Effective models for non-perfectly conducting thin coaxial cables

HAL Id: hal-02414849

https://hal.archives-ouvertes.fr/hal-02414849

Submitted on 16 Dec 2019

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Effective models for non-perfectly conducting thin coaxial cables

Geoffrey Beck, Sebastien Imperiale, Patrick Joly

To cite this version:

Geoffrey Beck, Sebastien Imperiale, Patrick Joly. Effective models for non-perfectly conducting thin

coaxial cables. Waves 2019 - 14th International Conference on Mathematical and Numerical Aspects

of Wave Propagation, Aug 2019, Vienna, Austria. �hal-02414849�

Effective models for non-perfectly conducting thin coaxial cables Geoffrey Beck

^1,∗

, Sébastien Imperiale

²

, Patrick Joly

¹

1

POEMS (CNRS-INRIA-ENSTA Paristech), Palaiseau, France

2

Inria & LMS, Ecole polytechnique, CNRS, Université Paris-Saclay, France

∗

Email: [email protected] Abstract

Continuing past work on the modelling of coax- ial cables, we investigate the question of the modeling of non-perfectly conducting thin coax- ial cables. Starting from 3D Maxwell’s equa- tions, we derive, by asymptotic analysis with respect to the (small) transverse dimension of the cable, a simplified effective 1D model. This model involves a fractional time derivatives that accounts for the so-called skin effects in highly conducting regions.

Keywords: Maxwell’s equations, Coaxial ca- bles, Asymptotic analysis

Statement of the problem

Figure 1: Section the coaxial cable. Σ

₊

and Σ

−

are the outer and inner boundary of S.

Denoting δ > 0 a small parameter, we consider a family of (thin) domains Ω

^δ

= G

_δ

Ω

where G

_δ

: (x

₁

, x

₂

, z) −→ (δx

₁

, δx

₂

, z).

and Ω is the disjoint union of a conducting do- main Ω

_c

and a dielectric one Ω

_d

,

Ω

_c

= C × R , Ω

_d

= S × R ,

where C = C

⁺

∪ C

⁻

, C

⁺

corresponding to the outer metallic shield and C

⁻

to the inner metal- lic wire and S is non-simply connected, see Fig- ure 1. Accordingly, we have, with obvious nota- tion

Ω

^δ

= Ω

^δ_d

∪ Ω

^δ_c

.

We are interested in the solution (E

^δ

, H

^δ

) of 3D Maxwell’s equation in Ω

^δ

:







ε

^δ

∂

_t

E

^δ

+ σ

^δ

E

^δ

− curl H

^δ

= j

^δ

, µ

^δ

∂

_t

H

^δ

+ curl E

^δ

= 0,

(1)

with zero initial data. In Ω

^δ_c

, (ε

^δ

, µ

^δ

) are con- stant equal to (ε

_c

, µ

_c

) and j

^δ

= 0. In Ω

^δ_d

, (ε

^δ

, µ

^δ

) do not depend on z and are obtained by a scal- ing in the transverse variable x

T

= (x

1

, x

2

) of fixed distributions in the reference domain Ω

_d

, for instance

ε

^δ

(x

_T

, z) = ε(x

_T

/δ).

The source term j

^δ

is defined similarly, moreover it is compactly supported, it has no longitudinal component and is divergence free. The conduc- tivity is weak in the dielectric Ω

^δ_d

, but very high in Ω

^δ_c

. More precisely

σ

^δ

(x

_T

, z) =

( δ

⁻⁴

σ

c

in Ω

^δ_c

, δ σ(x

_T

/δ), in Ω

^δ_d

.

(2) Note that the O(δ

⁻⁴

) magnitude of σ

^δ

in Ω

^δ_c

gives rise to a skin depth in O(δ

²

), small with respect to δ.

Our approach consists in obtaining the formal behaviour of the solution for small δ. To do so, we propose two distinct assymptotic expansions of the solution Ω

^δ_d

and Ω

^δ_c

that we match using transmission conditions. We present below our main results.

Electromagnetic field in the dielectric We introduce the following notations.

• ∇ for the 2D transverse gradient in x

T

, iden- tified to a 3D vector with third component 0,

• S

_Γ

:= S \ Γ where Γ is a cut that makes S

_Γ

simply connected (see Figure 1),

• [·]

_Γ

for the jump across Γ in the direction n,

• ∇ e is the 2D transverse gradient in S

Γ

,

• ∂

n

is the normal derivative, and ∂

τ

ψ = ∇ψ·τ e

the tangential derivative.

