finite-element electrical machine simulation...ijsol ds Ω = ∫ ⋅ r r Ωsol isol 18 dr.-ing....

Technische Universität Darmstadt, Fachbereich Elektrotechnik und InformationstechnikSchloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de

t für

Lecture Series

Finite-Element Electrical Machine Simulation

in the framework of the DFG Research Group 575„High Frequency Parasitic Effectsin Inverter-Fed Electrical Drives”

http://www.ew.e-technik.tu-darmstadt.de/FOR575

Dr.-Ing. Herbert De Gersemsummer semester 2006

Institut für Theorie Elektromagnetischer Felder

itut f

V07:Coupling to External Circuits

itut f

rIntroduction

zend-windingsend-rings

field-circuit coupling

FE/FIT modelgeometrical detailsferromagnetic saturation (non-linear!!)(motional) eddy currents

circuitexternal sources/loads, (e.g. power electronic equipment)parts outside the FE model(e.g. end windings/rings)representing (linear) parts for which an equivalentcircuit suffices (e.g. homopolar shaft flux)

itut f

rOverview

discrete magnetoquasistatic formulation (recapitulation)

solid conductors

stranded conductor model

circuit description

example

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rMagnetoquasistatics (1)

• neglect displacement currents with respect to conducting currents– Ampère-Maxwell

• magnetic vector potential– conservation of magnetic flux

• electric scalar potential (voltage)– Faraday-Lenz

∂∇× = +

0= +∇×rr

B A0B∇⋅ =r

B AEt t

∂ ∂∇× = − = −∇×

∂ ∂

∂= − −∇

τPloss

itut f

AmpèreH J∇× =r r

( )B E∇× =r r

∂∇× ∇× + = − ∇

123ν σ σ ϕ

1B H Hµν

= =r r r

J E=r r

σconductivity

permeability

reluctivity

Magnetoquasistatics (2)

source current density

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rPhysical Meaning

flux d= ⋅∫∫rr

SB Sφ

Stokes

d= ∇× ⋅∫∫r r

SA Sφ

definition magneticvector potential

drSdrs

induced voltage

= ⋅∫r r

SA sφ

indd dd

= − ⋅∫r r

Su A s

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rA-ϕ Formulation

H J∇× =r r

∂∇× = −

. 0B∇ =r

0B A= +∇×rr

∂= − −∇ϕ

H B= νr r

sJ E J= σ +r r r

( ) sAA Jt

∂∇× ν∇× +σ +σ∇ϕ =

0J∇⋅ =r

( ) 0At

⎛ ⎞∂−∇⋅ σ −∇⋅ σ∇ϕ =⎜ ⎟∂⎝ ⎠

magnetic vector potential, electric scalar potential

A∇×r A

− −∇∂

,A ϕr

ϕ uniqueandand not unique

( ),A ϕr

,A ctt

∂ψ⎛ ⎞+∇ψ ϕ− +⎜ ⎟∂⎝ ⎠

ris a solution

is a solution as well

itut f

rA* Formulation

H J∇× =r r

∂∇× = −

. 0B∇ =r *0B A= +∇×

∂= −

H B= νr r

sJ E J= σ +r r r

0J∇⋅ =r

modified magnetic vector potential

∂∇× ν∇× +σ =

t⎛ ⎞∂

−∇⋅ σ =⎜ ⎟⎜ ⎟∂⎝ ⎠

*A∇×r *A

runique not unique

*A +∇ψr

is a solution

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rDiscretisation (1)

spatial discretisation

∂∇× ∇× + = − ∇

123ν σ σ ϕ

jik= if=

sd d dj jj

i i idu

v vv vΩ Ω Ω

⎛ ⎞⎜ ⎟∇ × ⋅∇ × Ω + ⋅ Ω = ⋅ Ω⎜ ⎟⎝ ⎠

∑ ∫ ∫ ∫r r rrr rν σ

[ ]⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ =⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

