skin effect problems - inuTech€¦ · Driven Transient Skin Effect Problems Peter Böhm, inuTech GmbH. 23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005 Outline • Problem

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


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


??

Hoover-Dam (Nevada-Arizona)

KKW Krümmel (D) ICE3 train (D)

wall socket

From the Power Plant to the Wall Outlet


IGBT-Modul

GTO-Device

in

out

frequency

amplitude

amplitude

frequency

in

out

Load

Electronic High Power Systems


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


10 cm

Problem Definition


• 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)


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


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


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


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


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


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


???

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)


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


Scalar Cut of Current Density

Skineffect-regions

U


Scalar Cut of Current Density


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


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


( ) ( ) ( )

( )

( )

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


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)


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)


Node elements &current 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


• 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


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


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


• 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


Skin Effect State Static State

Current Driven Busbar


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


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


Thank you for attention !Thank you for attention !

Documents

skin effect problems - inuTech€¦ · Driven Transient Skin Effect Problems Peter Böhm, inuTech GmbH. 23. CADFEM Users´ Meeting, Bonn 09.-11. November 2005 Outline • Problem