### Introduction

Currently, electromagnetic shields are important objects of scientific and
technological development, due to the urgency of solving the problems of
ensuring electromagnetic compatibility of electronic equipment, as well as
protecting a person from electromagnetic radiation impact. One of the main
requirements for the characteristics of currently developed
electromagnetic shields is a high electromagnetic radiation attenuation value
(more than 30 dB) and a low value of electromagnetic radiation reflection
coefficient (less than -5 dB). These requirements are due to the necessity
of eliminating the occurrence of situations in which the electromagnetic
shield during its using becomes a source of secondary (re-reflected)
electromagnetic radiation that affects unshielded objects located near it.
Compliance with these requirements can be ensured only in the case
of the manufacture of electromagnetic shields in the form of
heterogeneous media containing electrically conductive materials. By the
latter, it is customary to understand media, which are a set of elements
characterized by various electrophysical properties, and in some cases
geometric properties. In this regard, electromagnetic shields in the form of heterogeneous
media include multilayer shields, as well as shields whose surface is
geometrically inhomogeneous (having roughness and / or gaps). The
second of these shields are more promising from the point of view of
development and research due to the fact that they are characterized by
lower weight and size parameters compared to the first.
The electromagnetic radiation reflection coefficient and attenuation of
such shields, as a rule, depend on the parameters of the geometric
inhomogeneities of their surfaces (roughness height, diameter and depth of
the slots).

The authors set a goal,
which was to establish the influence of the roughness height and the diameter
of the slots of the surfaces of electromagnetic shields on the processes
of their interaction with electromagnetic waves, the length of which
significantly exceeds the parameters of these inhomogeneities. Such processes
determine the electromagnetic radiation reflection coefficient
and attenuation of the shields. To achieve this goal, the following
tasks were solved:

1) an approach for
modeling the processes of electromagnetic radiation interaction with the
shields in the form of heterogeneous media has been developed;

2) the
electromagnetic shields as objects of study have been justified;

3) the initial and
boundary conditions for modeling the processes of electromagnetic
radiation interaction with the shields under investigation have
been determined;

4) modeling of the
processes of electromagnetic radiation interaction with the shields under investigation
has been performed depending on the diameter of the slits and the
roughness parameters of their surfaces.

### The experiment
methodology

For the numerical
simulation of the electromagnetic shields in the form of heterogeneous
media, as well as modeling the processes of electromagnetic radiation
interaction with such shields, the COMSOL Multiphysics 3.5 software package was
used. Modeling was implemented based on the following approach proposed by the
authors.

1. When modeling
the electromagnetic shields, a through counting scheme was used without
explicitly distinguishing the interface between adjacent media. The possibility
of this principle following is due to the fact that the continuity condition
for the total current is valid in each section of a layered medium [1].

2. Discretization
of electromagnetic radiation propagation medium and the location of
the shield interacting with this radiation was performed in such a way that the
nodes of the boundaries of the finite elements of the resulting mesh
simultaneously belong to media with different electrophysical properties. This
is necessary so that the condition of equality of the total currents and the
condition of equality of charge flows are satisfied at the interface
between the media. Moreover, during the calculation of the electric field
strength at the boundaries of the finite elements of the grid (*Е _{х}*)
it must be taken into account that

*E*(

_{x}*x*) experiences a discontinuity of a function of the first kind (i.e., a discontinuity of the electromagnetic field) due to the fact that

*E*

_{x}_{1}≠

*E*

_{x}_{2}.

Since *E _{x}*(

*x*) is a piecewise smooth and piecewise differentiable function, i. e., it has finite one-sided derivatives

*E*'

*(*

_{x+}*x*) и

*E*'

*(*

_{x–}*x*), then at the points of discontinuity

*x*the following conditions are satisfied:

_{i},

Based on the Dirichlet theorem, the function *E*(*x*)
at some discontinuity point ξ can be represented as follows:

At the interface between
media with different electrophysical properties, a double electric layer
(DEL) is always formed, the structure of which, as a rule, is unknown, but
it has a significant effect on electrokinetic phenomena. It is important to
note that in reality the function *E*(*x*), which is
characteristic of DEL, changes continuously, therefore equation (1) is valid
for the case when the thickness of the DEL, i.e., the thickness of the
interface, is much smaller than the size of each of the homogeneous elements
(layers) of the heterogeneous environment.

