Microwave effective electromagnetic response of sandwich like magnetic composite
In this study, the long wave approximations for the effective electromagnetic response of 2-D sandwich metamaterial structure, as 
 infinite two-component flat composite with cylindrical metallic inclusions contained by the same infinite two-component flat composites 
 with spheric...
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| Опубліковано в: : | Радіофізика та електроніка |
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| Дата: | 2011 |
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| Мова: | Англійська |
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Інститут радіофізики і електроніки ім. А.Я. Усикова НАН України
2011
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Microwave effective electromagnetic response of sandwich like magnetic composite / O.N. Rybin, A.I. Pitafi, S.P. Vyalkina // Радіофізика та електроніка. — 2011. — Т. 2(16), № 1. — С. 19-23. — Бібліогр.:17 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860249753836584960 |
|---|---|
| author | Rybin, O.N. Pitafi, A.I. Vyalkina, S.P. |
| author_facet | Rybin, O.N. Pitafi, A.I. Vyalkina, S.P. |
| citation_txt | Microwave effective electromagnetic response of sandwich like magnetic composite / O.N. Rybin, A.I. Pitafi, S.P. Vyalkina // Радіофізика та електроніка. — 2011. — Т. 2(16), № 1. — С. 19-23. — Бібліогр.:17 назв. — англ. |
| collection | DSpace DC |
| container_title | Радіофізика та електроніка |
| description | In this study, the long wave approximations for the effective electromagnetic response of 2-D sandwich metamaterial structure, as 
infinite two-component flat composite with cylindrical metallic inclusions contained by the same infinite two-component flat composites 
with spherical metallic inclusions are obtained. The expressions for the response are obtained by generalizing the expressions of the 
electromagnetic response of the infinite chain of infinitely long metallic cylinders periodically immersed in the flat magneto-dielectric 
matrix. The generalization has been done by following the approaches of S- and T-matrices. The case of ferrite like metallic saturated 
inclusions is considered in the study. The analytically obtained results are compared with the numerically calculated ones.
Изучено длинноволновое приближение для эффективного электромагнитного отклика двухмерной метаматериальной сэндвич-структуры, представленной как бесконечный двухкомпонентный плоский композит с цилиндрическими 
металлическими включениями, заключенный между Одинаковыми бесконечными двухкомпонентными композитами со 
сферическими металлическими включениями. Выражения для 
отклика получены путем обобщения выражений для 
электромагнитного отклика бесконечной цепочки бесконечно 
длинных металлических цилиндров, периодически вставленных в плоскую магнитодиэлектрическую матрицу. Обобщение сделано с помощью теории S- и T-матриц. Рассмотрен 
случай ферритоподобных металлических включений, намагниченных до насыщения. Проведена оценка точности полученных аналитических выражений путем сравнения результатов 
аналитического моделирования с результатами численного 
эксперимента.
Вивчено довгохвильове наближення для ефективного електромагнітного відгуку двовимірної метаматериальної сендвіч-структури, поданої як нескінчений двохкомпонентний композит з циліндричними металічними 
включеннями, що замкнений між однаковими нескінченними 
двохкомпонентними композитами, зі сферичними металічними включеннями. Вирази для відгуку отримано шляхом 
узагальнення виразів для електромагнітного відгуку нескінченного ланцюга нескінченно довгих металічних 
циліндрів, які періодично вставлені в плоску магнітодіелектричну матрицю. Узагальнення здійснено за допомогою 
S- і T-матриць. Розглянуто випадок феритоподібних включень, що намагнічені до насичення. Проведено оцінку 
точності отриманих аналітичних виразів, порівнюючи результати аналітичного моделювання з результатами чисельного 
експерименту.
