3d-modelling of an aperture illuminated by a hf electromagnetic source for emc application

8/2/2019 3D-Modelling of an Aperture Illuminated by a HF Electromagnetic Source For EMC Application

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3D-Modelling of an Aperture Illuminated

by a HF Electromagnetic Source

For EMC Application

Mohamed Djennah*, Zakia Abidi**and Franoise Rioux-Damidau*** Laboratory Signal and ImagePolytechnic School of Algiers - Algeria**IR4M, University Paris XI, UMR CNRS 8081, Bat 220, 91405 Orsay France

[email protected]

Abstract : In this paper, we present a numerical

method for determining the penetration of

electromagnetic fields through a small aperture in a

metal enclosure having inside it a conducting

component. To solve the problem of coupling between

the radiated electromagnetic field and the metallic

enclosure, we develop a variational formulation in the

interior region and an integral formulation in the

exterior region by using the equivalent electriccurrent on the exterior surface of the enclosure. The

numerical results obtained with this method show the

influence of the characteristic parameters (,,) andof the thickness of enclosure on the penetration of

electromagnetic field inside the enclosure.

I. INTRODUCTION

Nowadays, more and more of electronics is

embarked on board of terrestrial or air vehicles with

electronic components functioning at frequencies

higher and higher. The multiplication of the

feasibility of electromagnetic sources and of theirincreasing power thus generates new types of bad

function of the electrical appliance and problems of

incompatibility inter-equipment, which can go until

their destruction. Electromagnetic compatibility,

according to its "fundamental principles", is thusanalyzed in terms of emission (to not disturb the

environment) and of immunity (to not be

disturbed). The two aspects occur at the same time

in conduction (currents and parasite tensions) and

in radiation (electric and/or magnetic fields).

Consequently, in the scenario of an EMC problem,one will find three actors: a source of disturbance

(for example the equipment itself in the case of aproblem of emission, or its environment in the case

of immunity), a connection (for example by

conduction, radiation, cross talk) and a victim of

the coupled disturbances (for example the

equipment itself in the case of problem of

immunity).

In our study we are interested in the

radiated electromagnetic field. On figure 1 we

present a configuration of a problem model relatedto the EMC, The metal enclosure with a small

rectangular opening lit by the electromagnetic

radiation of a source located in the exterior region,

considered as harmonic in time.

Fig.1.Physical model

This problem was the subject of some

works by using various numerical models [1][2][3]

[4]. In our study, we first apply the finite elementmethod inside and on enclosure and next weexpress the external field by an integral

representation. This one is a function of the

equivalent electrical currents K carried by theexternal surface of the enclosure. These surface

currents represent the jump of the tangential

component of the magnetic field:

Khn ][ (1)

We realize the coupling of the two regions

by computing an integral-differential operator Rbinding the traces of tangential component of the

electric field and that of its rotational [5]:

)()( ecurlnenR (2)This term represents the border term in the

variational formulation of the problem. We

compute the operator R by using K as an

intermediate unknown factor between the interior

problem and the exterior problem.

II. FORMULATION

The physical system is defined by the following

equations:

Radiant Source

Aperture

Conducting wire

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- in the enclosure without the conducting wire :

0)(0

ecurlhi (3)

0)(0

hcurlei (4)

- in conducting wire:

0)( ecurlhc

i (5)

0)()( hcurleic

(6)

- in the exterior region without source:

0)(0

ecurlhi (7)

0)(0

hcurlei (8)

and moreover we take in all system:

0hdiv ; 0ediv (9)

With these equations, we take into account

the conditions at infinite:)/1(0 re (10)

)/1(0 rh (11)

We take ),,( as constant scalar

functions in each different region, where h and e are

respectively the magnetic and electric fields

intensities, is the electric conductivity, is the

electric permittivity and is the magneticpermeability.

