energy conversion

• The static coupling between the electric and the magnetic field bases on Amperes law.

• The dynamic coupling between the electric and the magnetic field bases on Faraday’s law.


Ampere’s circuit law• Ampere’s circuit law says that the circulation

of magnetic field intensity around any closed path is equal to the current flowing through the enclosed surface.

• The magnetomotive force, mmf, is analogous to emf and could be seen as the capability to produce magnetic flux through a circuit.


Faraday’s induction law• Faraday’s induction law is a relation between time-

varying electrical field and originated magnetic field or vice versa.

• The negative sign is a statement that the induced emf will force the current to flow in a direction that counteracts the change of magnetic flux. The statement that an induced current counteracts its’ originate is known as Lenz law.

• The electro motive force, emf, could be seen as the current producing capability in a circuit.


Coupled fields• Varying magnetic field induces electric field

• Electric field causes electric current and resistive losses and these losses thermal heating in the conductive material

dtdds

dtddl

SC

BE

22 EiEi tpec


Eddy currents in a plate

• It is assumed that induced current flow only in the horizontal direction (x) and the electric field is linearly dependent on y from the centre of the plate

dt

dByyEBywdtdyEwyE z

xzxx 22E

Ex(y), i

Ex(y), iy

Bz

z

x

w

d


Power loss in the plate• The instantaneous power loss per unit of volume

• The mean value of the instantaneous loss across the cross-section

221, y

dtdBytp z

ec

222/

2/

22

1211,

dtdBddywy

dtdB

dwytp z

d

d

zec


Magnetic saturation

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-3

-2

-1

0

1

2

3

4

5

time, t [sec]

norm

aliz

ed q

uant

ities

a(t)

/max

(a(t)

)

u(t)B(t)i(t)e(t)

• Electromagnetic devices are usually voltage driven, FEM formulation for electromagnetics current driven

• Electromagnet with geometric and material input• Electromagnetic transient• Useful modelling environment Matlab Simulink, modelling

examples in SimPowerSystems


Magnetic hysteresis


3φ transformers

• Construction: Shell and core type

• Connections: Y-Y,Y-Δ, …• Control: Tertiary windings


3φ transformer models in Simulink


Home assignments: A1 – A4


Home assignment A2

0 5 10 15 20 25 300

100

200

300

400

In/O

utpu

t pow

er, P

i,Pu [W

]

0 5 10 15 20 25 300.4

0.6

0.8

1

effic

ienc

y,

[-]

load current, Iout [A]0 5 10 15 20 25 30

0

5

10

15

20

Cop

per l

osse

s, P

cu [W

]

0 5 10 15 20 25 307

8

9

10

Cor

e lo

sses

, Pfe

[W]

load current, Iout [A]

0 5 10 15 20 25 300

1

2

3

Inpu

t cur

rent

, Iin

[A]

0 5 10 15 20 25 300

10

20

30

Out

put v

olta

ge U

2 [V]

load current, Iout [A]0 5 10 15 20 25 30

0

2

4

6

curre

nt d

ensi

ty, J

cm2 [A

/mm

2 ]

0 5 10 15 20 25 30-100

-50

0

Out

put v

olta

ge re

qula

tion

U

2 [-]

load current, Iout [A]

• U2 given, how you get transformer? • How to compare MEC and FEM?

– Magnetisation– Load characteristics


Home assignment A3• PMSM model based on

lumped equivalent circuits

– Magnetic– Thermal

• Compare with FEM– Shear stress– Sheet current density

• Try to improve your initial design

Documents

5. Electromagnetics · 2020-02-04 · Electromagnetics. Electric and magnetic circuits. Lund University / LTH / IEA / Avo Reinap / EIEN20 / 2020-02-04 2. L5: Electromagnetics •Previous