APPLICATION OF CAUCHY INTEGRAL TO APPROXIMATE

THE FIELD COMPONENTS AND CURRENT IN METAL &

DIELECTRICAL POLYGONS

Method of the Field (Current) Restoration Inside and Outside Flat Closed Contour

based on Its Known Values at the Contour

A. V. Ashikhmin, Yu. G. Pasternak, S. L. Podvalny and Yu. A. Rembovsky

Moscow p/o 30 ,IRCOS, JSC

Keywords: Analytic function of complex variable, Cauchy integral, field (current) approximation, metal and dielectrical

polygons.

Abstract: The Cauchy integral was proposed as a means to approximate the field components and current in metal and

dielectrical polygons. The report illustrates that such a technique can significantly scale down

electromagnetic issues solved in spatial- frequency domain. The technique also makes it possible to evaluate

the strength of electromagnetic field not only inside the considered closed contour, but outside it even in

case when there is no a prior information about object configuration and its physical characteristics.

1 INTRODUCTION

The purpose of this work is to illustrate applicability

of Cauchy integrals to description of

electromagnetic (EM) field or current in some

electromagnetic objects whose surface allows a

piecewise planar approximation. By example of

dielectrical and metal cubes, it was confirmed that

such a description could help restore the field

(current) values within an object surface (or object

section) for a wide frequency band. These simple

electromagnetic objects were selected because their

main properties had been studied very thoroughly by

a lot of scientists - both through numerical

computation of electrodynamics boundary values

and through real experiments as in (Mittra, 1977).

The Cauchy integral was then proposed as a means

to scale down the systems of linear algebraic

equations derived from electromagnetic vector

boundary equations and drawn in terms of space and

frequency. The report illustrates that such a

technique makes it possible to evaluate the strength

of electromagnetic field not only inside the

considered closed contour, but outside it even in

case when there is no a prior information about

object configuration and its physical characteristics.

It is proved that proposed technique can be used for

reducing systematic errors in measurement of

emitter’s angular coordinates by means of mobile

direction finders and increasing DF resolution and

accuracy.

There are a lot of highly effective methods for

electromagnetic field approximation (both for entire

electrodynamic objects and for their components,

e.g. in their finite elements) including the cases

when the object dimensions exceed the free-space

wavelength. Thus, to describe the field behavior via

modified finite elements method, we suggest using

Lagrange interpolation polynomials in this work

(Milan, Branislav, 2006). The finite elements

approach proposed by the authors (Milan, Branislav,

2006) allows to significantly reduce the number of

computations to describe the field, as compared with

regular polynomials. However, if you do not have a

priori information about the object geometry and

materials, it cannot restore the structure of the EM

field inside the object and outside of it. The authors

of this report suppose that the above problem can be

solved if the field/current component is taken as an

analytical complex variable function with a Cauchy

integral and this work is just an approach to that

solution.

153

V. Ashikhmin A., G. Pasternak Y., L. Podvalny S. and A. Rembovsky Y. (2007).

APPLICATION OF CAUCHY INTEGRAL TO APPROXIMATE THE FIELD COMPONENTS AND CURRENT IN METAL & DIELECTRICAL POLYGONS -

Method of the Field (Current) Restoration Inside and Outside Flat Closed Contour based on Its Known Values at the Contour.

In Proceedings of the Second International Conference on Wireless Information Networks and Systems, pages 153-156

DOI: 10.5220/0002144901530156

Copyright

c

SciTePress

2 THEORY

For practical purposes, it is advisable to treat the

analyzed surface

S

as a total of plane polygons of

different size. Suppose that there is a plane polygon

over a closed contour

L

.

Then, the Cauchy integral can be evaluated

through the known discrete values of

(

)

k

U

ξ

(

N

k

,...,2,1=

) using

L

(Privalov, 1984) integration

contour:

()

() ()

∑

∫

=

−

Δ⋅

⋅

⋅⋅

≈

−

⋅

⋅⋅

=

N

k

k

kk

L

z

U

iz

dU

i

zU

1

2

1

2

1

ξ

ξξ

πξ

ξξ

π

,

is an increment of the integration complex variable,

where the

kkk

yix

Δ

⋅

+Δ=Δ

ξ

point is center of the

intervals

22

kkk

yxl Δ+Δ=

whose length is

k

ξ

.

