[PPT] - Rigorous approach to the derivation of 1D models for wave PowerPoint Presentation

SLIDE 1

POems (INRIA)

Patrick Joly (INRIA)

Rigorous approach to the derivation

f 1D models for wave propagation in

electrical networks

EPI POEMS (UMR CNRS-ENSTA-INRIA)

Patrick Joly

with G. Beck (Poems, INRIA) and S. Imperiale (M3DISIM, INRIA) e Analysis and Numerics of Acoustic and Electromagnetic Problems RRICAM Workshop, Linz, October 2016

SLIDE 2

Motivation

2

POems (INRIA) Patrick Joly

Non destructive testing of electric networks (example : SNCF) Coaxial cables We need an efficient simulation tool, in time domain, based on a simplified 1D model that keeps in memory the complex 3D structure of the cable. Electromagnetic waves This is motivated by applications in non destructive testing (with CEA LIST) Defects often appear at junctions

SLIDE 3

Electromagnetic waves in thin cables

3

POems (INRIA) Patrick Joly

Surprisingly (I may be wrong), it does not seem to have been intensively studied from a rigorous mathematical point of view This is a subject that is surely considered as perfectly understood by electrical engineers (via modal analysis)

Hyo J. Eom. Electromagnetic theory for boundary value problems, (2004).

R. Dautray J. L. Lions Mathematical Analysis, Numerical method in Science and

Technology (1993).

Z. Menachem, Wave propagation in a curved waveguide with arbitrary

dielectric transverse profiles, (2007)

G. Beck, S. Imperiale, P

. Joly, Mathematical modeling of multi-conductor cables, (2014).

S. Imperiale, P

. Joly Error estimates for 1D asymptotic models in coaxial cables with heterogeneous cross-section (2012).

S. Imperiale, P

. Joly, Mathematical modeling of electromagnetic wave propagation in heterogeneous lossy coaxial cables with variable cross section, (2013)

G. Beck, Modélisation et analyse de réseaux électriques, PhD thesis (2016).

SLIDE 4

Part%I% Reduced%models% for%a%junction%of%wires

POems (INRIA) Patrick Joly

SLIDE 5

The%geometry%of% the%problem

POems (INRIA) Patrick Joly

SLIDE 6

x`

T = (x` 1, x` 2)

x`

3

x`

2

x`

1

Ω` = n O` + ⇥ x`

T + x` 3 e` 3

⇤ , x`

T ∈ S`

e`

3

The reference domain (1) the cables

6

POems (INRIA) Patrick Joly

S`

0 ≤ ` ≤ L O`

L + 1 semi-infinite cables will play a privileged role in our presentation

` = 0

SLIDE 7

x`

T = (x` 1, x` 2)

x`

3

x`

2

x`

1

Ω` = n O` + ⇥ x`

T + x` 3 e` 3

⇤ , x`

T ∈ S`

e`

3

The reference domain (1) the cables

6

POems (INRIA)

Perfect conductors dielectric materials

Patrick Joly

S`

0 ≤ ` ≤ L O`

is not simply connected Coaxial cables :

S`

SLIDE 8

∂ b Ki

∂ b Ke b K ∂ b K = ∂ b Ki ∪ ∂ b Ke ∪

L

[

`=0

S` x3 x1 x2

The reference domain (2) the junction

7

POems (INRIA) Patrick Joly

connected and bounded

SLIDE 9

b J = b K ∪

L

[

`=0

Ω`

8

POems (INRIA) Patrick Joly

ε(x`

T , x` 3) = ε`(xT )

µ(x`

T , x` 3) = µ`(xT )

ε, µ : b J − → R+ ε`, µ` : S` − → R+

The reference domain (3) the whole domain

SLIDE 10

Jδ = δ b J

A family of geometrical domains

9

POems (INRIA) Patrick Joly

δ δ δ

3D scaling

SLIDE 11

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ − rot Hδ = 0 Eδ × n = 0

+ (well prepared) initial conditions

Jδ ∂Jδ

Ω

` =

n O

` +

⇥ δ x`

T + x` 3 e` 3

⇤ , x`

T ∈ S`

J = K ∪

L

[

`=1

Ω

`

Kδ = δ b K

A family of geometrical domains

9

POems (INRIA) Patrick Joly

δ δ δ

2D scaling (transverse) 3D scaling

εδ(x) = ε(x/δ) µδ(x) = µ(x/δ)

ε(x) = ε`(x`

T /δ)

µ(x) = µ`(x`

T /δ)

O

`

SLIDE 12

Objective 1 : Propose an approximate model, posed with 1D unknowns on the limit graph Objective 2 : Propose a reconstruction of the 3D field from the 1D unknowns Objective 3 : Justify rigorously the approximate model via error estimates

Ω

` =

n O

` +

⇥ δ x`

T + x` 3 e` 3

⇤ , x`

T ∈ S`

J = K ∪

L

[

`=1

Ω

`

Kδ = δ b K

G =

L

Y

`=0

[0, +∞[

A family of geometrical domains

9

POems (INRIA) Patrick Joly

2D scaling (transverse) 3D scaling

εδ(x) = ε(x/δ) µδ(x) = µ(x/δ)

ε(x) = ε`(x`

T /δ)

µ(x) = µ`(x`

T /δ)

SLIDE 13

Objective 1 : Propose an approximate model, posed with 1D unknowns on the limit graph Objective 2 : Propose a reconstruction of the 3D field from the 1D unknowns Objective 3 : Justify rigorously the approximate model via error estimates

A family of geometrical domains

10

POems (INRIA) Patrick Joly

This is the 3D Maxwell counterpart of the work achieved for acoustics in the PhD thesis of A. Semin

P . Joly, A. Semin Construction and analysis of improved Kirchhoff conditions in a junction of thin slots (2008).

A. Semin Propagation d’ondes dans des jonctions de fentes minces

(Université Paris Sud (2008).