⎣ ⎦j i

jji ij

duk u m f

itut f

rDiscretisation (2)

sd d djj ip jq

qiz vz v v

duu c c J

dtΩ Ω Ω

⎛ ⎞⎜ ⎟⋅ Ω + ⋅ Ω = ⋅ Ω⎜ ⎟⎝ ⎠

∑ ∑∑ ∫ ∫ ∫r rrrr rν σ

FE, ,p qνM FE

, ,i jκM s,))

FE FEs

+ =aCM Ca M j) )))% ν σ

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rDiscretisation (3)

stL rtL

( )st rtdL Ldtφ

stRrtR

( )st rt agR R R+ + φ

air gap

itut f

FE FEs

+ =aCM Ca M j) )))% ν σ

Discretisation (4)

=b Ca)) )a)

along primary edges through primary faces

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FE FEs

aCM Ca M j)

) )))%14243ν σ

through dual faces= along primary edges

along dual edges= through primary faces

Discretisation (5)

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rDiscrete A-ϕ Formulation

=Ch j)))

= −bCe

0=Sb)) 0= +b Ca

= − − φae G)

ν=h M b)) )

sσ= +j M e j) )) ))

sddtν σ σ+ + φ =aCM Ca M M G j) )))%

0=Sj))% 0d

dtσ σ− − φ =aSM SM G)

magnetic vector potential, electric scalar potential

φG,φa)and

and not unique

unique

( ),φa)

, d ctdtψ⎛ ⎞+ ψ φ− +⎜ ⎟

⎝ ⎠a G)

is a solution

− − φa G)

itut f

rOverview

solid conductors

circuit description

example

itut f

rSolid Conductor Coupling

dE d udtφ

⋅ − = −∫l r

sol0 S

dE d u B dSdt

⋅ − = − ⋅∫ ∫l rr r

sol sol

sol0 0 S

dA dd d u A dSdt dt

− ∇ϕ⋅ − ⋅ − = − ∇× ⋅∫ ∫ ∫l l r

r rl l

sol sol

sol0 0 S

dA dd d u A ddt dt

− ∇ϕ⋅ − ⋅ − = − ⋅∫ ∫ ∫l l r

rl l l

u d= − ∇ϕ⋅∫l

soluvoltage drop solu

choice for ?∇ϕ

current soli

soli J dSΩ

= ⋅∫rr

itut f

rCoupling at Reference Plane

solusoli

( )sol sol sol

solAi J ds dS dSt

Ω Ω Ω

∂= ⋅ = − ∇ ⋅ − ⋅

∂∫ ∫ ∫r

r rr r σ ϕ σtotal current

∇ϕand do not necessary have to be continuous !Ar

∂= −∇ −

have to fulfill certain conditions

B A= ∇×rr

∇ϕchoose piecewise constantsuch that represents a jump at solid-conductor cross-sectionand such that

solΩϕ

u d= − ∇ϕ⋅∫l

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rCoupling at Reference Plane

app sol solu=e Q) %at primary edges

solusoli

appe)sol sol sol

Ti = Q j))

%at dual facets

solQ% = 2D incidence matrix

sol solsol sol

ju ij G

+ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

− ⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

CM C M M Q a

ν σ σ

sol sol soldudt

⎛ ⎞= −⎜ ⎟⎝ ⎠

aj M Q)))

s sol solu=j M Q))

%σ (not divergence-free)

(divergence-free)

sol sol solTG =Q M Q% % %σ

= conductance of the reference layer

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rCoupling at conductor volume

solusoli

it is possible to define the electric scalar potential such that thesource current is divergence-free

( ) 0−∇ ⋅ ∇ =σ ϕsolvewith boundary conditions

3. dense coupling !