|
| first_indexed | 2025-12-07T18:41:43Z |
| format | Article |
| fulltext |
ММІІККРРООХХВВИИЛЛЬЬООВВАА ЕЕЛЛЕЕККТТРРООДДИИННААММІІККАА
_________________________________________________________________________________________________________________
__________
ISSN 1028–821X Радіофізика та електроніка, 2011, том 2(16), № 1 © ІРЕ НАН України, 2011
UDC 537.8.029.6
O. N. Rybin, A. I. Pitafi
, S. P. Vyalkina
MICROWAVE EFFECTIVE ELECTROMAGNETIC RESPONSE
OF SANDWICH LIKE MAGNETIC COMPOSITE
Department of Information System and Technologies,
Kharkov State University of Food Technology and Trade, Kharkov Ukraine
333, Klochkovskaya St., Kharkov, 61051, Ukraine
Centre for Advanced Studies in Pure and Applied Mathematics,
Bahauddin Zakariya University
Multan, 60800 Pakistan
In this study, the long wave approximations for the effective electromagnetic response of 2-D sandwich metamaterial structure, as
infinite two-component flat composite with cylindrical metallic inclusions contained by the same infinite two-component flat composites
with spherical metallic inclusions are obtained. The expressions for the response are obtained by generalizing the expressions of the
electromagnetic response of the infinite chain of infinitely long metallic cylinders periodically immersed in the flat magneto-dielectric
matrix. The generalization has been done by following the approaches of S- and T-matrices. The case of ferrite like metallic saturated
inclusions is considered in the study. The analytically obtained results are compared with the numerically calculated ones.
Key words: metamaterials, electromagnetic response, effective parameters, Effective Medium Theory, microwave.
During the last decade, flat metamaterial
structures gained significant interest [14]. One
reason for this is the availability of existing methods
for measuring the complex permittivity and
permeability of bulk materials [58]. The second
reason is related to the pursuit for making the
characterization of metamaterials at «the effective
inter-atomic» level.
This study is dedicated to the
characterization of 2-D sandwich metamaterial
structure as infinite two-component flat composite
with cylindrical metallic inclusions contained by
same infinite two-component flat composites with
spherical metallic inclusions. The characterization of
the structure is based on obtaining its effective
electromagnetic response by deriving its long wave
approximations of expressions of the complex
effective dielectric and magnetic constants. The
prerequisite for the problem being considered is the
effective electromagnetic response of the infinite
chain of infinite circular metallic cylinders
periodically immersed in a magneto-dielectric matrix
with the thickness equals to the diameter of the
cylinders, as in [4]. Using the S- and T-matrices
approaches [9], we extend the results of [4] for the
case of the above chain structure containing a pair of
similar, two-component composite slabs as infinite
magneto-dielectric slabs periodically embedded with
spherical metallic inclusions. It is supposed that the
cylindrical inclusions and spherical ones are ferrite
like metallic inclusions magnetized up to the
saturation. Correctness of the analytically obtained
expressions is assessed with the numerically obtained
one by using Finite-Difference Time-Domain
method (FDTD).
1. Main Relations. Let us consider the
scatterer as 2-D infinite sandwich structure as shown
in Fig. 1 where h is the thickness of lateral slabs, d
is the spacing of chain of infinitely long metallic
cylinders of the radius a. It is supposed that and
are the dielectric and magnetic constants,
respectively, of the cylinders immersed in a magneto-
dielectric matrix with the dielectric and magnetic
constants and , respectively while 2r and 2r
are the dielectric and magnetic constants,
respectively, of the spheres of radius r immersed in
a magneto-dielectric matrix of the lateral slabs with
the dielectric and magnetic constants ~ and ,~
respectively.
Fig. 1. The scatterer
a
,
L
L
~,~ ~,~
h h
d
O. N. Rybin et al. / Microwave Effective Electromagnetic…
_________________________________________________________________________________________________________________
20
Let us consider the normal incidence of an
electromagnetic wave with frequency and
wavenumber k on the above sandwich structure.
Throughout our paper, we consider the initial plane
electromagnetic wave that is normally incident to the
flat boundaries of the structure. The wave has the
magnetic induction vector parallel to the axes of
cylinders while the electric intensity vector
perpendicular to it.