SLIDE 14

The%simplified% 1D%model

POems (INRIA) Patrick Joly

SLIDE 15

V

`(x` 3, t), I `(x` 3, t) : R+ × R+ → R

G

e0 e1

The 1D approximate model

12

POems (INRIA) Patrick Joly

C` ∂tV

` + ∂`I ` = 0

L` ∂tI

` + ∂`V ` = 0

x`

3 > 0, t > 0

V

0(0) − V `(0) = δ L

X

`=1

Z`,k ∂tI

k(0) L

X

`=0

I

`(0) = δ Y ∂tV 0(0)

V

` : 1D electric potentials

I

` : 1D electric currents

e2

∂` : longitudinal derivative

SLIDE 16

V

`(x` 3, t), I `(x` 3, t) : R+ × R+ → R

G

e0 e1

The 1D approximate model

12

POems (INRIA) Patrick Joly

C` ∂tV

` + ∂`I ` = 0

L` ∂tI

` + ∂`V ` = 0

x`

3 > 0, t > 0

V

0(0) − V `(0) = δ L

X

`=1

Z`,k ∂tI

k(0) L

X

`=0

I

`(0) = δ Y ∂tV 0(0)

V

` : 1D electric potentials

I

` : 1D electric currents

e2

Standard Kichhoff conditions

∂` : longitudinal derivative

SLIDE 17

V

`(x` 3, t), I `(x` 3, t) : R+ × R+ → R

G

e0 e1

The 1D approximate model

13

POems (INRIA) Patrick Joly

C` ∂tV

` + ∂`I ` = 0

L` ∂tI

` + ∂`V ` = 0

x`

3 > 0, t > 0

V

0(0) − V `(0) = δ L

X

`=1

Z`,k ∂tI

k(0) L

X

`=0

I

`(0) = δ Y ∂tV 0(0)

e2

Improved Kichhoff conditions

Junction defects

V

` : 1D electric potentials

I

` : 1D electric currents

∂` : longitudinal derivative

SLIDE 18

The 1D approximate model

14

POems (INRIA) Patrick Joly

C` ∂tV

` + ∂`I ` = 0

`

⇣ C` |V

`|2 + L` |I `|2⌘

+ δ 2 ⇣ Y |V

0(0)|2 +

Z I

0, I

⌘

Well-posedness and stability : there is conservation of the energy

I

0 :=

I

`(0)

1≤`≤L

SLIDE 19

The 1D approximate model

14

POems (INRIA) Patrick Joly

C` ∂tV

` + ∂`I ` = 0

`

⇣ C` |V

`|2 + L` |I `|2⌘

+ δ 2 ⇣ Y |V

0(0)|2 +

Z I

0, I

⌘

Well-posedness and stability : there is conservation of the energy

I

0 :=

I

`(0)

1≤`≤L

SLIDE 20

The%effective%coefficients% %%%%%%%%%(i)%%%%%%%%%%and

C`

L`

POems (INRIA) Patrick Joly

SLIDE 21

E` :=

u` ∈ L2

T (S`) / rotT u` = 0, divT

ε`u`
= 0, u` × n` = 0 on ∂S`

The 2D transverse electric potentials

16

POems (INRIA) Patrick Joly

E` = span ⇥ rT ϕ`

e

⇤ S` C` = Z

S`

ε` |rT ϕ`

e|2 dσ

S` ∂Sext

`

∂Sint

`

ϕ`

e = 0

ϕ`

e = 1

divT ⇣ ε` rT ϕ`

e

⌘ = 0

SLIDE 22

Topology of the reference domain

17

POems (INRIA) Patrick Joly

b J = b K ∪

L

[

`=1

Ω` b J is not simply connected

SLIDE 23

Ω` Γ` Ω0 ` 6= 0

18

POems (INRIA) Patrick Joly

S`

plane and orthogonal to

S0

plane and orthogonal to

Topology of the reference domain

For each

SLIDE 24

∀ 1 ≤ m ≤ L, b J \

L

[

`=1

Γ`

19

POems (INRIA) Patrick Joly

is simply connected

Γ`

Topology of the reference domain

Ω0

SLIDE 25

S0

20

POems (INRIA) Patrick Joly

S` Γ` Ω` Ω0 γ0,` γ`

Topology of the reference domain

SLIDE 26

S` \ γ` ∂S` γ` ∂nψ`

m = 0

⇥ µ ∂⌫ψ`

m

⇤

` = 0

⇥ ψ`

m

⇤

` = 1

The 2D transverse magnetic potentials

21

POems (INRIA) Patrick Joly

S` γ` divT ⇣ µ` rT ψ`

m

⌘ = 0

SLIDE 27

S` \ γ` ∂S` γ`

H` :=

u` ∈ L2

T (S`) / rotT u` = 0, divT

µ`u`
= 0, u` · n` = 0 on ∂S`

∂nψ`

m = 0

⇥ µ ∂⌫ψ`

m

⇤

` = 0

⇥ ψ`

m

⇤

` = 1

The 2D transverse magnetic potentials

21

POems (INRIA) Patrick Joly

S` γ` divT ⇣ µ` rT ψ`

m

⌘ = 0

SLIDE 28

S` \ γ` ∂S` γ`

H` :=

u` ∈ L2

T (S`) / rotT u` = 0, divT

µ`u`
= 0, u` · n` = 0 on ∂S`

∂nψ`

m = 0

⇥ µ ∂⌫ψ`

m

⇤

` = 0

⇥ ψ`

m

⇤

` = 1

The 2D transverse magnetic potentials

21

POems (INRIA) Patrick Joly

S` H` = span ⇥ e rT ψ`

m

⇤ γ` divT ⇣ µ` rT ψ`

m

⌘ = 0

SLIDE 29

S` \ γ` ∂S` γ`

H` :=

u` ∈ L2

T (S`) / rotT u` = 0, divT

µ`u`
= 0, u` · n` = 0 on ∂S`

∂nψ`

m = 0

⇥ µ ∂⌫ψ`

m

⇤

` = 0

⇥ ψ`

m

⇤

` = 1

The 2D transverse magnetic potentials

21

POems (INRIA) Patrick Joly

S` H` = span ⇥ e rT ψ`

m

⇤ e rT ψ`

m 2 L2 T (S`)

/ e rT ψ`

m = rT ψ` m in S` \ γ`

γ` divT ⇣ µ` rT ψ`

m

⌘ = 0

SLIDE 30

S` \ γ` ∂S` γ`

H` :=

u` ∈ L2

T (S`) / rotT u` = 0, divT

µ`u`
= 0, u` · n` = 0 on ∂S`

∂nψ`

m = 0

⇥ µ ∂⌫ψ`

m

⇤

` = 0

⇥ ψ`

m

⇤

` = 1

The 2D transverse magnetic potentials

21

POems (INRIA) Patrick Joly

S` H` = span ⇥ e rT ψ`

m

⇤ e rT ψ`

m 2 L2 T (S`)