Below are computational modeling results

illustrating applicability of Cauchy integration for

approximation of field/current components in the

simplest electrodynamic objects - dielectrical or

steel cube.

The dash lines in Figs. 1, 2 reflect the actual

(derived through rigorous numerical solutions of

diffraction problems) frequency profiles of the real

and imaginary part of

z

E

-component (total EM

field) and

z

J

-component (current density on the

metal surface). The solid lines show the restored

profiles derived through Cauchy integrals.

Let’s consider diffraction of a plane polarized

EM-wave on a 110x110x110 mm dielectrical cube

where (

,10=

r

ε

003.0=

э

tg

δ

) - Figure 1. The

coordinates of the normalized wave vector

→→→

=

000

/ kkk

n

were taken as

()

577.0;577.0;577.0 −−−

. Projection of the

normalized vector

→→→

=

...

/

падпадпад

n

EEE

onto the

coordinate axis

);;(

z

y

x

were (-0.408; -0.408;

0.816) respectively. The sight point coordinates

were taken as (45; 45; -55) mm (near the bottom

cube face angle).

The results of numerical experiments to restore

the current density on the “shady” surface of the

metal cube are shown in Figure 2. For the above

profiles, the values of the normalized wave vector

coordinates

→→→

=

000

/ kkk

n

amounted to (-0.667; -

0.667; -0.333); and projections of that vector

→→→

=

...

/

падпадпад

n

EEE

to

);;(

z

y

x

axis were taken as

(

)

943.0;236.0;236.0

−

−

. The surface current

density was to be restored in the middle of a “shady”

face of the cube (

55

−

=

y

mm); and the sight point

coordinates were taken as (0; -55; 0) mm.

With this, we would like to draw attention to the

actual and imaginary part of the

z

E

- component

(total EM field, Figure 1). There, the Cauchy

integral can “scan” almost all the frequency band in

question and “trace” even the slightest resonances

around such frequencies as 1.4, 1.5 and 1.85 GHz.

The available errors between the actual and

estimated profiles are, most probably, caused by

numerical integration errors and by the nature of the

function describing the field structure (current) – it

is not purely analytical.

a)

b)

Figure 1: Restoration of the Field at the Bottom of a

Dielectric Cube, section

55

−

=

z

mm).

WINSYS 2007 - International Conference on Wireless Information Networks and Systems

154

a)

b)

Figure 2: Restoration of the Current Surface Density on

the “Shady” Side of a Metal Cube (

55−=

z

mm).

Now let us consider a method for restoration of

the EM field outside a closed contour and try to

reduce the angular coordinates systematic error in

emitters’ position finding by means of a mobile

antenna array - see Figure 3.

Figure 3: Electromagnetic object: carrier enclosure +

antenna array.

Suppose that an EM wave is falling onto an

array, then the general vector of the voltage

amplitudes, which emerged on the antenna resistors,

is

[]

T

N

UUUUU ,...,,,

321

=

→

. In this case, the values of

the function describing the field at the circle with

R

radius comprising the elements of the array and,

consequently,

N

electrically short symmetrical

vibrators will be as follows:

[]

() ()()

[]

∑

+

=

−

+⋅−⋅⋅=⋅⋅=

1

1

1

1/1expexp

N

n

n

NniBiRzU

ξξ

.

The values of

n

B

complex factors are determined

through solution of the following algebraic

equations with several complex unknowns:

(

) ()

⎥

⎦

⎤

⎢

⎣

⎡

+

⋅++

⎥

⎦

⎤

⎢

⎣

⎡

+

+=

+

N

N

N

ikB

N

N

ikBBU

Nk

1/2

exp...

1/2

exp

121

ππ

,

The values of the

R

function measured and

approximated at the outer circle with

U

radius shall

conform to Cauchy or Poisson integral formula:

So we obtain an a first kind Fredholm integral

equation related to unknown

[

]

()

ψ

⋅

⋅ irU

внешнее

exp

function, describing the scalar field on

r

-radius:

[]

() ()

[]

∑

+

=

−

+−=

1

1

1

1/)1(expexp

K

k

kвнешнее

KkiXirU

ψψ

,

And this function is fully determined by

X

-factors.