( )sol sol sol

solAi J ds dS dSt

Ω Ω Ω

∂= ⋅ = − ∇ ⋅ − ⋅

∂∫ ∫ ∫r

r rr r σ ϕ σ

total current

∂= − ∇ −

123123

σ ϕ σ

itut f

rCoupling at conductor volume

electrokinetic solution:

planeT = −SM S SM e)% % %σ σφ

s app sol solu= =j M e M Q)) )

sol sol soldudt

⎛ ⎞= −⎜ ⎟⎝ ⎠

aj M Q)))

solusoli

appe)app plane sol sol

T u= + =e e S Q) ) % φ

sol solsol sol

ju ij G

+ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

− ⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

CM C M M Q a

)% ν σ σ

ωrelatively dense

itut f

rNumerical Example

SSOR-COCG coupling matrices number of iterations

solution time (s)

single-phase transformer

(2D)(3D)

198127

three-phase transformer

(2D)(3D)

756465

145176

sol str,Q Q% %

sol str,Q Q

sol str,Q Qsol str,Q Q% %

itut f

rConductor Models

solusoli

zl 2D FEM

solJsol

zl 2D FEM

strJ str

1wS strSstrN

itut f

rStranded Conductor Model (1)

assumptions• homogeneous current distribution• no eddy currents

notice (model)• there will be an induced voltage !!• current not constant when cross-section not constant

winding function [1/m2]• computed geometrically• by field solution (lecture V10)

str,qtr

str, str, ( )q q qJ t i t=r r

s str,1

J t i t=

= ∑r r

itut f

( )0, 0, ( , )=r

zJ J x yin 2D:

sl/2U z

= +r r

sl/2U z

= −r r

U+in Ω

U-in Ωsl/2S

str, 0Ut =r

U+ U-in \ \Ω Ω Ω

itut f

induced voltage ~ flux linkage• which flux is linked?

for a single path

for a coil• integrating along the coil• average at the coil cross-section

= ⋅∫r r

SA sφ

str, str, dq qA t VΩ

= ⋅∫r r

itut f

str, str, dq qA t VΩ

= ⋅∫r r

ψin 2D:

str, t tsl/2 sl/2

1 1d dU U

U z zN A N AS S

+ −Ω Ω

= Ω− Ω∫ ∫ψ

itut f

s dii vJΩ

= ⋅ Ω∫j r) r)

s str,1

J t i t=

= ∑r r

str , ,

d ( )q

i q qiq

t i tv= Ω

= ⋅ Ω∑ ∫

str str=j P i))

itut f

str,str, str,

qq q q

du R i

str, str, dq

q qA t VΩ

= ⋅∫r r

str, ,

str, str, str, dq

jq q q j q

du R i v t

= + ⋅ Ω∑ ∫

1442443

str str str strT d

au R i P)

itut f

field model+ stranded conductors+ voltage sources

str strstr str

−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

CM C P ai uP R

symmetrisation: multiply the circuit equations by

no eddy-current term !

itut f

rOverview

solid conductors

circuit description

example

itut f

rCoupling Requirements

keeps the FE matrixpart unchangedsparsitypreconditioners (multigrid)possible benefits thanks to structured grids (FIT)

preserves symmetryKrylov subspace solvers for symmetric systems (CG, MINRES, QMR)storage

preserves positive definitenesssolvers (CG)preconditioners (IC)

unknowns nodal voltages(+a few currents)

loop currents(+a few voltages)

twig voltageslink currents

no (yes) no (yes) yes

yes yes yes

yes (no) yes (no) no [yes]

T x fK By gB C

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦

(compacted) modified nodal analysis

(compacted) loop analysis

hybridanalysis

itut f

rCircuit Description (1)

1. Trace a tree through the circuitpriorities for tree

branches1.

voltage sources2.

solid conductors (field-circuit coupling)3.

capacitors (largest capacitance first)4.

resistors (largest conductance first)5.

inductors (smallest inductance first)

6.stranded conductors (field-circuit coupling)

7.current sources

highest priority, preferably twig

voltage sources

solid conductors (coupled)

capacitors (largest capacitance first)

resistors (largest conductance first)

inductors (smallest inductance first)

stranded conductors (coupled)

current sources

smallest priority, preferably link

Priority list

1 Ωn1

twiglink

starting from the circuit node n0#,the twigs are selected in the order1. voltage source 10 V2. resistor 1 Ω3. resistor 3 Ω

itut f

2. Determine fundamental cutsets and fundamental loops

fundamental cutsetfundamental loop

The orientation of the fundamental cutset/loop is determined by the orientation of the corresponding twig/link

A fundamental cutset is formed by 1 twigand the unique set of links completing the set of branches which would upon removal result in two disconnected circuit parts.