/ e rT ψ`

m = rT ψ` m in S` \ γ`

L` = Z

S`

µ` |e rT ψ`

m|2 dσ

γ` divT ⇣ µ` rT ψ`

m

⌘ = 0

SLIDE 31

The%effective%coefficients% %%%%%%%%%(i)%%%%%%%%%%and

Y Z

POems (INRIA) Patrick Joly

SLIDE 32

The%effective%coefficients% %%%%%%%%%(i)%%%%%%%%%%and

Y Z

For (quite relative) simplicity, we present the case where the product is constant in each section

ε`µ` S`

the product

POems (INRIA) Patrick Joly

SLIDE 33

div

ε rΦe
= 0

Φe = 0 ∂ b Jext ∂ b Jint b J x`

3 → +∞

23

POems (INRIA) Patrick Joly

Φe = 1

exponentially fast

The 3D electric potential

Ω` Ω0

SLIDE 34

div

ε rΦe
= 0

Φe = 0 ∂ b Jext ∂ b Jint b J Φe(., x`

3) ∼ ϕ` e

x`

3 → +∞

23

POems (INRIA) Patrick Joly

Φe = 1

exponentially fast

The 3D electric potential

Ω` Ω0

SLIDE 35

div

ε rΦe
= 0

Φe = 0 ∂ b Jext ∂ b Jint b J Φe(., x`

3) ∼ ϕ` e

x`

3 → +∞

23

POems (INRIA) Patrick Joly

Φe = 1

exponentially fast

The 3D electric potential

Ω` Ω0 = ⇒ ∂`Φe ∈ L2(Ω`)

SLIDE 36

div

ε rΦe
= 0

Φe = 0 ∂ b Jext ∂ b Jint b J

24

POems (INRIA) Patrick Joly

Φe = 1

The 3D electric potential

E :=

u ∈ L2( b

J ) / rot u = 0, div

εu
= 0, u × n = 0 on ∂ b

J

E = span ⇥ rΦe ⇤ dim E = 1

SLIDE 37

div

ε rΦe
= 0

Φe = 0 ∂ b Jext ∂ b Jint b J

25

POems (INRIA) Patrick Joly

Φe = 1

The 3D electric potential

Y = Z

b K

ε |rΦe|2 dx + 2

L

X

`=0

Z

Ω`

ε |∂`Φe|2 dx Ω` b K

SLIDE 38

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

b J \ Γ` ∂nΨ`

m = 0

⇥ Ψ`

m

⇤

` = 1

∂ b J ⇥ µ ∂⌫Ψ`

m

⇤

` = 0

26

POems (INRIA) Patrick Joly

Γ` Ω`

The 3D magnetic potentials

SLIDE 39

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

b J \ Γ` ∂nΨ`

m = 0

⇥ Ψ`

m

⇤

` = 1

∂ b J ⇥ µ ∂⌫Ψ`

m

⇤

` = 0

26

POems (INRIA) Patrick Joly

Γ` Ω`

The 3D magnetic potentials

H :=

u ∈ L2( b

J ) / rot u = 0, div

µu
= 0, u · n = 0 on ∂ b

J dim H = L

SLIDE 40

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

b J \ Γ` ∂nΨ`

m = 0

⇥ Ψ`

m

⇤

` = 1

∂ b J ⇥ µ ∂⌫Ψ`

m

⇤

` = 0

26

POems (INRIA) Patrick Joly

Γ` Ω`

The 3D magnetic potentials

SLIDE 41

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

b J \ Γ` ∂nΨ`

m = 0

⇥ Ψ`

m

⇤

` = 1

∂ b J ⇥ µ ∂⌫Ψ`

m

⇤

` = 0

27

POems (INRIA) Patrick Joly

Γ` Ω`

The 3D magnetic potentials

SLIDE 42

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

b J \ Γ` ∂nΨ`

m = 0

⇥ Ψ`

m

⇤

` = 1

at ∞ ∂ b J ⇥ µ ∂⌫Ψ`

m

⇤

` = 0

27

POems (INRIA) Patrick Joly

Γ` Ω`

The 3D magnetic potentials

SLIDE 43

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

SLIDE 44

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

e rΨ`

m(·, x0 3) ⇠ e

rT ψ0

m

SLIDE 45

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

b J \ Γ` ∂nΨ`

m = 0

⇥ Ψ`

m

⇤

` = 1

Ψ`

m(., xk 3) ∼ 0

k 6= 0, `

at ∞ ∂ b J ⇥ µ ∂⌫Ψ`

m

⇤

` = 0

27

POems (INRIA) Patrick Joly

Γ` Ω` ∼ ≡

exponentially fast

The 3D magnetic potentials

e rΨ`

m(·, x` 3) ⇠ e

rT ψ`

m

e rΨ`

m(·, x0 3) ⇠ e

rT ψ0

m

SLIDE 46

Ω0 γ` γ0,` div

µ rΨ`

m

= 0

b J \ Γ` ∂nΨ`

m = 0

⇥ Ψ`

m

⇤

` = 1

∂ b J ⇥ µ ∂⌫Ψ`

m

⇤

` = 0

28

POems (INRIA) Patrick Joly

Γ` Ω`

The 3D magnetic potentials

∂jΨ`

m ∈ L2(Ωj)

rΨ`

m

SLIDE 47

∂ b Jext ∂ b Jint Ω0 γ` γ0,` div

µ rΨ`

m

= 0

b J \ Γ` ∂nΨ`

m = 0

⇥ Ψ`

m

⇤

` = 1

b K

29

POems (INRIA) Patrick Joly

Γ` Ω`

The 3D magnetic potential

Z`k = Z

b K

µ rΨ`

m · rΨk m dx + 2 L

X

j=0

Z

Ωj

µ ∂jΨ`

m ∂jΨk m dx

SLIDE 48

Reconstruction%of%the%3d%fields% (1)%in%the%cables

POems (INRIA) Patrick Joly

SLIDE 49

S`

31

POems (INRIA) Patrick Joly

S`

The 3D reconstruction (1) inside cables

S` \ γ` ∂S` γ` S` divT ⇣ ε−1

`

rT ψ`

e

⌘ = 0

SLIDE 51

O(1)