If the thickness of the DEL is much smaller than the dimensions of
each of the homogeneous elements (layers) of the inhomogeneous medium,
then equation (1) can also be obtained and justified on the basis of the
condition for a linear change in *E*(*x*) in the region of
the DEL. The thickness of the DEL depends on the kind of contacting substances
and can be tens of angstroms. According to modern concepts, the outer lining of
a DEL consists of two parts: the first is formed by ions closely attracted
to the surface of one of the elements of an inhomogeneous medium obtained
after its discretization (a “dense” or “Helmholtz” layer with a
thickness *h*), and the second by ions located at distances from
this surface, exceeding the radius of the ion, and the number of these
ions decreases with distance from the interface (“diffuse layer”).
The potential in the dense and diffuse parts of the DEL is distributed
according to the exponential law [2], that is, the linearity
condition *E*(*x*) is violated. However, if the thickness of the
DES is much smaller than the size of each of the homogeneous elements
(layers) of the inhomogeneous medium, then *E*(*x*) can be
expanded in a power series and, therefore, it can be considered approximately
linearly. This allows us to conclude that condition (1) and the stated
principle of modeling electromagnetic screens in the form of inhomogeneous
media are valid.

3. When modeling the processes of electromagnetic radiation
interaction with electromagnetic radiation, electromagnetic waves in the form
of soliton-like signals are used. The rationality of following the stated
principle is due to the fact that the penetration depth of such signals in
complex environments is higher than other signals currently used in modeling
(rectangular, sawtooth, etc.). This is due to the fact that soliton-like
signals are less prone to “noise” arising at the interfaces between homogeneous
elements (layers) of an inhomogeneous medium. Soliton-like signals are described
by the following expressions [3,4]:

Figure 1 presents the graphical dependencies corresponding to
the above expressions and showing the forms of soliton-like signals in the
horizontal and vertical planes [2].

Figure 1: Form of a soliton-like
signal in horizontal (a)and vertical (b) planes.

The broadening of the spectral lines is due to the non-stationary
process of interaction of the radiating atom with the particles
surrounding it – other atoms and molecules, ions and electrons. Therefore,
functions of the form (2) and (3) continuously fill the frequency range ω – 2Δ
≤ ω ≤ ω + 2Δ, and it takes some time to establish the signal (transient).
It is easy to verify that *E _{x}*(

*t*),

*E*(

_{y}*t*) and their time derivatives satisfy the necessary matching conditions, since they take values equal to zero in the case of pulse propagation in a medium with zero initial conditions [5].

It should also be noted that the wave group is a kind of
oscillatory circuit with distributed parameters, in which forced
oscillations are not established immediately, but lichens after some time of
the appearance of external EMF [6]. After the pulse reaches its maximum value,
it monotonically decreases to zero, therefore, in formulas (2) and (3), the
authors used the hyperbolic cosine of the harmonic function.

The following electromagnetic shields were selected as objects of
study:

– shields based on silver, the surfaces of which are rough
(shields 1);

– shields in the form of copper plates, the surfaces of which have
slots (shields 2).

The choice of these materials is due to their widespread use at present for the manufacture of electromagnetic shields. In particular, silver coatings, due to their high conductivity and good adhesive properties, are used for electromagnetic shielding of cases and assemblies of high-precision electronic equipment, and copper and copper-containing plates due to their low cost compared to other metal materials in the process of electromagnetic shielding of rooms.

### Results and their discussion

During the simulation, it was assumed that the
process of changing the electromagnetic field strength in the area of the
screens is described by the following expression:

The initial conditions are described using the following
expressions:

*E*(*t*) was
determined according to formulas (2) and (3) with the following parameters:
ω = 10^{14} s^{–1}, Δ = 10^{4} s^{–1},
λ = 10^{15} m, θ = 10^{6}, *m* = 1.
The values of the integrals were calculated approximately using the
Simpson formula. Otherwise, at each iteration it will be necessary to calculate
the integrals (2), (3), due to which the calculation time will increase
significantly.

Figures 2 and 3 show the areas in which the process of interaction of electromagnetic radiation with the selected shields was simulated.

