23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005 Finite Element Analysis of Voltage Finite Element Analysis of Voltage - - and Current and Current - - Driven Transient Skin Effect Problems Driven Transient Skin Effect Problems Peter Böhm, inuTech GmbH
31
Embed
skin effect problems - inuTech€¦ · Driven Transient Skin Effect Problems Peter Böhm, inuTech GmbH. 23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005 Outline • Problem
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Finite Element Analysis of VoltageFinite Element Analysis of Voltage-- and Currentand Current--Driven Transient Skin Effect ProblemsDriven Transient Skin Effect Problems
Peter Böhm, inuTech GmbH
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Outline
• Problem of distributed parasitic effects• Maxwell equations in the QSA• Potential formulations• Node-/ edge elements• A new simulator based on the C++ library DIFFPACK• Results
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
??
Hoover-Dam (Nevada-Arizona)
KKW Krümmel (D) ICE3 train (D)
wall socket
From the Power Plant to the Wall Outlet
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
IGBT-Modul
GTO-Device
in
out
frequency
amplitude
amplitude
frequency
in
out
Load
Electronic High Power Systems
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Switching losses of semiconductor devices
Power converter with dc commutation:
max
With PWM:
from 0 up to AC ACI Iω ω
Cd
Lσσσσ L
UACUd
i
Te
u
Ta
Ud
t
UAC
The Pulse Width Modulation Method
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
10 cm
Problem Definition
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
• parasitic effects are inevitable• can be reduced to a minimumby optimizing the geometry
• application-dependent optimization targets (e.g.)• maximum current homogeneity -> j(r,t)• minimum self/mutual inductivity -> Lij(t)• minimum overvoltages -> U(t)• minimum turn-on delay -> I(t)• minimum local electrothermal heating
Problem Definition (cont’d)
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
ii
L L=∑
At switching moment:DC-current - Modul - „DC“-current
wiringL
0 switchsin with I I tω ω ω= ⋅ =
wiringL
moduleL
motorLCd
Ud
L(motor)L(wiring) L(modul) L(wiring)
Why to Optimize Lmodul
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Maxwell equations:Constitutive Equations:
Boundary conditions:
Initial conditions:
Interface conditions:
B H
D E
J E
µεσ
=
=
=
r r
r r
r r
E
H
0 on
0 on
E n
H n
× = Γ× = Γ
r rr r nc
nc
0 on
0 onc c n n
c c n n
H n H n
B n B n
× + × = Γ⋅ + ⋅ = Γ
r rr rr rr r
( )0 0 n c in and B t B= Ω Ωr r
0
DH J
t
BE
tB
D ρ
∂∇ × = +∂
∂∇ × = −∂
∇ ⋅ =∇ ⋅ =
rr r
rr
r
r
Maxwell Equations in the QSA
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
A,V-A - Voltage Ramp
10-7
10-5
10-3
10-1
time [s]
40
60
80
100V
olta
ge [
V]
Terminal Voltageexample busbar1
terminal voltage
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Scalar Cut of Current Density
Skineffect-regions
U
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Scalar Cut of Current Density
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
10-7
10-5
10-3
10-1
time [s]
0
2e+05
4e+05
6e+05
8e+05
1e+06
1e+06
curr
ent
[A]
Terminal Currentexample busbar1
terminal current
10-8
10-6
10-4
10-2
100
time [s]
0
2e-08
4e-08
6e-08
8e-08
1e-07
indu
ctiv
ity [H
]
self inductivityexample busbar1
inductivity
10-8 10-6 10-4 10-2 100
time [s]
0
0.05
0.1
0.15
0.2
0.25
Res
ista
nce
[Ohm
]
Resistanceexample busbar1
resistance
Example A,V-A
I (t) ??? L = 2W / I2 => unsatisfying results
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
time [s]
0
200
400
600
800
1000
curr
ent
[A
]
Terminal Currentexmaple viertelstab
applied terminal current
T-T0-Φ - Current Ramp
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
( ) ( ) ( )
( )
( )
0 0
0
0
1 1
0 in ,
0
c
T T grad T Tt t t
T T gradt
T gradt
µ µ µσ σ
µ µ µ
µ µ
∂ ∂ ∂ ∇× ∇× + − Φ = −∇× ∇× − ∂ ∂ ∂
∂ ∇⋅ + − Φ = Ω∂∂ ∇⋅ − Φ =∂
r r r r
r r
rn in Ω
( )
( )
0 E1 E2
cn
0 c n
10 , T T-grad 0 on ,
0 on
0 or T -grad 0 on Γ , Γ
T n n
T n
n
µσ
µ
∇× × = + Φ ⋅ = Γ Γ
× = Γ
Φ = Φ ⋅ =
Φ
r r rr r
r r
r r
( )0 cn and T -grad are continuous on nµ Φ ⋅ Γr r
Boundaryconditions:
Basicequations:
T-T0-Φ-Method
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
0 c
0 cn outer
0 E1 E2
10 in
on ( )
10 on and
S
T
T n H n
T n
σ
σ
∇× ∇× = Ω
× = × Γ Γ
∇× × = Γ Γ
r
r rr rA
r r
How to construct 0T
r
How to get
cn on SH n× Γr r
02
ˆ
4
I dl rdB
r
µπ
×= ⋅r
r
Biot-Savart-Field:
dl∫r
K ˆii ll e∆ ⋅∑
T-T0-Φ-Method (cont’d)
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
2.08e+05
2.08e+05
2.08e+05
2.08e+05
2.08e+05
2.08e+05
curlT 0 [A/m2]
curlT 0 [A/m2]
T-T0-Φ-Method (cont’d)
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Node elements ¤t driven problems
I(soll) I(gl=0) I(gl=2)Contact 1 200 A 180 A 200 AContact 2 100 A 60 A 100 AContact 3 200 A 100 A 200 AContact 4 100 A 90 A 100 AIterations (CG) 112 328
artificialsources
ContactJ const≈r
T0-Problem using Node Elements
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
• possible (and not even obvious) wrong results at reentrant corners.
• bad approximation of field singularities at edges and corners. • node elements require special care about enforcing boundary conditions at material interfaces, conducting surfaces, corners, ...
• node elements impose continuity of all three spatial components but material interfaces with different magnetic permeabilities allow only the tangential component to be continuous
• occurence of spurious modes (only eigenvalue problems)• (need for a gauged formulation, e.g. Coulomb gauge )
Disadvantages using Node Elements
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Edge elements:• allow normal component of a vector to be discontinuous
• better at handling field singularities• better essential b.c. handling• eliminate spurious modes
1
ˆ ˆ ˆM
e e e e e eeh xj j x yj j y zj j z
j
A A N e A N e A N e=
= + +∑r
1
Me e eh j j
j
A A N=
=∑r r
N1N2
N3
N4
N5 N6
Nx1
Ny1
Nz1
Nx2
Ny2
Nz2
Nx4
Ny4
Nz4
Nx3
Ny3
Nz3
Node Elements versus Edge Elements
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Problem of T,T0, ΦΦΦΦ-Method:
But excellent resolved magnetic field is essential basic for inductance calculation=> 2. order scalar elements
0
mag
lower order approximation for B
lower order approximation for W
less good results for L
H T T= + − ∇Φ⇒
⇒
⇒
r r r
Bad Approximation of B-field at Γcn
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
• test different potential based formulations• A,V • T,T0-ΦΦΦΦ• A with decoupled grad ϕϕϕϕ
• test different element formulations• node-based• vector-based
• test different gauging strategies • allow for combinations of vector- and scalar-elements of