O(δ2)

+ δ h ψ`

e

⇣x`

T

δ ⌘ − ψ`

m

⇣x`

T

δ ⌘i C` ∂tV

`(x` 3, t) e` 3

33

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x`

3

x`

3

+ δ h ϕ`

m

⇣x`

T

δ ⌘ − ϕ`

e

⇣x`

T

δ ⌘i L` ∂tI

`(x` 3, t) e` 3

E

app(x` T , x` 3, t) := rT ϕ` e

⇣x`

T

δ ⌘ V

`(x` 3, t)

H

app(x` T , x` 3, t) := e

rT ψ`

m

⇣x`

T

δ ⌘ I(x`

3, t)

SLIDE 52

Reconstruction%of%the%3d%fields% (1)%in%the%junction

POems (INRIA) Patrick Joly

SLIDE 53

+ δ

L

X

`=1

E` ⇣x δ ⌘ ∂tI

`(0, t)

+ δ H0 ⇣x δ ⌘ ∂tV δ

0(0, t)

e

35

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

Eδ

app(x, t) := rΦe

∼ δ

SLIDE 55

36

POems (INRIA) Patrick Joly

The 3D electric potential

rot H0 = ε rΦe div

µ H0
= 0

H0 · n = 0 b J ∂ b J rot E` = µ e rΨ`

m

div

ε E`
= 0

E` × n = 0 b J ∂ b J

h ek

3 ⇥

E` ⇥ ek

3

i (· , xk

3) ⇠

Z`k + xk

3 Lk

rT ϕk

e

h e`

3 ⇥

E` ⇥ e`

3

i (· , x`

3) ⇠ x` 3 L` rT ϕ` e

k 6= `

Ωk Ω`

h e0

3 ⇥

H0 ⇥ e0

3

i (· , x0

3) ⇠

Y + x0

3 C0

rT ψ0

m

h e`

3 ⇥

H0 ⇥ e`

3

i (· , x`

3) ⇠ x` 3 C` rT ψ` m

Ω` Ω0

` 6= 0

e e

SLIDE 56

Basic%ingredients%%

f%the%analysis

POems (INRIA) Patrick Joly

SLIDE 57

?

38

POems (INRIA) Patrick Joly

Matched asymptotics

x`

3

1D-like behaviour 3D behaviour

One cannot expect a uniform (in space) asymptotic expansion

SLIDE 58

?

38

POems (INRIA) Patrick Joly

Matched asymptotics

x`

3

1D-like behaviour 3D behaviour

One cannot expect a uniform (in space) asymptotic expansion

∼ δ

1 2

∼ δ

1 2

SLIDE 59

39

POems (INRIA) Patrick Joly

Asymptotic expansions : the ansatz

x`

3

E(x`

T , x` 3, t) = b

E

`

⇣x`

T

δ , x`

3, t

⌘ + δ b E

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b E

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . . H(x`

T , x` 3, t) = b

H

`

⇣x`

T

δ , x`

3, t

⌘ + δ b H

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b H

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . .

Eδ(x, t) = b E

0⇣x

δ , t ⌘ + δ b E

1⇣x

δ , t ⌘ + δ2 b E

2⇣x

δ , t ⌘ + . . . Hδ(x, t) = b H

0⇣x

δ , t ⌘ + δ b H

1⇣x

δ , t ⌘ + δ2 b H

2⇣x

δ , t ⌘ + . . .

3D scaling 2D scaling

SLIDE 60

39

POems (INRIA) Patrick Joly

Asymptotic expansions : the ansatz

x`

3

E(x`

T , x` 3, t) = b

E

`

⇣x`

T

δ , x`

3, t

⌘ + δ b E

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b E

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . . H(x`

T , x` 3, t) = b

H

`

⇣x`

T

δ , x`

3, t

⌘ + δ b H

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b H

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . .

Eδ(x, t) = b E

0⇣x

δ , t ⌘ + δ b E

1⇣x

δ , t ⌘ + δ2 b E

2⇣x

δ , t ⌘ + . . . Hδ(x, t) = b H

0⇣x

δ , t ⌘ + δ b H

1⇣x

δ , t ⌘ + δ2 b H

2⇣x

δ , t ⌘ + . . .

3D scaling 2D scaling The two asymptotic expansions must match in the overlaping zone

SLIDE 61

40

POems (INRIA) Patrick Joly

x`

3

E(x`

T , x` 3, t) = b

E

`

⇣x`

T

δ , x`

3, t

⌘ + δ b E

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b E

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . . H(x`

T , x` 3, t) = b

H

`

⇣x`

T

δ , x`

3, t

⌘ + δ b H

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b H

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . .

Eδ(x, t) = b E

0⇣x

δ , t ⌘ + δ b E

1⇣x

δ , t ⌘ + δ2 b E

2⇣x

δ , t ⌘ + . . . Hδ(x, t) = b H

0⇣x

δ , t ⌘ + δ b H

1⇣x

δ , t ⌘ + δ2 b H

2⇣x

δ , t ⌘ + . . .

Asymptotic expansions : the ansatz

SLIDE 62

40

POems (INRIA) Patrick Joly

x`

3

E(x`

T , x` 3, t) = b

E

`

⇣x`

T

δ , x`

3, t

⌘ + δ b E

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b E

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . . H(x`

T , x` 3, t) = b

H

`

⇣x`

T

δ , x`

3, t

⌘ + δ b H

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b H

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . .

Eδ(x, t) = b E

0⇣x

δ , t ⌘ + δ b E

1⇣x

δ , t ⌘ + δ2 b E

2⇣x

δ , t ⌘ + . . . Hδ(x, t) = b H

0⇣x

δ , t ⌘ + δ b H

1⇣x

δ , t ⌘ + δ2 b H

2⇣x

δ , t ⌘ + . . .

b H0, b H1, . . .

b E0, b E1, . . . b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

Asymptotic expansions : the ansatz

SLIDE 63

40

POems (INRIA) Patrick Joly

x`

3

E(x`

T , x` 3, t) = b

E

`

⇣x`

T

δ , x`

3, t

⌘ + δ b E

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b E

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . . H(x`

T , x` 3, t) = b

H

`

⇣x`

T

δ , x`

3, t

⌘ + δ b H

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b H

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . .

x

t

separates 3D scaling and

Eδ(x, t) = b E

0⇣x

δ , t ⌘ + δ b E

1⇣x

δ , t ⌘ + δ2 b E

2⇣x

δ , t ⌘ + . . . Hδ(x, t) = b H

0⇣x

δ , t ⌘ + δ b H

1⇣x

δ , t ⌘ + δ2 b H

2⇣x

δ , t ⌘ + . . .

b H0, b H1, . . .

b E0, b E1, . . . b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

Asymptotic expansions : the ansatz

SLIDE 64

40

POems (INRIA) Patrick Joly

x`

3

E(x`

T , x` 3, t) = b

E

`

⇣x`

T

δ , x`

3, t

⌘ + δ b E

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b E

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . . H(x`

T , x` 3, t) = b

H

`

⇣x`

T

δ , x`

3, t

⌘ + δ b H

1 `

⇣x`

T

δ , x`

3, t

⌘ + δ2 b H

2 `

⇣x`

T

δ , x`

3, t

⌘ + . . .

(x`

3, t)

(xT ) 2D scaling separates

x

t

separates 3D scaling and

Eδ(x, t) = b E

0⇣x

δ , t ⌘ + δ b E

1⇣x

δ , t ⌘ + δ2 b E

2⇣x

δ , t ⌘ + . . . Hδ(x, t) = b H

0⇣x

δ , t ⌘ + δ b H

1⇣x

δ , t ⌘ + δ2 b H

2⇣x

δ , t ⌘ + . . .

b H0, b H1, . . .

b E0, b E1, . . . b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

Asymptotic expansions : the ansatz

SLIDE 65

Identifying the terms with the same power in (starting with ) ,

ne obtains a cascade of equations

δ−1

δ

41

POems (INRIA) Patrick Joly

x`

3

(x`

3, t)

(xT ) 2D scaling separates Step 0

b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

One first gets that the fields and do not have any longitudinal component

b E0

`

b H0

`

b H0

` = b

H0

`,T

b E0

` = b

E0

`,T

The terms of the expansion inside the cables

− → rotT b E0

3,T = 0

− → rotT b H0

3,T = 0

Concerning the longitudinal fields

SLIDE 66

rot b H0 = 0, div

ε b

H0 = 0, b H0 · n = 0

rotT b H0

` = 0, divT

µ` b

H0

`

= 0, b

H0

` · n` = 0

rotT b E0

` = 0, divT

ε` b

E0

`

= 0, b

E0

` × n` = 0

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

⇒

rotb E0 = 0, div

εb

E0 = 0, b E0 × n = 0

b E0 = b U 0(t) rΦe(x)

b H0 =

L

X

`=1

b J0

`(t) rΨ` m(x)

42

POems (INRIA) Patrick Joly

x`

3

x

t

separates 3D scaling and (x`

3, t)

(xT ) 2D scaling separates b H0

` = b

I0

`(x3, t) e

rT ψ`

m(xT )

⇒

b H0, b H1, . . .

b E0, b E1, . . .

⇒ ⇒

Step 0 Step 0

The terms of the expansion inside the cables

Concerning the transverse fields Identifying the terms with the same power in (starting with ) ,

ne obtains a cascade of equations

δ−1

δ

b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

SLIDE 67

(x`

3, t)

(xT )

⇒

C` ∂t b V 0

` + ∂`b

I0

` = 0

43

POems (INRIA) Patrick Joly

x`

3

2D scaling separates

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

Step 1

Z

S`

e`

3 ⇥ e

rT ψ`

e

· rT ϕ`

e = 1

e

ε` ∂t b E0

` + e` 3 × ∂` b

H0

`,T − −

→ rotT b H1

`,3 = 0

C` = Z

S`

ε` |rT ϕ`

e|2 dσ

The terms of the expansion inside the cables

b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

Identifying the terms with the same power in (starting with ) ,

ne obtains a cascade of equations

δ−1

δ

? rT ϕ`

e in L2(S`)

One tests against

rT ϕ`

e in S`

SLIDE 68

(x`

3, t)

(xT )

⇒

C` ∂t b V 0

` + ∂`b

I0

` = 0

44

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x`

3

2D scaling separates

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

Step 1

µ` ∂t b H0

` − e` 3 × ∂` b

E0

`,T + −

→ rotT b E1

3,T = 0

⇒

L` ∂tb I0

` + ∂` b

V 0

` = 0

e

ε` ∂t b E0

` + e` 3 × ∂` b

H0

`,T − −

→ rotT b H1

`,L = 0

Identifying the terms with the same power in (starting with ) ,

ne obtains a cascade of equations

δ−1

δ

b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

One tests against in S`

e rT ψ`

m

SLIDE 69

45

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x`

3

x

t

separates 3D scaling and

b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

e

L` ∂tb I0

` + ∂` b

V 0

` = 0

C` ∂t b V 0

` + ∂`b

I0

` = 0

Equations at the

rigin are missing

SLIDE 70

rot b H0 = 0, div

ε b

H0 = 0, b H0 · n = 0

rotb E0 = 0, div

εb

E0 = 0, b E0 × n = 0

b E0 = b U 0(t) rΦe(x)

b H0 =

L

X

`=1

b J0

`(t) rΨ` m(x)

45

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x`

3

x

t

separates 3D scaling and

b H0, b H1, . . .

b E0, b E1, . . .

⇒ ⇒

Step 0

b E0

`, b

E1

`, . . .

b H0

`, b

H1

`, . . .