A fundamental loop consists of 1link and the unique path through the tree closing the loop.

Property: priority(twig) ≥ priority(branch), ∀branch ∈ fundamental cutset

Property: priority(link) ≤ priority(branch), ∀branch ∈ fundamental loop

itut f

3. Construct the fundamental cutset and fundamental loop matrices

⎥⎥⎥

⎢⎢⎢

⎡−=

⎥⎦

⎤⎢⎣

⎡−−−−

fundamental cutset matrixn1

fundamental loop matrix

remark: Tlntw,twln, DB −=

itut f

4. Partition the fundamental incidence matrices

lnotwv,D

⎥⎥⎥

⎢⎢⎢

⎡−=

lnitwv,D

lnitwo,D

lnotwo,D

twv twigs at which the voltage is known (voltage sources)two twigs at which an unknown voltage is assigned (“free twigs”)twu eliminated twigs (“eliminated twigs”) (not in this example)lnu eliminated links (“eliminated links”) (not in this example)lno links at which an unknown current is assigned (“free links”)lni links at which the current is known (current sources)

itut f

5. Write impedance/admittance matrices and voltage/current vectors

⎥⎦

⎤⎢⎣

3/11/1

[ ]4lno =Z

[ ]2lni =i

[ ]twv 10u =

3. voltage vectorfor the voltage sources

2. impedance matrix for the free links

1. admittance matrixfor the free twigsn1

4. current vectorfor the current sources

itut f

1. write the Kirchhoff current law for each fundamental cutsetassociated with a free twig

2. write the Kirchhoff voltage law for each fundamental loop associated with a free link

intuitive approach:6. Write system of equations

b1 0 1 00 0.333 1 21 1 4 10

⎡ ⎤ ⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥ ⎢ ⎥

= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦

two two,lno two,lni lnitwo

lno,two lno lno,twv twv lno

Y D D iuB Z B vi

−⎡ ⎤ ⎡ ⎤⎡ ⎤=⎢ ⎥ ⎢ ⎥⎢ ⎥− − ⎣ ⎦⎣ ⎦ ⎣ ⎦

remark: symmetric because Tlnotwo,twolno, DB −=

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7. Solve system of equations & propagate the circuit solution

de 0.5 A

2 A-2.5 A

-2.5 A

2 V10 V

-7.5 V

10 V 9.5 V

b 0.57.5

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥

= −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

solution

twig currents:

lnilnitwo,lnolnotwo,two iDiDi

link voltages

lno lno,two two lno,twv twv u B u B u= − −−−=

lnilnitwv,lnolnotwv,twv iDiDi −−= lni lni,two two lni,twv twv u B u B u= − −

itut f

rParticularities

n4 n53 V

1. Distinct circuit parts

2. Dangling nodesselfloop

a branch to a dangling node always a twigassociated fundamental cutsetonly contains the twig

4 Ω 3. Self-loopsa self-loop

is always a linkthe associated fundamental looponly contains the link2 V

itut f

rConsistency Check (1)

1. Fundamental loop consisting of voltage sources

v1 v0Problem: a voltage source is

necessarily selected as link

Treatment: check the Kirchhoffvoltage law in the associated fundamental loop

` e.g. v0-v1+v2 = 0 ??if valid

omit the voltage source linkif not valid

the circuit has no solution

itut f

rConsistency Check (2)

2. Fundamental cutset consisting of current sources

Problem: a current source is necessarily selected as twig

Treatment: check the Kirchhoffcurrent law in the associated fundamental cutsete.g. i0 + i1 - i2 = 0 ??

if validreplace the current source twig by a short-circuit connection

if not validthe circuit has no solution

itut f

rPartial Transformation (1)

1. Stranded conductor being selected as twig

i1istr

Problem: a stranded conductor (current-driven branch) is necessarily selected as twig

Property: priority(twig) ≥ priority(branch), ∀branch ∈ associated fundamental cutset

Treatment: apply the Kirchhoff current law in the associated fundamental cutset to express the stranded-conductor current in terms of link currentse.g. istr = i1 + i2