We obtain that, for small δ and all x

_T

∈ S

^δ

, E

^δ

(x

_T

, z, t) ∼ V

^δ

(z, t) ∇ϕ

_e

(x

_T

/δ)

+ δ R

_t

0

V

^δ

(z, s) ds

∇ϕ

_r

(x

_T

/δ), + δ ∂

_z

V

^δ

(z, t) (ϕ

_e

− ϕ

_m

)(x

_T

/δ) e

_z

,

H

^δ

(x

_T

, z, t) ∼ I

^δ

(z, t) ∇ψ

_m

(x

_T

/δ), + δ R

t

0

∂

1 2

t

I

^δ

(z, s) ds

∇ψ

_r

(x

_T

/δ), + δ ∂

_z

I

^δ

(z, t) (ψ

_e

− ψ

_m

)(x

_T

/δ) e

_z

, where e

_z

= (0, 0, 1)

^t

. Moreover:

i) The potential ϕ

e

∈ H

¹

(S) satisfies,

div ε ∇ ϕ

e

= 0 (S), ϕ

e

= 0 (Σ

+

), ϕ

e

= 1 (Σ

−

), and the same for ϕ

_m

with µ

⁻¹

instead of ε.

ii) The potential ψ

_m

∈ H

¹

(S

_Γ

) satisfies div µ ∇ ψ

m

= 0 (S

Γ

), ∂

n

ψ

m

= 0 (∂S), and [ψ

m

]

Γ

= 1, [∂

n

ψ

m

]

Γ

= 0. The same holds for ψ

_e

with ε

⁻¹

instead of µ. Moreover

Z

S

µ ψ

e

= Z

S

µ ψ

m

= 0.

iii) The function ϕ

r

∈ H

₀¹

(S) is the solution of div ε ∇ϕ

_r

= − div σ ∇ϕ

_e

.

iv) The function ψ

_r

∈ H

¹

(S) satisfies div µ ∇ψ

_r

= 0 (S), µ ∂

n

ψ

r

=−

r µ

c

σ

_c

∂

_τ²

ψ

m

(∂S).

v) The electric potential V

^δ

(z, t) and current I

^δ

(z, t) are 1D unknowns governed by general- ized telegrapher’s equations:







C ∂

_t

V

^δ

+ δ G V

^δ

+ ∂

_z

I

^δ

= j, L ∂

t

I

^δ

+ δ R ∂

1 2

t

I

^δ

+ ∂

z

V

^δ

= 0,

(3)

where j(z, t) is an effective source term, j(z, t) =

Z

S

j(x

_T

, z, t) · ∇ϕ

_e

(x

_T

), (4) and ∂

1 2

t

is the square root derivative in the sense of Caputo

∂

1 2

t

u(t) = 1

√ π Z

t

0

∂

τ

u(τ )

√ t − τ dτ.

As in [1], the capacity C, inductance L and con- ductance G are given by:

C = Z

S

ε

∇ϕ

_e

2

, L = Z

S

µ e ∇ψ

_m

2

, G = Z

S

σ

∇ϕ

_e

2

. Moreover, we obtain an explicit expression for the resistance R, which takes into account skin effects:

R = Z

∂S

r µ

c

σ

_c

∂

τ

ψ

m

2

. (5)

This generalizes formulas of the literature (see [2], chapter 13) already derived in very simple cases.

Electric field in the outer conductor In the rescaled con- ducting domain C

+

the electromagnetic fields are described using tangential and normal coordinates (τ, ν ). The penetra- tion depth `

^δ

of the fields is in O(δ).

L

⁺

being the length of Σ

₊

, one shows that there exists a 3D field

E

⁺

: [0, L

⁺

] × R

⁺

× R × R

⁺

→ R

³

such that in C

₊^δ

× R and for small δ,

E

^δ

(x

_T

(τ, ν ), z, t) ∼ δ

²

E

⁺

(τ /δ, ν/δ

²

, z, t), (and a similar property holds for the magnetic field). The important fact is that the component E

_z⁺

is solution of the 1D heat equation

µ

c

σ

c

∂

t

E

_z⁺

− ∂

_ν²

E

_z⁺

= 0, (6) and thus satisfies, at the boundary ν = 0:

∂

_ν

E

_z⁺

+ √ µ

_c

σ

_c

∂

1 2

t

E

_z⁺

= 0.

The above equation is used when writing trans- mission conditions across Σ

^δ₊

× R .

This explains the appearance of ∂

1 2

t

in the effec- tive model (3).

References

[1] G. Beck, S. Imperiale, P. Joly Mathematical modelling of multi conductor cables, Disc. &

Cont. Dynamical Systems, (2014).

[2] W. H. Hayt, J. A. Buck, Engineering Elec-

tromagnetics, 6th Edition, (2001).