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

e

L` ∂tb I0

` + ∂` b

V 0

` = 0

C` ∂t b V 0

` + ∂`b

I0

` = 0

Equations at the

rigin are missing

SLIDE 71

46

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x

t

separates 3D scaling and

rotb E0 = 0, div

εb

E0 = 0, b E0 × n = 0 rot b H0 = 0, div

ε b

H0 = 0, b H0 × n = 0

⇒ ⇒

Matching conditions : step 0

b V 0

`(0) − b

V 0

0(0) = 0, 1 ≤ ` ≤ L

⇒

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

e

L` ∂tb I0

` + ∂` b

V 0

` = 0

C` ∂t b V 0

` + ∂`b

I0

` = 0

b E0 = b U 0(t) rΦe(x)

b H0 =

L

X

`=1

b J0

`(t) rΨ` m(x)

rΦe(x) ⇠ rT ϕ`

e(xT )

∞

Cable n`

b V 0

`(0, t) = b

U 0(t), 0 ≤ ` ≤ L

Equations at the

rigin are missing

SLIDE 72

47

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x

t

separates 3D scaling and

rotb E0 = 0, div

εb

E0 = 0, b E0 × n = 0 rot b H0 = 0, div

ε b

H0 = 0, b H0 × n = 0

⇒ ⇒

Matching conditions : step 0

b V 0

`(0) − b

V 0

0(0) = 0, 1 ≤ ` ≤ L

⇒

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

e

L` ∂tb I0

` + ∂` b

V 0

` = 0

C` ∂t b V 0

` + ∂`b

I0

` = 0

b E0 = b U 0(t) rΦe(x)

b H0 =

L

X

`=1

b J0

`(t) rΨ` m(x)

b V 0

`(0, t) = b

U 0(t), 0 ≤ ` ≤ L

Equations at the

rigin are missing

SLIDE 73

47

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x

t

separates 3D scaling and

rotb E0 = 0, div

εb

E0 = 0, b E0 × n = 0 rot b H0 = 0, div

ε b

H0 = 0, b H0 × n = 0

⇒ ⇒

Matching conditions : step 0

b V 0

`(0) − b

V 0

0(0) = 0, 1 ≤ ` ≤ L

⇒

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

e

L` ∂tb I0

` + ∂` b

V 0

` = 0

C` ∂t b V 0

` + ∂`b

I0

` = 0

b E0 = b U 0(t) rΦe(x)

b H0 =

L

X

`=1

b J0

`(t) rΨ` m(x)

b V 0

`(0, t) = b

U 0(t), 0 ≤ ` ≤ L

Equations at the

rigin are missing

SLIDE 74

47

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x

t

separates 3D scaling and

rotb E0 = 0, div

εb

E0 = 0, b E0 × n = 0 rot b H0 = 0, div

ε b

H0 = 0, b H0 × n = 0

⇒ ⇒

Matching conditions : step 0

b V 0

`(0) − b

V 0

0(0) = 0, 1 ≤ ` ≤ L

⇒

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

e

L` ∂tb I0

` + ∂` b

V 0

` = 0

C` ∂t b V 0

` + ∂`b

I0

` = 0

b E0 = b U 0(t) rΦe(x)

b H0 =

L

X

`=1

b J0

`(t) rΨ` m(x)

b V 0

`(0, t) = b

U 0(t), 0 ≤ ` ≤ L b I0

`(0, t) = −b

J`(t), 1 ≤ ` ≤ L

Equations at the

rigin are missing

SLIDE 75

47

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x

t

separates 3D scaling and

rotb E0 = 0, div

εb

E0 = 0, b E0 × n = 0 rot b H0 = 0, div

ε b

H0 = 0, b H0 × n = 0

⇒ ⇒

Matching conditions : step 0

b V 0

`(0) − b

V 0

0(0) = 0, 1 ≤ ` ≤ L

⇒

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

e

L` ∂tb I0

` + ∂` b

V 0

` = 0

C` ∂t b V 0

` + ∂`b

I0

` = 0

b E0 = b U 0(t) rΦe(x)

b H0 =

L

X

`=1

b J0

`(t) rΨ` m(x)

b V 0

`(0, t) = b

U 0(t), 0 ≤ ` ≤ L b I0

`(0, t) = −b

J`(t), 1 ≤ ` ≤ L

b I0

0(0, t) = L

X

`=1

b J0

`(t),

Equations at the

rigin are missing

SLIDE 76

47

POems (INRIA) Patrick Joly

The 3D reconstruction (1) inside cables

x

t

separates 3D scaling and

rotb E0 = 0, div

εb

E0 = 0, b E0 × n = 0 rot b H0 = 0, div

ε b

H0 = 0, b H0 × n = 0

⇒ ⇒

Matching conditions : step 0

b V 0

`(0) − b

V 0

0(0) = 0, 1 ≤ ` ≤ L

⇒

b E0

` = b

V 0

`(x3, t) rT ϕ` e(xT )

b H0

` = b

I0

`(x3, t) rT ψ` m(xT )

e

L` ∂tb I0

` + ∂` b

V 0

` = 0

C` ∂t b V 0

` + ∂`b

I0

` = 0

b E0 = b U 0(t) rΦe(x)

b H0 =

L

X

`=1

b J0

`(t) rΨ` m(x)

b V 0

`(0, t) = b

U 0(t), 0 ≤ ` ≤ L b I0

`(0, t) = −b

J`(t), 1 ≤ ` ≤ L

b I0

0(0, t) = L

X

`=1

b J0

`(t),

L

X

`=0

b I0

`(0, t) = 0

⇒

}

Equations at the

rigin are missing

SLIDE 77

Part%II% Losses%via%% skin%effect

We come gack to the case of a single cable

x3

x1

x2

POems (INRIA) Patrick Joly

SLIDE 78

penetration depth

Choice of scaling for the asymptotic analysis

E, H 6= 0

σδ = σ · δ−4 δ

>

Taking account losses via skin effect

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ − rot Hδ = 0 µc ∂tHδ + rot Eδ = 0 εc ∂tEδ + σδ Eδ − rot Hδ = 0 δ ` σ >

>1

` δ

> >

⇒

no longer perfectly conducting ( )

σ < +∞

At given frequency ω

` ∼

2/!µc

1

2

`δ ∼

2/!µc

1

2 · 2

POems (INRIA) Patrick Joly

SLIDE 79

`

penetration depth

σ >

>1

` δ

> >

⇒

At given frequency ω

` ∼

2/!µc

1

2

Choice of scaling for the asymptotic analysis

E, H 6= 0

σδ = σ · δ−4 `δ ∼

2/!

1

2 · 2

δ

t

F

half derivative operator in the Caputo sense

The resistance is given by :

R = Z

∂S

rµc σ |e rψm ⇥ n|2

div (µ rψm) = 0

∂nψm = 0

⇥ ψm ⇤ = 1

R

A second order model is provided by the modified telegrapher’s model

( C ∂tV δ + ∂3Iδ = 0, L ∂tIδ + δ R ∂1/2

t

Iδ + ∂3V δ = 0.