The simulation results of electromagnetic radiation interaction
with the shield 1 are presented in Figure 4. The animated simulation
result of electromagnetic radiation interaction with the shield 1 is available
for viewing by reference https://youtu.be/_Rh2dRbqKpg.

From Figure 4 it follows that the greatest intensity of
the electromagnetic field interacting with the shields 1 is observed at the
points located in the recesses and at the tops of the roughnesses of their
surface. In this case, the tension at the points located in the
recesses of the surface is higher than at the points located on the tops of its
roughness.

The simulation results of electromagnetic radiation interaction
with the shields 2 are presented in Figure 5. The animated simulation
result of electromagnetic radiation interaction with the shields 2 is
available for viewing by reference

https://www.youtube.com/playlist?list=PLt1iDS0NcZFL4T8_eq2N80qVYvzRrm3nK.

From Figure 5 it follows that an increase in the slots diameter of
the shields 2 leads to an increase in the propagation distance
through these slots of the electromagnetic field interacting with
such shields. The highest electromagnetic field intensity was recorded at the
points of the boundaries of the slots lying on the front line of the
electromagnetic field and the most distant from the source of this field. It
was established that the magnitude of the electromagnetic field strength at
these points doesn’t depend on the diameter of the investigated shields
slots if the value of this diameter varies from 500 nm to 6 μm.

According to Blokhintsev DI [7], to illustrate the
uncertainty relation, the diffraction of an ensemble of particles with a
given momentum from a slot in the shield is considered. It was assumed that the
wavelength and, at the same time, the total momentum of the particle
don’t change during diffraction. In this paper, it is noted that with
a decrease in the depth of the slot, the nature of the wave field behind
the shield is much more complicated, because the field, when the
electromagnetic wavelength is much less than the diameter of the slot, can no
longer be characterized by certain values of the wavelength.

In the case of the simulation performed by the authors, the length
of electromagnetic waves is much larger than the slots diameter. Based on
the evaluation of the simulation results, it was determined that the
diffraction length of electromagnetic waves can increase. Moreover, it is
determined not only by the slots diameter, but also by its depth. The
wave packet in quantum mechanics for non-monochromatic waves, as noted
in Quantum Mechanics [7], doesn’t satisfy the matching conditions. In
application packages known to the authors for solving electrodynamic problems,
the matching conditions are also not considered and not taken into account.
Numerical simulation in the presence of strong discontinuities in the
electromagnetic field is in this case an incorrect task.

### Conclusions

1. The proposed approach, unlike analogues, is characterized by the following features:

– based on the use
of electromagnetic waves in the form of soliton-like signals, which take into
account the temporal dispersion of electrical induction (taking into account
the unsteady broadening of their spectral lines and integration over the
continuous spectrum), which eliminates the need to neglect spatial derivatives,
as well as use spatial nonlocal relations to take into account the influence
of surface charge, surface current and spatial dispersion of electric induction
at the interfaces of adjacent media;

– based on the
creation of an “unsharp” interface between adjacent media using the Dirichlet
theorem for piecewise smooth functions and with the condition of continuity of
the total current.

2. Based on the simulation results carried out using the proposed approach, we can conclude that the maximum electromagnetic field interacting with the shields whose surface is geometrically inhomogeneous is observed at the points of this surface that are most distant from the electromagnetic source (in the recesses). This allows us to conclude that the development of technologies for heterogenizing the surfaces of electromagnetic shields is promising in order to reduce the values of their electromagnetic radiation reflection coefficients while maintaining the attenuation values.

### Foundation

This work was supported by the Belarusian Republican Foundation for Basic
Research, project No. F19-096.

### References

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Soliton-like pulse signal in a medium with strong discontinuities in the
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(In Russian)

**Corresponding
author:** Boiprav OV, Educational Institution “Belarusian State University
of Informatics and Radioelectronics”, Minsk, Belarus, E-mail: boiprav@tut.by

**Citation: **Aliseyko
MA, Boiprav OV, Grinchik NN, Tarasevich AV. Modeling the interaction of
solit-like pulse signals with electromagnetic shields in the form of
heterogeneous media (2020) Edelweiss Chem Sci J 3: 1-5.

#### Keywords

Artificial intelligence, Information and communications
technology, Crystallography, Out-of-scope prediction.