~ small and independent Schur complements

itut f

rPartial Transformation (2)

2. Solid conductor being selected as link

u1 usol

Problem: a solid conductor (voltage-driven branch) is necessarily selected as link

Property: priority(link) ≤ priority(branch), ∀branch ∈ associated fundamental loop

Treatment: apply the Kirchhoff voltage law in the associated fundamental loop to express the solid-conductor voltage in terms of twig voltagee.g. usol = u1 + u2

~ small and independent Schur complements

itut f

rTopological Changes (1)

1. Switching elements closes (switch, diode, thyristor,…)

Problem: the priority of a branch increases during (transient) simulation

Treatment: consider associated fundamental loop and possibly change link/twig-mode with the branch with the lowest priority

1 11/R G=

2 21/R G=

( )U t

1 11/R G=

2 21/R G=

( )U t

0 00 0

G uG u

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

G uR i U

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦ ⎣ ⎦

itut f

rTopological Changes (2)

2. Switching element opens (switch, diode, thyristor,…)

Problem: the priority of a branch decreases during (transient) simulation

Treatment: consider associated fundamental cutset and possibly change link/twig-mode with the branch with the highest priority

1 11/R G=

2 21/R G=

( )U t

1 11/R G=

2 21/R G=

( )U t

0 00 0

G uG u

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦

G uR i U

⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦ ⎣ ⎦

itut f

rField-Circuit Coupling

1. magnetoquasistatic PDE in terms of the magnetic vector potential2. Kirchhoff current law (applied for fundamental cutsets)3. Kirchhoff voltage law (applied for fundamental loops)+ branch relations for solid and stranded conductors+ branch relations for resistors, capacitors, inductors, ...

two lno

two two two two,lno two two,lni lni

lno lno,twu twulno lno,two lno lno

ν σ σ

⎡ ⎤+ α − − ⎡ ⎤⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥− χ + χα χ = −χ⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ χ⎣ ⎦− −χ −χ −χα ⎣ ⎦⎢ ⎥⎣ ⎦

CM C M M Q Q aQ M G C D u D i

i B uQ B R L

symmetrisation factorfactor determined by the time integrator

itut f

rOther Applications

dielectricum

electrode

line of symmetry

magneticshort-circuitconnection

external electric circuit

3. electrokinetic field+ magnetic circuit+ electric circuit

1. magnetic field + magnetic circuit

2. magnetic field + analytical model+ electric circuit

itut f

rOverview

solid conductors

circuit description

example

itut f

rTransformer (1)

secondary coil(600 turns)

primary bars

symmetry plane (magnetic BC)

flux walls(electric BC)

iron yoke parts

itut f

rTransformer (2)

C1C2 D1 D2

C1 C2 D1 D22D FE

itut f

rTransformer (3)

no-load operation

D1 8.4 mV 89u = ∠ o

D2 8.1 mV 89u = ∠ oloadR = ∞

real time instant

C1C2 D1 D2

C1 6.4 mV 90u = ∠ o

C2 5.0 V 90u = ∠ o

imaginary time instant

C1C2 D1 D2

itut f

rTransformer (4)

load operation

D1 1.0 mV 72u = ∠ o

D2 1.2 mV 74u = ∠ oload 0.1R = Ω

real time instant

C1C2 D1 D2

C1 0.20 V 85u = ∠ o

C2 0.23 V 60u = ∠ o

imaginary time instant

C1C2 D1 D2

itut f

rTransformer (5)

load resistance (Ohm)

secondary current (A)

Technische Universität Darmstadt, Fachbereich Elektrotechnik und InformationstechnikSchloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de

t für

Lecture Series

Finite-Element Electrical Machine Simulation

http://www.ew.e-technik.tu-darmstadt.de/FOR575NEXT LECTURE : THURSDAY, July 6th 2006

V08: Modelling and Simulation of Induction Machines

Dr.-Ing. Herbert De Gersemsummer semester 2006

Institut für Theorie Elektromagnetischer Felder

finite-element electrical machine simulation...ijsol ds Ω = ∫ ⋅ r r Ωsol isol 18 dr.-ing....

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