III. VARIATIONAL PROBLEM

With the object of evaluating the

electromagnetic field inside the enclosure and in

particular of calculating the induced voltage in the

conducting wire, we formulate our problem in

electric field:

0)()(1

eiiecurlcurl

(12)

Multiplying equation (13) by a test function e andintegrating it on the volume of the domain occupied

by the enclosure p, we have:

deecurlndecurlecurl '))(()'()(

0')( deeii (13)

where is the border of and n is the unit outward

normal to .

The total electric field e is composed theoretically

of two parts, one of reaction er and the other of

source e s:rseee (14)

With this decomposition, we can write (13) as an

integral equation where er

is the unknown in the

first term and where second member is function ofes.

In another way, the conducting wire intervenes as

an integral term in the problem formulation. Wesuppose that the ray of the wire is very small

compared with the dimensions of the enclosure and

as the length is small we can take it as one or two

edges as a sequence of mesh edges. We can write

the wire term in the variational formulation as :

S

Wire Sdehnt

edenT ')(')( (15)

nhnen )( (16)where S is the surface enclosing the wire and isthe surface impedance of the wire.

Equation (14) becomes:

w

lkji

w

Wire ldeeZt

T'1

(17)

with

22

1

a

jZw (18)

where a is the wire radius, and Z is the wire

impedance per unit length in ( /m )

IV. CALCULATION THE BORDER TERM

The integral term on in the integralequation (15) is called border term. At this stage, it

should be noted on the first hand that we cannot

solve the problem without identifying the jump ofthe tangential component of the magnetic field on

On the other hand, the solution of the interiorproblem could not be the solution of the problem

set in the open space R3. Only the border termenables us to take account of the behavior of thefield in the exterior region. To treat the borderterm, we first of all will formulate the external

problem and will express its solution according to

the traces of the field on the border . The system

of equations (7)-(10) governs the behavior of the

magnetic field in the exterior region. We indicate

by e0 the trace of the tangential component of

electric field on the solution of exterior problemcan be written in the following integral form:

where K represents the jump of the tangential

component of the magnetic field:

Khn (20)The equivalent surface current K, introduced on ,

is fictitious. It is also an intermediate unknown

used to couple the exterior and the interior

problems. To keep the electric field e like only

)19(),()()( 0

dyyxGyKxe



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unknown actor on , we will have to express K

according to e:

)(eRhn . (21)Our aim now is to calculateR. This computation isbased on a variational formulation:

(21)

V. NUMERICAL IMPLEMENTATION

In order to numerically solve the problem, the

volume of study is cut in tetrahedral elements and

the electric field vector E is described in terms of

basic functions Wij associated to the edges of these

elements.

N

i

iij eWe1

(22)

abbaijW (23)

where N is the total number of edges of the mesh,

Wijis the vector basis function associated with edge

( ij ) and eij is the problem unknown whichrepresents the circulation of the electric field along

the edge (ij). i is the barycentric coordinate oftetrahedra associated with the node i.

We have developed the currents Kon the basic

function:

ii gradxnx )()( (24)

and we have: i

ii xPxK )()( (25)

where:

i : described the vertex of .

Pi :the value ofKin vertex i.

i: the barycentric coordinate.

n(x): normal vector on .

Our variational formulation can be rewriten like the

following linear forms:

VVGR SeM (26)

CCGCSeM (27)

where: 21 MjRMM VG

RMMGC 1

R : is a full matrix with dimension (nbat nbat); it

represents the edge term of the approached

variational problem.

M1 and M2. are two matrices; their dimensions are

(nbat nbat); an element of these matrices is zero

only if (ij) and (kl) do not belong to the same

tetrahedron.

