Post on 17-Apr-2018
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
• 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
Typical geometrical length << shortest wavelength encountered
( )eddy switch
EDJ E
t t
ε εσ τσ
∂∂⇒ ⇒
∂ ∂
rrr r
= = =
Displacement current can be neglected
material dielectric relaxation timecopper 1.5e-19 saluminum 2.5e-19 slow dopted Si (3 S/m) 3.5e-11 sglas (1e-14 to 1e-12 S/m) 4.5 ... 4500 s
20 MHz⇒⇒⇒⇒15 m 200 MHz
2 GHz
Quasistationary Approximation
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
0nE =×
0nE
nBnH ,
,curl −=E
vgrad
t t= − −
∂ ∂A
E
curlB A and c n
c
in
in
curl= +0J T Tgrad0H T T cin
grad0H T nin
( )
**1 1
with weak gauge 0
pot
AA j
t
A
µσ σ
σ
∂ − ∆ = −∂
∇ ⋅ =ρ*
* and A
B A Et
∂= ∇ × = −∂
Potential Formulations for Skineffect Problems
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Voltage drivensolid conductor
Current drivensolid conductor
V1
V2
I1
I2
Ii
Ii
t
V
Thin highly conductingthreads sunk in a bulk
(no stranded conductor)
t
I
t
V
t
I
t
Ii
t
Iges
The Three Basic Regimes
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
n
10
0 in ,
10 in
c
AA grad
t t
Agrad
t t
A
νσ σµ
νσ σ
µ
∂ ∂∇× ∇× + + = ∂ ∂
∂ ∂−∇ + = Ω ∂ ∂
∇× ∇× = Ω
rr
r
r
( ) ( )
( )
( )
E1
0
E2
c n
cn
0 , on
0 , 0 on
10 or 0 on
1 and are continuous on
t
A n t d
A n t
A n A n
A n A n
ν ϕ τ τ
ν
µ
µ
× = = Γ
× = = Γ
× = ∇× × = ∂ Ω + Ω
× ∇× × Γ
∫r r
r r
r rr r
r rr r
Basicequations:
Boundaryconditions:
A,V-A-Method
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
???
2 1 appliedUϕ ϕ− =
By path integral: applied
AU Edr
tϕ ∂= − = ∇ +
∂∫ ∫r
r r
( )( ) ( )( ),
0
,
c c n
B r tJ r t dp t d HdB d
dtσΩ Ω +Ω
= Ω + Ω∫ ∫ ∫r
r r
( )
( ) ( ) ( )1
with:
c c n
c n c n c n c n c n
c c c E E
BE J d Hd
t
B A A A AHd Hd Hd H n d Jd
t t t t t
Ap t E J d J d J d J n
tϕ ϕ ϕ
+
Ω Ω +Ω
Ω +Ω Ω +Ω Ω +Ω ∂ Ω +Ω Ω +Ω
Ω Ω Ω Γ
∂= ⋅ Ω + ⋅ Ω∂
∂ ∂ ∂ ∂ ∂⋅ Ω = ∇× ⋅ Ω = ⋅∇× Ω + × ⋅ Γ = ⋅ Ω ∂ ∂ ∂ ∂ ∂
∂= + ⋅ Ω = −∇ ⋅ Ω = ∇ ⋅ Ω − ⋅ ∂
∫ ∫
∫ ∫ ∫ ∫ ∫
∫ ∫ ∫
rr r r
r r r rrr r r r rr
rr r r r r r
Ñ
( ) ( ) ( ) ( )2
1
nc
E
d
u t J np t u t i t
+
Γ
Γ
= − ⋅ = ⋅
∫
∫r r
Ñ
Ñ
By energy consideration:
A,V-A-Method (cont’d)
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
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
different order
Own Developed Diffpack® Simulator
Non-standard Application=> Diffpack SimulatorDiffpack Simulator
23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005
Skin Effect State Static State
Current Driven Busbar
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]
1.00e-08
1.05e-08
1.10e-08
1.15e-08
1.20e-08
L [H
]
Inductivityexample viertelstab
Omega_c & Omega_nOmega_n
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
time [s]
1.0e-06
1.0e-05
1.0e-04
1.0e-03
1.0e-02
1.0e-01
1.0e+00
1.0e+01
1.0e+02
Vol
tage
[V
]
Terminal voltageexample viertelstab
terminal voltage
Example T-T0-Φ
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]
1.00e-08
1.05e-08
1.10e-08
1.15e-08
1.20e-08
L [H
]
Inductivityexample viertelstab
Omega_c & Omega_nOmega_n
Skin State
Static State
EddyCurrent
State
Three Different States