Finite-Element Electrical Machine Simulation...iJsol dS Ω = ∫ ⋅ r r Ωsol isol 18 Dr.-Ing....
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Technische Universität Darmstadt, Fachbereich Elektrotechnik und Informationstechnik Schloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de Dr.-Ing. Herbert De Gersem Institut für Theorie Elektromagnetischer Felder Lecture Series Finite-Element Electrical Machine Simulation in the framework of the DFG Research Group 575 „High Frequency Parasitic Effects in Inverter-Fed Electrical Drives” http://www.ew.e-technik.tu-darmstadt.de/FOR575 Dr.-Ing. Herbert De Gersem summer semester 2006 Institut für Theorie Elektromagnetischer Felder
Transcript of 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
Dr.-
Ing.
Her
bert
De
Ger
sem
In
stitu
t für
The
orie
Ele
ktro
mag
netis
cher
Fel
der
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
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V07:Coupling to External Circuits
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rIntroduction
x
y
zend-windingsend-rings
mℜ
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)
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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
DH Jt
∂∇× = +
∂
rr r
0= +∇×rr
B A0B∇⋅ =r
B AEt t
∂ ∂∇× = − = −∇×
∂ ∂
rrr
Ar
ϕ
∂= − −∇
∂
rr AE
tϕ
Welec
Wmagn
τPloss
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r
AmpèreH J∇× =r r
( )B E∇× =r r
ν σ
( )sJ
AAt
∂∇× ∇× + = − ∇
∂ r
rr
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
rA
rB
drSdrs
φ
induced voltage
d∂
= ⋅∫r r
SA sφ
indd dd
∂
= − ⋅∫r r
Su A s
t
S∂S
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rA-ϕ Formulation
H J∇× =r r
BEt
∂∇× = −
∂
rr
. 0B∇ =r
0B A= +∇×rr
AEt
∂= − −∇ϕ
∂
rr
H B= νr r
sJ E J= σ +r r r
( ) sAA Jt
∂∇× ν∇× +σ +σ∇ϕ =
∂
rr r
0J∇⋅ =r
( ) 0At
⎛ ⎞∂−∇⋅ σ −∇⋅ σ∇ϕ =⎜ ⎟∂⎝ ⎠
r
magnetic vector potential, electric scalar potential
A∇×r A
t∂
− −∇∂
r
,A ϕr
ϕ uniqueandand not unique
( ),A ϕr
,A ctt
∂ψ⎛ ⎞+∇ψ ϕ− +⎜ ⎟∂⎝ ⎠
ris a solution
is a solution as well
∇ϕ
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rA* Formulation
H J∇× =r r
BEt
∂∇× = −
∂
rr
. 0B∇ =r *0B A= +∇×
rr
*AEt
∂= −
∂
rr
H B= νr r
sJ E J= σ +r r r
0J∇⋅ =r
modified magnetic vector potential
( )*
*s
AA Jt
∂∇× ν∇× +σ =
∂
rr r
*0A
t⎛ ⎞∂
−∇⋅ σ =⎜ ⎟⎜ ⎟∂⎝ ⎠
r
*A∇×r *A
runique not unique
*Ar
*A +∇ψr
is a solution
is a solution as well
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rDiscretisation (1)
spatial discretisation
( )sJ
AAt
∂∇× ∇× + = − ∇
∂ r
rr
123ν σ σ ϕ
jik= if=
sd d dj jj
jj
i i idu
u Jd
vt
v vv vΩ Ω Ω
⎛ ⎞⎜ ⎟∇ × ⋅∇ × Ω + ⋅ Ω = ⋅ Ω⎜ ⎟⎝ ⎠
∑ ∫ ∫ ∫r r rrr rν σ
jim=
[ ]⎡ ⎤⎡ ⎤ ⎡ ⎤ ⎡ ⎤+ =⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦
⎣ ⎦j i
jji ij
duk u m f
dt
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rDiscretisation (2)
sd d djj ip jq
jq
pijp
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,))
ij)
ja
FE FEs
ddt
+ =aCM Ca M j) )))% ν σ
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rDiscretisation (3)
stL rtL
( )st rtdL Ldtφ
+ +
mV
m=V
stRrtR
( )st rt agR R R+ + φ
φ
air gap
air gap
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FE FEs
ddt
+ =aCM Ca M j) )))% ν σ
Discretisation (4)
=b Ca)) )a)
along primary edges through primary faces
jb))
ja)
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FE FEs
ddt
+ =
h
aCM Ca M j)
) )))%14243ν σ
h)
j))
through dual faces= along primary edges
along dual edges= through primary faces
Discretisation (5)
jj))
jh)
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rDiscrete A-ϕ Formulation
=Ch j)))
%
ddt
= −bCe
)))
0=Sb)) 0= +b Ca
)) )
ddt
= − − 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
Ca)
φG,φa)and
and not unique
unique
( ),φa)
, d ctdtψ⎛ ⎞+ ψ φ− +⎜ ⎟
⎝ ⎠a G)
is a solution
is a solution as well
ddt
− − φa G)
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agne
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rOverview
discrete magnetoquasistatic formulation (recapitulation)
solid conductors
stranded conductor model
circuit description
example
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rSolid Conductor Coupling
sol
sol0
dE d udtφ
⋅ − = −∫l r
l
sol
sol0 S
dE d u B dSdt
⋅ − = − ⋅∫ ∫l rr r
l
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
sol
sol0
u d= − ∇ϕ⋅∫l
l
soluvoltage drop solu
choice for ?∇ϕ
current soli
sol
soli J dSΩ
= ⋅∫rr
solΩ
soli
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rCoupling at Reference Plane
solusoli
∇ϕ
( )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
AEt
∂= −∇ −
∂
rr
ϕ
only
and
have to fulfill certain conditions
B A= ∇×rr
∇ϕchoose piecewise constantsuch that represents a jump at solid-conductor cross-sectionand such that
solΩϕ
1.