As it has been shown in [4], the long-wave
approximation ( 1, kdka ) of S-parameters for the
above slab sandwich metamaterial structure for the
case 21~
r , 21~
r are defined by the
equalities:
;42
1~
1011
bb
kd
S (1)
,42
1
1
~
1021
bb
kd
S (2)
where
00
0
1
itI
b
;
kd
I
2
0 ;
0
00
a
i
Nt
;
ka ;
3
2
0
4
)(
202.1
4
781.1
log
2
kdkd
N
;
1
781.1
2
log21
4
1
2
1
1
24
1
4 2
2
2
22
2
0
K
KK
i
a
;
K ;
1
1
4 ikdt
kd
b
;
;
3
2
22
8
)(
202.1
1
)(3
4
kd
kd
N
;
1
201
a
i
NNt
;
4
2
1
i
a
.
781.1
2
log432
8
313
8
2
2
22
2
KKKK
KK
Here is defined by the metal model as
0
1)(
i
+ , (3)
where is the conductivity of the inclusion metal,
0 is the permittivity of vacuum.
Let us consider the above slab metamaterial
structure as a four-terminal network of sub-slabs with
the field functions FI, BI, FII, BII, Fig. 2 where eff
and eff are, respectively, the complex effective
dielectric and magnetic constants of the considered
slab for the case 21~
r , 21~
r , while eff~
and eff~ are, respectively, the complex effective
dielectric and magnetic constants of the lateral slabs.
Fig. 2. Representation of the presented metamaterial slab as a four-
terminal network of sub-slabs
The appropriate expressions of the complex
effective constants eff and eff are obtained in [4]
and are:
,
)
~~
(1
~~
log
1
1
2
,
)
~~
(1
~~
log
1
1
2
2111
2111
2111
2111
SS
SS
a
ic
SS
SS
a
ic
eff
eff
(4)
where 11
~
S , 21
~
S are S-parameters of the considered
metamaterial slab for the case 1~ , 1~ ;
1
1
eff
eff
Z
Z
,
eff
eff
effZ
is the normalized
complex effective characteristic impedance of the
metamaterial slab for the case 21~
r ,
21~
r , c is the velocity of light in vacuum.
The expressions of the complex effective
constants eff~ and eff~ are given by [10]:
,
2
3
1)(~
,
2
3
1)(~
1
1
s
e
e
s
reff
s
m
m
s
reff
F
bf
bf
F
F
bf
bf
F
(5a)
,
cossin1
cossin2
2
f (5b)
where
33 34 LrFs is the metal volume fraction
of the lateral slabs; L is the constant of unit cell of the
lateral slabs; 21 rrmb ; 1r is the magnetic
constant of matrix of the lateral slabs; 21 rreb ,
FII
BII
FI
BI
effeff ~,~ effeff ,
effeff ~,~
h h a2
O. N. Rybin et al. / Microwave Effective Electromagnetic…
_________________________________________________________________________________________________________________
21
1r is the electric constant of matrix of the lateral
slabs and 2r is exactly defined by the right side in
Eq. (3); .2rLc
If the complex constants eff and eff , and
S-parameters of the above four-terminal network are
known for the case 21~
r , 21~
r , then its
S-parameters for the case of arbitrary values of the
constants ~, ~ can be obtained by T-matrix of the
four-terminal network defined via the matrix equality
in [9]:
.