POems (INRIA) Patrick Joly

SLIDE 81

The transmission problem

σδ = σ · δ−4

dielectric medium sheath

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ − rot Hδ = 0 µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0 Eδ × n = Eδ × n Hδ × n = Hδ × n

Γ

ν τ

Taking account losses via skin effect

H. Haddar, P

. Joly , H. M. Nguyen, Generalized impedance boundary conditions for scattering by strongly absorbing obsracles : the scalar case (2005).

H. Haddar, P

. Joly , H. M. Nguyen, Generalized impedance boundary conditions for scattering from strongly absorbing obsracles : the case of Maxwell’s equations (2008).

K. Schmidt, A. Thöns-Zueva, P

. Joly, Asymptotic analysis for acoustics in viscous gases close to rigid walls (2014).

POems (INRIA) Patrick Joly

SLIDE 82

The transmission problem

σδ = σ · δ−4

dielectric medium sheath

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ − rot Hδ = 0 µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0 Eδ × n = Eδ × n Hδ × n = Hδ × n

Γ

ν τ

Taking account losses via skin effect

POems (INRIA) Patrick Joly

x3

SLIDE 83

The transmission problem

σδ = σ · δ−4

dielectric medium sheath

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ − rot Hδ = 0 µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0 Eδ × n = Eδ × n Hδ × n = Hδ × n

Γ

ν τ

Eδ(xT , x3) = b E0 ⇣xT δ , x3 ⌘ + δ b E1 ⇣xT δ , x3 ⌘ + δ2 b E2 ⇣xT δ , x3 ⌘ + ...

b En : S ⇥ R 7! R3

Taking account losses via skin effect

POems (INRIA) Patrick Joly

x3

SLIDE 84

The transmission problem

σδ = σ · δ−4

dielectric medium sheath

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ − rot Hδ = 0 µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0 Eδ × n = Eδ × n Hδ × n = Hδ × n

Γ

ν τ

Eδ(xT , x3) = b E0 ⇣xT δ , x3 ⌘ + δ b E1 ⇣xT δ , x3 ⌘ + δ2 b E2 ⇣xT δ , x3 ⌘ + ...

b En : S ⇥ R 7! R3

Taking account losses via skin effect

En : [0, L] ⇥ [0, +1[ ⇥ R 7! R3

Eδ(xT , x3) = E0 ⇣τ δ , ν δ2 , x3 ⌘ + δ E1 ⇣τ δ , ν δ2 , x3 ⌘ + δ2 E2 ⇣τ δ , ν δ2 , x3 ⌘ + ...

POems (INRIA) Patrick Joly

x3

SLIDE 85

The transmission problem

σδ = σ · δ−4

dielectric medium sheath

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ − rot Hδ = 0 µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0 Eδ × n = Eδ × n Hδ × n = Hδ × n

Γ

ν τ

Eδ(xT , x3) = b E0 ⇣xT δ , x3 ⌘ + δ b E1 ⇣xT δ , x3 ⌘ + δ2 b E2 ⇣xT δ , x3 ⌘ + ...

b En : S ⇥ R 7! R3 lim

ν→+∞ En(τ, ν, x3) = 0

Taking account losses via skin effect

En : [0, L] ⇥ [0, +1[ ⇥ R 7! R3

Eδ(xT , x3) = E0 ⇣τ δ , ν δ2 , x3 ⌘ + δ E1 ⇣τ δ , ν δ2 , x3 ⌘ + δ2 E2 ⇣τ δ , ν δ2 , x3 ⌘ + ...

POems (INRIA) Patrick Joly

x3

SLIDE 86

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0

Looking inside the conductor (1)

Eδ(xT , x3) = E0 ⇣τ δ , ν δ2 , x3 ⌘ + δ E1 ⇣τ δ , ν δ2 , x3 ⌘ + δ2 E2 ⇣τ δ , ν δ2 , x3 ⌘ + ...

POems (INRIA) Patrick Joly

SLIDE 87

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0

Looking inside the conductor (1)

E0 (ˆ τ, ˆ ν, x3) = E1 (ˆ τ, ˆ ν, x3) = 0

The first two terms in the second equation give

(δ−4, δ−3)

Eδ(xT , x3) = E0 ⇣τ δ , ν δ2 , x3 ⌘ + δ E1 ⇣τ δ , ν δ2 , x3 ⌘ + δ2 E2 ⇣τ δ , ν δ2 , x3 ⌘ + ...

POems (INRIA) Patrick Joly

SLIDE 88

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0

Looking inside the conductor (1)

E0 (ˆ τ, ˆ ν, x3) = E1 (ˆ τ, ˆ ν, x3) = 0

The first two terms in the second equation give

(δ−4, δ−3)

In other words, the electric field vanishes at second order in the conductor

Eδ(xT , x3) = E0 ⇣τ δ , ν δ2 , x3 ⌘ + δ E1 ⇣τ δ , ν δ2 , x3 ⌘ + δ2 E2 ⇣τ δ , ν δ2 , x3 ⌘ + ...

POems (INRIA) Patrick Joly

SLIDE 89

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0

Looking inside the conductor (1)

E0 (ˆ τ, ˆ ν, x3) = E1 (ˆ τ, ˆ ν, x3) = 0

The first two terms in the second equation give

(δ−4, δ−3)

In other words, the electric field vanishes at second order in the conductor

Eδ × n = Eδ × n

From the transmission condition , one gets

Eδ(xT , x3) = E0 ⇣τ δ , ν δ2 , x3 ⌘ + δ E1 ⇣τ δ , ν δ2 , x3 ⌘ + δ2 E2 ⇣τ δ , ν δ2 , x3 ⌘ + ...

POems (INRIA) Patrick Joly

SLIDE 90

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ + σδ Eδ − rot Hδ = 0

Looking inside the conductor (1)

E0 (ˆ τ, ˆ ν, x3) = E1 (ˆ τ, ˆ ν, x3) = 0

The first two terms in the second equation give

(δ−4, δ−3)

In other words, the electric field vanishes at second order in the conductor

Eδ × n = Eδ × n

From the transmission condition , one gets

E0 × n = E1 × n = 0

Eδ(xT , x3) = E0 ⇣τ δ , ν δ2 , x3 ⌘ + δ E1 ⇣τ δ , ν δ2 , x3 ⌘ + δ2 E2 ⇣τ δ , ν δ2 , x3 ⌘ + ...