2.
3.
solΩ
sol
sol0
u d= − ∇ϕ⋅∫l
l
4.
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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
sol solsol sol
0T
ju ij G
+ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥
− ⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
CM C M M Q a
Q M
)% %
% %
ν σ σ
σ
ω
ω
sol sol soldudt
⎛ ⎞= −⎜ ⎟⎝ ⎠
aj M Q)))
%σ
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
appEr
1.ϕ
it is possible to define the electric scalar potential such that thesource current is divergence-free
( ) 0−∇ ⋅ ∇ =σ ϕsolvewith boundary conditions
2.
3. dense coupling !
4.
( )sol sol sol
solAi J ds dS dSt
Ω Ω Ω
∂= ⋅ = − ∇ ⋅ − ⋅
∂∫ ∫ ∫r
r rr r σ ϕ σ
total current
se
JJ
AJt
∂= − ∇ −
∂rr
rr
123123
σ ϕ σ
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rCoupling at conductor volume
electrokinetic solution:
planeT = −SM S SM e)% % %σ σφ
s app sol solu= =j M e M Q)) )
σ σ
sol sol soldudt
⎛ ⎞= −⎜ ⎟⎝ ⎠
aj M Q)))
σ
solusoli
appe)app plane sol sol
T u= + =e e S Q) ) % φ
sol
sol solsol sol
0T
ju ij G
+ −⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥
− ⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
CM C M M Q a
Q M
)% ν σ σ
σ
ω
ωrelatively dense
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rNumerical Example
SSOR-COCG coupling matrices number of iterations
solution time (s)
single-phase transformer
(2D)(3D)
198127
1512
three-phase transformer
(2D)(3D)
756465
145176
sol str,Q Q% %
sol str,Q Q
sol str,Q Qsol str,Q Q% %
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rConductor Models
solusoli
zl 2D FEM
solE
solJsol
solid
con
duct
orm
odel
stru
stri
zl 2D FEM
strE
strJ str
1wS strSstrN
stra
nded
con
duct
orm
odel
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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
str
s str,1
( )n
q qq
J t i t=
= ∑r r
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rStranded Conductor Model (2)
( )0, 0, ( , )=r
zJ J x yin 2D:
tstr,
sl/2U z
Nt eS
= +r r
U+Ω
U-Ω
tstr,
sl/2U z
Nt eS
= −r r
U+in Ω
U-in Ωsl/2S
str, 0Ut =r
U+ U-in \ \Ω Ω Ω
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rStranded Conductor Model (3)
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
d∂
= ⋅∫r r
SA sφ
str, str, dq qA t VΩ
= ⋅∫r r
ψ
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rStranded Conductor Model (4)
str, str, dq qA t VΩ
= ⋅∫r r
ψin 2D:
U+Ω
U-Ω
sl/2S
str, t tsl/2 sl/2
1 1d dU U
U z zN A N AS S
+ −Ω Ω
= Ω− Ω∫ ∫ψ
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rStranded Conductor Model (5)
s dii vJΩ
= ⋅ Ω∫j r) r)
str
s str,1
( )n
q qq
J t i t=
= ∑r r
str
str , ,
str,1
d ( )q
i q
n
i q qiq
t i tv= Ω
= ⋅ Ω∑ ∫
P
j)) r
14424
r
43
str str=j P i))
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rStranded Conductor Model (6)
str,str, str,
qq q q
du R i
dt= +
ψ
str, str, dq
q qA t VΩ
= ⋅∫r r
ψ
str, ,
str, str, str, dq
j q
jq q q j q
j
du R i v t
dtΩ
= + ⋅ Ω∑ ∫
P
a) rr
1442443
str str str strT d
dt= +
au R i P)
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rStranded Conductor Model (7)
field model+ stranded conductors+ voltage sources
str
str strstr str
0Tj
−⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦
CM C P ai uP R
)% ν
ω
1j
−ω
symmetrisation: multiply the circuit equations by
no eddy-current term !