2221
1211
II
II
I
I
B
F
TT
TT
B
F
(6)
Let us obtain the elements of T-matrix in
Eq. (6). The elements can be obtained by multiplying
the field functions matrix for each sub-slab region of
the four-terminal network as follows:
,
0
0
)1()1(
)1()1(
0
0
)1()1(
)1()1(
0
0
~
~
11
11
2
2
11
11
~
~
2221
1211
hki
hki
effeffeff
effeffeff
aik
aik
effeffeff
effeffeff
hki
hki
eff
eff
eff
eff
eff
eff
e
e
RRR
RRR
e
e
RRR
RRR
e
e
TT
TT
(7)
where ck effeffeff ~~~
is the wavenumber in the
lateral slabs,
1
1
effeff
effeff
effR
is the complex
effective reflection coefficient at the plane
boundaries between the first and second slabs
(while effR is the complex effective reflection
coefficient at the plane boundaries between the
second and third slabs); ck effeffeff is the
wavenumber in the second sub-slab. Multiplying
matrices, gives:
,
1
1
2221
1211
2
2221
1211
aa
aa
RTT
TT
eff
(8)
where ;)(
~
2222
11
hkiaik
eff
aik effeffeff eeRea
;2sin212 akiRa effeff ;1221 aa
.)(
~
2222
22
hkiaik
eff
aik effeffeff eeRea
Taking in account the formula Eq. (8) and
the fact that
,
1
2221
1211
12
2211
2
12
12
11
12
22
12
TT
TT
S
SSS
S
S
S
S
S
(9)
gives:
.
2sin2
,
)1(
2
~
2
11
2
~
22
21
aik
eff
aik
eff
hki
eff
aik
eff
aik
hki
eff
effeff
eff
effeff
eff
eRe
akeiR
S
eRe
eR
S
(10)
Having the expressions of S-parameters of
the considered sandwich structure we can obtain its
expressions of the effective dielectric and magnetic
constants via the formulas obtained in [6]:
,
11
11
,
11
11
2
2
2
2
KK
KK
n
KK
KK
n
avav
avav
(11)
where
;...,2,1,0,
2
2ln
l
ha
liT
nav
(12)
;
11
1
2
2111
2
2111
KKSS
KKSS
T (13)
;
2
1
11
2
21
2
11
S
SS
K
(14)
and the phase is defined by:
ieTT . (15)
In our study, we consider the case 0l in
Eq. (12). In this case, the sandwich slab’s thickness
( ah 22 ) is less than the wavelength of initial
electromagnetic wave in the slab [6].
2. The Effective Parameters: The Results
of Modeling and Simulation. In this section we
evaluate the accuracy and applicability range of
mathematical model presented by the expressions
Eqs. (10)(15) for the case when 1d mm,
25.0a mm, 5.0L mm. In our case, the matrix
media are air to prevent losses due to the magneto-
dielectric ( ~~1 ). Cobalt was used as
material of the inclusions. In the case of saturation
magnetization for such Cobalt: 2502 r . All of the
calculations are done at the frequency equals to 1 GHz.
Let us compare the modeling results
obtained through the model (10)(15) with the
numerically obtained results. The numerical results
are obtained by FDTD simulations for evaluating the
complex S-parameters that are being used for
calculating the complex effective parameters via the
formulas are analogous to that of Eqs. (11)(15). The
numerical experiments are done by using the free
Meep FDTD software package. The comparisons are
shown in Fig. 34 where the real parts of the
effective constants are a function of metal volume
O. N. Rybin et al. / Microwave Effective Electromagnetic…
_________________________________________________________________________________________________________________
22
fraction of the lateral slabs ( 22 LrF ) for
different values of the lateral slabs thickness .h As, it
can be seen from the graphs; good agreement
between the analytical and numerical results is
observed on the low values of volume fraction of the
spherical inclusions ( 3.0F ) while it is clear that the
analytical results are not in good agreement with the
numerical ones as the volume fraction increases.
