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2020 International Scientific-Practical Conference
Problems of Infocommunications. Science and Technology
PIC S&T
'2020
978-x-xxxx-xxxx-x/20/$31.00 ©2020 IEEE
Parametric synthesis of 3D structure of SRR
element of the metamaterial
Ihor Sliusar
Department of information systems and technologies
Poltava State Agrarian Academy
Poltava, Ukraine
islyusar2007@ukr.net
Vadym Slyusar
Central Research Institute of Armaments and Military
Equipment of Armed Forces of Ukraine
Kyiv, Ukraine
swadim@ukr.net
Yurii Utkin
Department of information systems and technologies
Poltava State Agrarian Academy
Poltava, Ukraine
1008utkin@gmail.com
Olena Kopishynska
Department of information systems and technologies
Poltava State Agrarian Academy
Poltava, Ukraine
17elenak@gmail.com
Abstract—Variants of split ring resonators (SRR) models
are proposed. They are considered as unit cells of DNG
metamaterials. The synthesized SRR variants are based on
the implementation of 3D geometry. Efficiency assessment is
carried out on the base of determination of the relative
bandwidth δ fDNG, in which the DNG properties are observed.
Since metamaterials are complex composite structures,
instead of analytical calculations, the Finite Element Method
(FEM) is used to estimate the electromagnetic properties of
SRR. The adequacy of the proposed SRR model is confirmed
by the coincidence of the obtained results in the particular
case with the corresponding estimates of the base model. At
the same time, the influence of the geometric parameters of
the SRR components (orientation and size of the conductor;
axial rotation of the rings, subtract material) on the
effectiveness of the decisions was investigated. The use of 3D-
based constituent elements in SRRs allowed us to achieve a
10 times increase in δ fDNG compared to the base prototype.
Keywords—double negative; metamaterial; split ring
resonator
I. INTRODUCTION
Telecommunication miniaturization trends have
intensified the search for approaches to the creation of
Eletrically Small Antennas (ESA) [1]. At the same time,
the technologies of microstrip antennas have reached their
limits in terms of reducing the dimensions of microwave
devices. As a result, one of the promising areas of ESA
design is the use of metamaterials.
As is known [2, 3], metamaterials as substrates for
printed miniature antennas can reduce the size of the
emitters, increase their passband and radiation efficiency.
Metamaterials that contain only one negative
electromagnetic parameter (εr or μ) are called Single
Negative (SNG). However, the greatest interest is shown in
the creation of metamaterials with a negative refractive
index of electromagnetic waves, which are characterized
simultaneously by negative values of the dielectric constant
and magnetic permeability (εr < 0, µ < 0) [2, 3]. Such
materials are called Double Negative (DNG) [2, 3].
At the same time, the main task of improving of the
metastructures is the synthesis of such environment that
would have minimal losses and almost insignificant
dispersion properties, as well as provide a wide frequency
band corresponding to DNG.
II. ANALYSIS OF RECENT STUDIES AND PUBLICATIONS,
WHICH DISCUSS THE PROBLEM
To identify a metamaterial as a DNG, it is necessary to
evaluate its electromagnetic properties. Such an estimate
comes down to an analysis of the sign of the real part of the
quantities εr and µ. The following relations should be used
[4]:
εr = n / z и µ = n ∙ z, (1)
where n is the refractive index; z is the wave impedance.
In turn, to obtain the values of n and z, you can use the
formulas [5]:
, (2)
, (3)
where k0 = 2πf/c, с – speed of light, f – frequency, d –
linear size of the metamaterial unit cell, Sxy – S-parameters
of dispersion matrix [5].
As a classic example of the metastructure unit cell, on
the basis of which a negative refractive index can be
achieved (accordingly, a DNG frequency range can exist),
the Split Ring Resonator (SRR) should be mentioned [2, 3].
Such a structure is described more in detail in [6].
The systematization of existing sources related to this
topic indicates that the studies of the joint use of microstrip
or patch antennas and metamaterials are dominant [4, 7-
11].