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rOverview
discrete magnetoquasistatic formulation (recapitulation)
solid conductors
stranded conductor model
circuit description
example
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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
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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
n0#
n3
n4
4 Ω
3 Ω
10 V
-2 A
twiglink
starting from the circuit node n0#,the twigs are selected in the order1. voltage source 10 V2. resistor 1 Ω3. resistor 3 Ω
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rCircuit Description (2)
2. Determine fundamental cutsets and fundamental loops
fundamental cutsetfundamental loop
n1
n0#
n3
n4
a
b
c
de
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
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rCircuit Description (3)
3. Construct the fundamental cutset and fundamental loop matrices
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
11111
111D
⎥⎦
⎤⎢⎣
⎡−−−−
=111
1111B
fundamental cutset matrixn1
n0#
n3
n4
a
b
c
de
fundamental loop matrix
remark: Tlntw,twln, DB −=
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rCircuit Description (4)
4. Partition the fundamental incidence matrices
n1
n0#
n3
n4
a
b
c
de
lnotwv,D
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
11111
111D
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)
38
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rCircuit Description (5)
5. Write impedance/admittance matrices and voltage/current vectors
⎥⎦
⎤⎢⎣
⎡=
3/11/1
twoY
[ ]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
n0#
n3
n4
a
b
c
de
4. current vectorfor the current sources
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rCircuit Description (6)
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
n1
n0#
n3
n4
a
b
c
de
b1 0 1 00 0.333 1 21 1 4 10
c
d
uui
⎡ ⎤ ⎡ ⎤ ⎡ ⎤−⎢ ⎥ ⎢ ⎥ ⎢ ⎥
= −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥− − −⎣ ⎦ ⎣ ⎦ ⎣ ⎦
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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rCircuit Description (7)
7. Solve system of equations & propagate the circuit solution
n1
n0#
n3
n4
a
b
c
de 0.5 A
0.5 A
2 A-2.5 A
-2.5 A
2 V10 V
0.5 V
-7.5 V
2.5 V
0 V
10 V 9.5 V
7.5 V
b 0.57.5
0.5c
d
uui
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥
= −⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦
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= − −
41
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rParticularities
n4 n53 V
5 Ω
1. Distinct circuit parts
2. Dangling nodesselfloop
n0#
n2n3
n1
7 Ω
dang
ling
node
a branch to a dangling node always a twigassociated fundamental cutsetonly contains the twig
-1 A
4 Ω 3. Self-loopsa self-loop
is always a linkthe associated fundamental looponly contains the link2 V
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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
v2
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rConsistency Check (2)
2. Fundamental cutset consisting of current sources
i1i0
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
i2
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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
i2
~ small and independent Schur complements
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rPartial Transformation (2)
2. Solid conductor being selected as link
u1 usol
u2
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
46
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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
1 1
2 2
0 00 0
G uG u
⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
1 1
2 2
1 01
G uR i U
⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦ ⎣ ⎦
47
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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
1 1
2 2
0 00 0
G uG u
⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣ ⎦
1 1
2 2
1 01
G uR i U
⎡ ⎤ ⎡ ⎤ ⎡ ⎤=⎢ ⎥ ⎢ ⎥ ⎢ ⎥− ⎣ ⎦⎣ ⎦ ⎣ ⎦
49
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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
0T
T
ν σ 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
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rOther Applications
appφ
appΘ
0Λ
1Λ
2Λ
0φ
1φ
2φ
1,wΓ
2,wΓ
0,gΓ
0,wΓ
appφ
appΘ
0Λ
1Λ
2Λ
0φ
1φ
2φ
1,gΓ
2,gΓ
1,wΓ
2,wΓ
dielectricum
electrode
line of symmetry
magneticshort-circuitconnection
~
external electric circuit
U CL
R
3. electrokinetic field+ magnetic circuit+ electric circuit
1. magnetic field + magnetic circuit
2. magnetic field + analytical model+ electric circuit
57
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rOverview
discrete magnetoquasistatic formulation (recapitulation)
solid conductors
stranded conductor model
circuit description
example
58
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rTransformer (1)
secondary coil(600 turns)
primary bars
symmetry plane (magnetic BC)
flux walls(electric BC)
iron yoke parts
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rTransformer (2)
C1C2 D1 D2
C1 C2 D1 D22D FE
Rload
770 A
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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
61
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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
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rTransformer (5)
10-5
100
105
0
1
2
3
4
5
6
load resistance (Ohm)
seco
ndar
y po
wer
con
sum
ptio
n (W
)
10-4
10-2
100
102
0
1
2
3
4
5
6
secondary current (A)
seco
ndar
y vo
ltage
(V)
Technische Universität Darmstadt, Fachbereich Elektrotechnik und InformationstechnikSchloßgartenstr. 8, 64289 Darmstadt, Germany - URL: www.TEMF.de
Dr.-
Ing.
Her
bert
De
Ger
sem
In
stitu
t für
The
orie
Ele
ktro
mag
netis
cher
Fel
der
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
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