Moreover, it can be observed from the graphs, better
accuracy of the proposed mathematical model is
obtained for the case of long lateral slabs
( 5, llLh ) that is in good agreement with the main
idea of Effective Medium Theory (EMT) according
to which, the wave length of incident electromagnetic
wave must be sufficiently longer than the thickness
of scatterer [1113]. At the same time, the numerical
calculations have shown that the proposed analytical
model Eqs. (10)(15) is ineffective in the case when
.ar
0.05 0.10 0.15 0.20 0.25 0.30 0.35
1.62
1.64
1.66
1.68
1.70
1.72
1.74
1
2
R
ea
l
P
ar
t
o
f
th
e
E
ff
ec
ti
v
e
M
ag
n
et
ic
C
o
n
st
an
t
Volume Fraction
Fig. 3 Change of real part of the effective dielectric constant of the
metamaterial slab versus the metal volume fraction F of spherical
inclusions at 1 GHz: the line 1 is for the numerical simulations, the
line 2 is for the analytical modeling
0.05 0.10 0.15 0.20 0.25 0.30 0.35
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1
2
R
ea
l
P
ar
t
o
f
th
e
E
ff
ec
ti
v
e
D
ie
le
ct
ri
c
C
o
n
st
an
t
Volume Fraction
Fig. 4. Change of real part of the effective magnetic constant of the
metamaterial slab versus the metal volume fraction F of spherical
inclusions at 1 GHz: the line 1 is for the numerical simulations, the
line 2 is for the analytical modelling
As we can observe from the graphs in
Fig. 34, the enhancement of the real part of the
complex effective dielectric constant takes place with
respect to the value of the dielectric constant of the
matrix while the enhancement of the real part of the
complex effective magnetic constant is negligible.
Furthermore, the real part of the effective constants
increases with the volume fraction of the inclusions
this is the well known result in [4, 14]. Anyway,
comparing these results with the results in [4], we
conclude that the main contribution in the
enhancement of the constants is because of the
cylindrical inclusions. It is also interesting to mention
that the magnetization of the inclusions of the
considered structure does not lead to the phenomena
such as ultra-low index and negative magnetic
constant while the same was found in [15, 16]. This
means that the last two phenomena are normally
observed in the case of the inclusions of the same
geometry. Anyway the considered metamaterial
structure can be used can for designing high directive
patch antennas because of the enhancement of
effective dielectric constant [17].
In our paper, we are not going to present the
graphs for imaginary parts of the effective constants.
This is because both the analytical modeling and
numerical simulations have shown that the values of
the above mentioned parameters are not increasing in
the order of 310 which indicates low losses of the
considered metamaterial structure, that, in turn,
indicates a low level of damping of electromagnetic
waves in the presented metamaterial slab.
Conclusion. In this paper, the electromagnetic
response problem for 2-D sandwich metamaterial
structure as infinite two-component slab composite
with cylindrical metallic inclusions contained by
same infinite two-component slab composites with
spherical metallic inclusions is analytically solved in
the microwave frequency range. The above response
is found by means of applying S- and T-parameters
approaches to the solution of electromagnetic
response problem for the infinite chain of infinitely
long metallic cylinders periodically immersed in the
slab magneto-dielectric matrix. The case of Cobalt
inclusions magnetized to saturation is considered.
The enhancement of real part of the complex
effective dielectric constant with respect to the value
of dielectric constant of the matrix was found as a
function of the volume fraction of cylinder inclusions
while the enhancement of real part of the complex
effective magnetic constant is neglected.
A good agreement between the analytical
calculations and the numerical ones was found on the
low values of the metal volume fraction of spherical
inclusions.
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14. Zouganelis G. Two layer magnetodielectric metamaterial with
enhanced dielectric constant as a new ferrite like material /
G. Zouganelis, O. Rybin // Japanese Journal of Applied
Physics. – 2006. – 45, N 44. – P. L1175–L1178.
15. Rybin O. Microwave Effective Medium Approximation for
Two-Component Metamaterial with Saturated Cylindrical
Ferrite Like Metal Inclusions / O. Rybin, T. Nawaz //
Telecommunications and Radio Engineering. – 2009. – 68,
N 7. – P. 567575.