POems (INRIA) Patrick Joly

SLIDE 91

µδ ∂tHδ + rot Eδ = 0 εδ ∂tEδ − rot Hδ = 0

Eδ(xT , x3) = b E0 ⇣xT δ , x3 ⌘

+ δ b E1 ⇣xT δ , x3 ⌘ + δ2 b E2 ⇣xT δ , x3 ⌘ + ...

E0 × n = E1 × n = 0

µ ∂t b H1

T − e3 × ∂x3 b

E1

T + −

→ rotT b E2

3 = 0

µ ∂t b H0

T − e3 × ∂x3 b

E0

T + −

→ rotT b E1

3 = 0

as well as the equations (that provide the second telegrapher’s equation) With the same process as for , we still get

σ = +∞

b H0

T = I0 e

rT ψm b E0

T = V 0 rT ϕe

b E1

T = V 1 rT ϕe

b H1

T = I1 e

rT ψm

⇒

?

Looking inside the dielectric (1)

⇒ C ∂tV 0 + ∂3I0 = 0

POems (INRIA) Patrick Joly

SLIDE 92

=

iωµc

−1/2√σ b E2

3(ˆ

τ, 0, x3)

εδ ∂tEδ + σδ Eδ − rot Hδ = 0

b E2

3(ˆ

τ, ˆ ν, x3) = b E2

3(ˆ

τ, 0, x3) e−ˆ

ν√iωµcσ

from which we easily deduce

b H0

τ(ˆ

τ, 0, x3) =

iωµc

−1∂ˆ

ν b

E2

3(ˆ

τ, 0, x3)

so that, on the boundary

Coning back inside the conductor (2)

Transmission conditions then provide b

E2

3 =

iωµc/σ

1

2 b

H0

τ

( ˆ ν → +∞ )

(∗)

∂2

ˆ ν b

E2

δ0

POems (INRIA) Patrick Joly

SLIDE 93

+ Z

S

!

rotT b E2

3 · e

rψm = 0

L ∂tI1 + ∂x3V 1 Remembering and , we get

b E1

T = V 1 rT ϕe

b H1

T = I1 e

rT ψm

Coming back inside the dielectric (2)

The last integral is evaluated using Green’s formula

· e rT ψm µ ∂t b H1

T − e3 × ∂x3 b

E1

T + −

→ rotT b E2

3 = 0

Z

S

+

= ⇣iωµc σ ⌘ 1

2 Z

∂S

b H0

T ⇥ n

·

⇣ e rT ψm ⇥ n ⌘

Z

S

!

rotT b E2

3 · e

rψm = Z

∂S

b E2

3 · e

rψm ⇥ n

b E2

3 =

iωµc/σ

1

2 b

H0

τ

POems (INRIA) Patrick Joly

SLIDE 94

+ Z

S

!

rotT b E2

3 · e

rψm = 0

L ∂tI1 + ∂x3V 1 Remembering and , we get

b E1

T = V 1 rT ϕe

b H1

T = I1 e

rT ψm

Coming back inside the dielectric (2)

The last integral is evaluated using Green’s formula

· e rT ψm µ ∂t b H1

T − e3 × ∂x3 b

E1

T + −

→ rotT b E2

3 = 0

Z

S

+

= ⇣iωµc σ ⌘ 1

2 Z

∂S

b H0

T ⇥ n

·

⇣ e rT ψm ⇥ n ⌘

= ⇣iωµc σ ⌘ 1

2 ⇣ Z

∂S

e

rT ψm ⇥ n

2 ⌘

I0 Z

S

!

rotT b E2

3 · e

rψm = Z

∂S

b E2

3 · e

rψm ⇥ n

b H0

T = I0 e

rT ψm

POems (INRIA) Patrick Joly

SLIDE 95

+ Z

S

!

rotT b E2

3 · e

rψm = 0

L ∂tI1 + ∂x3V 1 Remembering and , we get

b E1

T = V 1 rT ϕe

b H1

T = I1 e

rT ψm

Coming back inside the dielectric (2)

The last integral is evaluated using Green’s formula

· e rT ψm µ ∂t b H1

T − e3 × ∂x3 b

E1

T + −

→ rotT b E2

3 = 0

Z

S

+

= ⇣iωµc σ ⌘ 1

2 Z

∂S

b H0

T ⇥ n

·

⇣ e rT ψm ⇥ n ⌘

= ⇣iωµc σ ⌘ 1

2 ⇣ Z

∂S

e

rT ψm ⇥ n

2 ⌘

I0 = R ∂1/2

t

I0 Z

S

!

rotT b E2

3 · e

rψm = Z

∂S

b E2

3 · e

rψm ⇥ n

b H0

T = I0 e

rT ψm

POems (INRIA) Patrick Joly

SLIDE 96

The modified telegrapher’s model

C ∂tV 0 + ∂3I0 = 0 C ∂tV 1 + ∂3I1 = 0 L ∂tI0 + ∂3V 0 = 0

L ∂tI1 + R ∂1/2

t

I0 + ∂3V 1 = 0 L ∂t

I0 + δ I1

+ δ R ∂1/2

t

I0 = 0

+ ∂3

V 0 + δ V 1

C ∂t

V 0 + δ V 1

+ ∂3

I0 + δ I1

= 0

δ × δ ×

Noticing that leads to δ R ∂1/2

t

I0 = δ R ∂1/2

t

I0 + δ I1

− δ2 R ∂1/2

t

I1

POems (INRIA) Patrick Joly

SLIDE 97

The modified telegrapher’s model

C ∂tV 0 + ∂3I0 = 0 C ∂tV 1 + ∂3I1 = 0 L ∂tI0 + ∂3V 0 = 0

L ∂tI1 + R ∂1/2

t

I0 + ∂3V 1 = 0 L ∂t

I0 + δ I1

+ δ R ∂1/2

t

I0 = 0

+ ∂3

V 0 + δ V 1

C ∂t

V 0 + δ V 1

+ ∂3

I0 + δ I1

= 0

( C ∂tV δ + ∂3Iδ = 0, L ∂tIδ + δ R ∂1/2

t

Iδ + ∂3V δ = 0.

δ × δ ×

Noticing that leads to δ R ∂1/2

t

I0 = δ R ∂1/2

t

I0 + δ I1

− δ2 R ∂1/2

t

I1

POems (INRIA) Patrick Joly

SLIDE 98

Thank%you% for%your%attention

POems (INRIA) Patrick Joly