Unfortunately, metamaterials based on circular or
rectangular SRRs are characterized by a narrow DNG
frequency band, high level of electromagnetic losses, etc
[6]. To make up for flaws, it is possible to form structures,
PIC S&T
'2020
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Kharkiv, Ukraine
in which perpendicular arrangement of metamaterial cells
based on printed SRRs is used [12] or several layers based
on them [6] are implemented.
However, to reveal all the possibilities for integrating
ESA technologies and metamaterials, it is advisable to
conduct an open optimization of the parameters of SRR
structural elements, for example, through the use of 3D
geometry. A similarity of such structures may be a design
option from [13].
Considering that metamaterials are complex composite
structures, analytical solutions to assess their properties
become inapplicable. As a result, the role of fixed assets in
the process of metamaterials designing is assigned to
numerical modeling. In this case, the most common is the
use of the Finite Element Method (FEM), successfully
tested by the authors, for example, in [14-16].
III. THE AIM OF RESEARCH
Thus, the aim of the work is to increase the efficiency
of metamaterial using SRR by modifying its geometry.
IV. THE MAIN RESULTS OF THE STUDY
In this paper, a printed SRR [7] was used as a
prototype, the model of which is presented in HFSS [7,
17] – Fig. 1.
Fig. 1. Printed SRR as a base model.
The indicated SRR variant contains a composite epoxy
material substrate (in HFSS this material is FR4). On one
of its surfaces there are metal split “rings”, and on the
other – conductor. Since the model of such an SRR is one
unit cell, which should be in a multi-element system, the
formation of limit conditions in the form of a “Master /
Slave” boundary on the x- and y-edges of the model will
be most acceptable (Fig. 2.a) [7]. This area is very critical
for determining the correct location of the excitation ports,
which should be located quite far from the near fields
induced on the SRR structure. This ensures that the
dispersion parameters are calculated correctly. Since the
exact calculation of the S-parameter phase is important for
an efficient parameter estimation [7], the ports are
displayed on the inner surface of the box describing the
boundary conditions (blue arrows in Fig. 2.b). Herein, a
modification of wave ports is used, which is called
“Floquet Ports” [7], since in these ports the harmonicа of
Floquet with zero indices propagates (it is often called the
main harmonicа). It has a field structure that matches the
field of the incident wave. In this case, higher order
harmonics are not excited.
а)
b)
Fig. 2. Formation of the calculation area: a) – “Master / Slave” boundary
conditions; b) – Floquet Ports [7].
Based on the considered model, the authors
synthesized the SRR variant based on the implementation
of 3D geometry (Fig. 3). The following situations were
considered as introduced assumptions:
– SRR “rings” and conductor made of copper; the
orientation of the wire and the cuts of the “rings” of the
SRR are identical to the base model;
– the geometric dimensions of the SRR components
according to Ox and Oz coincide with the same parameters
of the base model;
– subject to change – the dimensions of the SRR
components along the Oy axis;
– to correctly compare the properties of the basic and
synthesized SRR models, its dimensions should not extend
beyond the figure (parallelepiped) describing the boundary
conditions of the basic model;
– analysis is carried out in the range 0.1 ÷ 20 GHz;
2020 International Scientific-Practical Conference
Problems of Infocommunications. Science and Technology
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'2020
– conditions for the calculations coincide with the
base model;
– performance assessment is based on determining the
relative bandwidth, for which the conditions εr < 0, µ < 0
are valid.
Fig. 3. Synthesized model No. 1 of metamaterial cell.
There are several options for interpreting of the concept
of “relative bandwidth”. Following [14, 15], we further use
the definition that corresponds to the expression:
, (4)
where f1 and f2 are the limit values of the frequencies of the
range characterized by the condition: {Re(εr) < 0 and
Re(µ) < 0}, Δf = f2 – f1.
To check the adequacy of the proposed model No. 1
(Fig. 3) with respect to the base, the substrate material
“FR4 epoxy” (εr = 4.4) was selected and the first
calculation was carried out with the values of the variables:
t_copper = Wide = 0,017 mm. The results obtained
coincided with the corresponding estimates of the base
model (δ fDNG = 0.19). At the same time, a new graph
(Fig. 4) was formed as the result, which shows the
frequency dependences of Re(εr) and Re(µ.