16. Rybin O. Unusual Properties of Two-Component
Metamaterial Medium With Metal Ferrite Like Inclusions In
the Microwave Frequency Range / O. Rybin, T. Nawaz,
A. Pitafi // The Seventh International Kharkov Symposium on
Physics and Engineering of Microwaves, Millimeter and
Submillimeter Waves: proc. / Karazin Kharkov National
University. Kharkov, 2010. – P. 1-3.
17. Study of ultra low index (0 < ε < 1) metamaterial using a patch
antenna / G .Zouganelis, F. Soma, O. Rybin, H. Ohsato // 18th
Fall Meeting of The Ceramic Society of Japan: proc. / Osaka
Prefecture University. Osaka, 2005. – P. 148.
О. Н. Рыбин, А. И. Питафи, С. П. Вялкина
ЭФФЕКТИВНЫЙ СВЧ-ОТКЛИК
СЭНДВИЧ-ПОДОБНОГО
МАГНИТНОГО КОМПОЗИТА
Изучено длинноволновое приближение для эффектив-
ного электромагнитного отклика двухмерной метаматериаль-
ной сэндвич-структуры, представленной как бесконечный
двухкомпонентный плоский композит с цилиндрическими
металлическими включениями, заключенный между Одина-
ковыми бесконечными двухкомпонентными композитами со
сферическими металлическими включениями. Выражения для
отклика получены путем обобщения выражений для
электромагнитного отклика бесконечной цепочки бесконечно
длинных металлических цилиндров, периодически вставлен-
ных в плоскую магнитодиэлектрическую матрицу. Обобще-
ние сделано с помощью теории S- и T-матриц. Рассмотрен
случай ферритоподобных металлических включений, намагни-
ченных до насыщения. Проведена оценка точности получен-
ных аналитических выражений путем сравнения результатов
аналитического моделирования с результатами численного
эксперимента.
Ключевые слова: метаматериалы, электромагнит-
ный отклик, эффективные параметры, метод эффективной
среды, СВЧ-диапазон.
О. М. Рибін, А. І. Пітафі, С. П. Вялкіна
ЕФЕКТИВНИЙ НВЧ-ВІДГУК
САНДВІЧ-ПОДІБНОГО
МАГНІТНОГО КОМПОЗИТУ
Вивчено довгохвильове наближення для ефектив-
ного електромагнітного відгуку двовимірної метаматериаль-
ної сендвіч-структури, поданої як нескінчений двох-
компонентний композит з циліндричними металічними
включеннями, що замкнений між однаковими нескінченними
двохкомпонентними композитами, зі сферичними металіч-
ними включеннями. Вирази для відгуку отримано шляхом
узагальнення виразів для електромагнітного відгуку не-
скінченного ланцюга нескінченно довгих металічних
циліндрів, які періодично вставлені в плоску магніто-
діелектричну матрицю. Узагальнення здійснено за допомогою
S- і T-матриць. Розглянуто випадок феритоподібних вклю-
чень, що намагнічені до насичення. Проведено оцінку
точності отриманих аналітичних виразів, порівнюючи резуль-
тати аналітичного моделювання з результатами чисельного
експерименту.
Ключові слова: метаматеріали, електромагнітний
відгук, ефективні параметри, метод ефективного середовища,
НВЧ-діапазон.
Рукопись поступила 30.11.10 г.