To solve this integral equation is an incorrect

mathematical problem and therefore we had to use

Tikhonov regularization (Bakhvalov, Zhidkov,

Kobelkov; 1987). We have also applied a

collocation method (Bakhvalov, Zhidkov, Kobelkov;

1987) and thus reduced the initial integral equation

to a system of linear algebraic equations with

complex variables.

According to the numerical analysis of the

bearing frequency dependence (provided that the

bearing was measured with an array with Radius

5.0

=

R

m and that the real azimuth angle was

0

_

45=

fallEMW

ϕ

for the falling wave), the maximum

systematic errors was 16

0

at 90 MHz: the measured

bearing value was 61

0

. The developed technique can

significantly reduce the amount of DF systematic

errors caused by scattered waves from the carrier.

Thus, when restoring the P&A structure of the scalar

field on the circle with

5.1

=

r

m, the error will be

reduced from 16

0

to 1

0

regardless of the carrier's

geometry and its location relative to the array

(Figure 4). Moreover, since the radius of a virtual

Nk ,...,2,1

=

() ( ) ( )

[]

[]

()

()

()

[]

()

.

cos2

1/)1(exp

2

1

cos2

exp

2

1

1/1exp

22

22

2

0

1

1

1

22

22

2

0

1

1

1

ψ

ϕψ

ψ

π

ψ

ϕψ

ψ

π

ϕ

π

π

d

RrRr

Rr

KkiX

d

RrRr

Rr

irU

NniBRU

K

k

k

внешнее

N

n

n

+−−

−

⋅

⋅

⎭

⎬

⎫

⎩

⎨

⎧

+−=

=

+−−

−

=

=+−=

∫

∑

∫

∑

⋅

+

=

−

⋅

+

=

−

APPLICATION OF CAUCHY INTEGRAL TO APPROXIMATE THE FIELD COMPONENTS AND CURRENT IN

METAL & DIELECTRICAL POLYGONS

155

array is increased by three times and since the

number of its elements rises from 12 to 36, the

scanning resolution of the azimuth angular

coordinates will also rise.

Figure 4: Application of the proposed method for

reduction of direction finding systematic errors and

increase of azimuth resolution on a mobile carrier (the full

line shows the array pattern as in Figure 3 and the dash

line shows the array pattern for the array with the triple

diameter).

3 CONCLUSION

The EM field and current surface density

components in metal-dielectric structures can be

approximated by means of Cauchy integrals treated

by the method of average and based on the finite

aggregate of scalar field values taken through the

integration contour. With this, it can also be possible

to restore the phase and frequency characteristics of

the field or current within the integration contour.

Besides, these characteristics will be more exact

than those of amplitude and frequency and will

cover a wider frequency band.

In most of the reviewed cases, the field (current)

approximation accuracy will decrease as the sight

point approaches to the integration contour (if the

grid pitch is fixed). To increase such approximation

around the integration contour, a tapered grid can be

used. Its pitch will decrease as the sight point

approaches the contour.

Approximation of the field (or current) through

Cauchy integral can be applied to scale down

electromagnetic issues solved in terms of space and

frequencies and to allow for diffusers affecting

antenna systems and their directional properties.

It is proved that proposed technique can be used

for reducing systematic errors in measurement of

emitter’s angular coordinates by means of mobile

direction finders and increasing DF resolution and

accuracy.

REFERENCES

Mittra, R., Computer Techniques for Electromagnetics,

Moscow, MIR Publishers, 1977.

Milan, M., Ilic, Branislav, M., Notaros. Higher Order

Large-Domain Hierarchical FEM Technique For

Electromagnetic Modeling Using Legendre Basis

Functions On Generalized Hexahedra, Revised paper

for Electromagnetics, February 11, 2006

www.bnotaros.umassd.edu.

Privalov, I., I., Introduction to Complex Variable

Functions” Moscow, Nauka Publishers, 1984.

Bakhvalov, N., S., Zhidkov, N., P., Kobelkov; G., M.,

Numerical Methods, Moscow, Nauka Publishers,

1987.

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156