VI. NUMERICAL RESULTS :

A. Representation of a grid mesh

Fig.2. Tetrahedral grid of the enclosure

B. Representation of the fields:

Fig.3. Magnetic field h, f= 1 MHz

Fig.4. Electric field ef= 1 MHz

Number of internal nodes 489

Number of frontier nodes 302

Number of internal edges 3190

Number of frontier edges 900

Number of faces 600

Number of tetrahedral 3000

dyyxGyKnxene ),()()(00

dyyxGyKxe ),()()(0

dydx

yx

xKyKKeKe

r

e

)()(

4

1,,

0



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C. ISO-VALUE OF ELECTROMAGNETIC FIELD

ACCORDING TO PLAN (YOZ) (REAL PART):

Fig.5.f=1GHz, DD=0.7, RHO=1.667210-8

Js = 100, R=0.2 , VN1= -1, VN2= 0, VN3=0

The real part of the electric field according to plan(YOZ) X=0.1

Fig.6.f=1GHz, DD=0.7, RHO=1.667210-8Js = 100, R=0.2 , VN1= -1, VN2= 0, VN3=0

Fig.7. Representation of total magnetic field

in f=50 MHz

Fig.19. Variation of induced voltage in conducting wire

according to distance DD between source and aperture

CONCLUSION

In this article we have some numerical

results to evaluate the penetration of the

electromagnetic field in a metal enclosure. Weconsidered a radiant electromagnetic source in front

of an aperture of the enclosure at a limited distance

DD. We realize a coupling between the boundary

integral method outside and the finite element

method inside the enclosure to solve our problem.

The numerical results show the contribution of each

parameter of the problem in the evaluation of the

electromagnetic energy penetrating in the

enclosure, On the one hand, according to theelectromagnetic parameters: (electric conductivity

and permittivity () and magnetic permeability

() of the enclosure) and in addition, according tothe density of the current source, the geometrical

position and the frequency of the radiant source.

The numerical results show the variation in the

values and the form of electromagnetic energy in

each point inside the enclosure while varying at

each time only one parameter. Considering the

number of parameters we cannot expose all the

results in this paper.

REFERENCES

[1] W.P.Carpes, L. Pichon, RazekAnalysis of the

Coupling of an Incident Wave with a Wire inside a

Cavity using an FEM in Frequency and Times

Domains IEEE Transaction on ElectromagneticCompatibility.Vol 44, No3 August 2002.

[2] F.Paladian, P.Bonnet, M.Klingler A frequency-

domain prediction model using measured scattering

parameters of electrically short lines to determine

the per unit length parameters matrices of

multiconductor transmission lines 14th

International Zurich Symposium & Technical

Exhibition on Electromagnetic Compatibility,

Zrich (Suisse), February 2001, Actes du Colloque,

pp.293-298.

[3] D. Lecointe, W. Tabbara, J. Lasserre Aperture

Coupling of Electromagnetic Energy to a Wire

inside a Rectangular Metallic Cavity IEEE AP-S

Antennas Propagation Soc. Int. Symp, Vol. 3,

pp.1571-1574, 1992.

[4] T. Yang, J. L. Volakis Coupling to Wires in

Cavity Enclosure Using Iterative Algorithm

Radiation Laboratory, EECS, Dept. The University

of Michigan, ElectroScience Labo. The Ohio State

University, Columbus, OH43212, 2004.[5] M. Djennah, F. Rioux-Damidau, B. Bandelier

Computation of electric charges and eddy currents

with an e formulation Journal IEEE Transaction

Magnetic 1997, Vol 32, pp 1322-250 1 2 3 4 5 6 7 8 9 10-2

0

2

4

6

8

10

12

14

16

18

DD - Distance entre la source et l'ouverture -

Valeurdetensioninduiteda

nsleConducteur

Rfem

Ifem

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

55.00

60.00

65.00

-0. 50 -0. 40 -0.30 -0.20 -0. 10 0.00 0.10 0. 20 0.30 0.40 0. 50

-0.50

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-0.30

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-0.10

0.00

0.10

0.20

0.30

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-0.30

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-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.300.35

0.40

0.45

0.50

0.55

0.60

- 0.50 - 0.40 - 0.30 -0.2 0 - 0.1 0 0.00 0.10 0.20 0 .30 0. 40 0.50

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

-800.00

-700.00

-600.00

-500.00

-400.00

-300.00

-200.00

-100.00

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100.00

200.00

300.00

400.00

500.00

600.00

700.00

H

T[A/m]

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3d-modelling of an aperture illuminated by a hf electromagnetic source for emc application

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