|
| id | nasplib_isofts_kiev_ua-123456789-78033 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1028-821X |
| language | English |
| last_indexed | 2025-12-07T18:41:43Z |
| publishDate | 2011 |
| publisher | Інститут радіофізики і електроніки ім. А.Я. Усикова НАН України |
| record_format | dspace |
| spelling | Rybin, O.N. Pitafi, A.I. Vyalkina, S.P. 2015-03-10T15:33:48Z 2015-03-10T15:33:48Z 2011 Microwave effective electromagnetic response of sandwich like magnetic composite / O.N. Rybin, A.I. Pitafi, S.P. Vyalkina // Радіофізика та електроніка. — 2011. — Т. 2(16), № 1. — С. 19-23. — Бібліогр.:17 назв. — англ. 1028-821X https://nasplib.isofts.kiev.ua/handle/123456789/78033 537.8.029.6 In this study, the long wave approximations for the effective electromagnetic response of 2-D sandwich metamaterial structure, as 
 infinite two-component flat composite with cylindrical metallic inclusions contained by the same infinite two-component flat composites 
 with spherical metallic inclusions are obtained. The expressions for the response are obtained by generalizing the expressions of the 
 electromagnetic response of the infinite chain of infinitely long metallic cylinders periodically immersed in the flat magneto-dielectric 
 matrix. The generalization has been done by following the approaches of S- and T-matrices. The case of ferrite like metallic saturated 
 inclusions is considered in the study. The analytically obtained results are compared with the numerically calculated ones. Изучено длинноволновое приближение для эффективного электромагнитного отклика двухмерной метаматериальной сэндвич-структуры, представленной как бесконечный двухкомпонентный плоский композит с цилиндрическими 
 металлическими включениями, заключенный между Одинаковыми бесконечными двухкомпонентными композитами со 
 сферическими металлическими включениями. Выражения для 
 отклика получены путем обобщения выражений для 
 электромагнитного отклика бесконечной цепочки бесконечно 
 длинных металлических цилиндров, периодически вставленных в плоскую магнитодиэлектрическую матрицу. Обобщение сделано с помощью теории S- и T-матриц. Рассмотрен 
 случай ферритоподобных металлических включений, намагниченных до насыщения. Проведена оценка точности полученных аналитических выражений путем сравнения результатов 
 аналитического моделирования с результатами численного 
 эксперимента. Вивчено довгохвильове наближення для ефективного електромагнітного відгуку двовимірної метаматериальної сендвіч-структури, поданої як нескінчений двохкомпонентний композит з циліндричними металічними 
 включеннями, що замкнений між однаковими нескінченними 
 двохкомпонентними композитами, зі сферичними металічними включеннями. Вирази для відгуку отримано шляхом 
 узагальнення виразів для електромагнітного відгуку нескінченного ланцюга нескінченно довгих металічних 
 циліндрів, які періодично вставлені в плоску магнітодіелектричну матрицю. Узагальнення здійснено за допомогою 
 S- і T-матриць. Розглянуто випадок феритоподібних включень, що намагнічені до насичення. Проведено оцінку 
 точності отриманих аналітичних виразів, порівнюючи результати аналітичного моделювання з результатами чисельного 
 експерименту. en Інститут радіофізики і електроніки ім. А.Я. Усикова НАН України Радіофізика та електроніка Мікрохвильова електродинаміка Microwave effective electromagnetic response of sandwich like magnetic composite Эффективный СВЧ-отклик сэндвич-подобного магнитного композита Ефективний НВЧ-відгук сандвіч-подібного магнітного композиту Article published earlier |
| spellingShingle | Microwave effective electromagnetic response of sandwich like magnetic composite Rybin, O.N. Pitafi, A.I. Vyalkina, S.P. Мікрохвильова електродинаміка |
| title | Microwave effective electromagnetic response of sandwich like magnetic composite |
| title_alt | Эффективный СВЧ-отклик сэндвич-подобного магнитного композита Ефективний НВЧ-відгук сандвіч-подібного магнітного композиту |
| title_full | Microwave effective electromagnetic response of sandwich like magnetic composite |
| title_fullStr | Microwave effective electromagnetic response of sandwich like magnetic composite |
| title_full_unstemmed | Microwave effective electromagnetic response of sandwich like magnetic composite |
| title_short | Microwave effective electromagnetic response of sandwich like magnetic composite |
| title_sort | microwave effective electromagnetic response of sandwich like magnetic composite |
| topic | Мікрохвильова електродинаміка |
| topic_facet | Мікрохвильова електродинаміка |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/78033 |
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