Fig. 4 Evaluation of the characteristics of model SRR No. 1: a) – Re(εr);
b) – Re(µ).
Further, the dependence of δ fDNG on the geometric
dimensions of the conductor and SRR rings along the Oy
axis was studied. First, the situation of a synchronous
increase in their size was considered (Table I), and then the
size of the SRR rings was fixed and the dimensions of the
conductor were changed along the Oy axis. The
corresponding results are presented in Table II. The
greatest effect is observed at t_copper = Wide = 1 mm, for
which δ fDNG = 0.335 was obtained (Fig. 5). It can be
assumed that a further increase in the size of the cell will
lead to the expansion of the DNG region.
Fig. 5. DNG SRR estimate at t_copper = Wide = 1 mm: a) – Re(εr);
b) – Re(µ).
The analysis indicates an increase in the width of the
DNG frequency range as the dimensions of the SRR
constituent elements along the Oy axis increase.
TABLE I. DNG BANDWIDTH ESTIMATION
Variable structures,
mm
Boundary
frequencies, GHz
Bandwidth
t_copper
Wide
f1
f2
∆f, GHz
δ fDNG
0.017
0.017
12.32
14.30
1.98
0.149
0.0358
0.0358
12,32
14.30
1.98
0.149
0.1
0.1
12.17
14.70
2.53
0.19
0.2
0.2
11.92
14.89
2.97
0.222
0.3
0.3
11.73
14.89
3.16
0.237
0.4
0.4
11.51
14.86
3.35
0.254
0.5
0.5
11.34
14.86
3.52
0.269
0.6
0.6
11.21
14.92
3.71
0.284
0.7
0.7
11.04
14.89
3.85
0.297
0.8
0.8
10.92
14.97
4.05
0.313
0.9
0.9
10.78
14.97
4.19
0.325
1
1
10.67
14.97
4.3
0.335
TABLE II. THE RESULTS OF CALCULATIONS OF THE FREQUENCY
DOMAIN DNG WHEN CHANGING THE SIZE OF THE CONDUCTOR AND
FIXING THE SIZE OF THE SRR RINGS ALONG THE AXIS OY
Variable structures,
mm
Boundary
frequencies, GHz
Bandwidth
t_copper
Wide
f1
f2
∆f, GHz
δ fDNG
0.1
0.7
11.33
12.22
0.89
0.076
0.2
11.26
12.96
1.7
0.14
0.4
11.20
14.13
2.93
0.213
0.6
11.10
14.86
3.76
0.29
The next step in the study was to assess the effect of the
substrate material. In addition to the classic FR4, several
material options were considered: Rogers RO3003 (εr = 3);
Rogers RO3006 (εr = 6.15); Rogers RO3010 (εr = 10.2);
and vacuum. By the way, using of vacuum as the substrate
material allows us to transform model No. 1 to the variant
shown in Fig. 6.
Herein, the values of the variables
t_copper = Wide = 0.2 mm were taken. Corresponding
estimates of the length of the DNG frequency domain are
shown in Fig. 7-10 and are presented in Table III.
The obtained results suggest the possibility of a slight
decrease in the frequency range of the DNG domain due to
the use of a substrate material with a higher εr value.
PIC S&T
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Fig. 6. Version of the model SRR without substrate.
TABLE III. ESTIMATION OF THE DNG FREQUENCY DOMAIN WHEN
THE SRR SUBSTRATE MATERIAL CHANGES
Material
(εr)
Variable
structures, mm
Boundary
frequencies,
GHz
Bandwidth
t_copper
Wide
f1
f2
∆f, GHz
δ fDNG
Vacuum (1)
0.2
0.2
14.10
17.59
3.49
0.22
RO3000
(tm) (3)
0.2
0.2
12.61
15.71
3.1
0.22
FR4_epox
y (4.4)
0.2
0.2
11.92
14.89
2.97
0.22
RO3006
(tm) (6.15)
0.2
0.2
11.45
14.22
2.77
0.22
RO3010
(tm) (10.2)
0.2
0.2
10.63
12.94
2.31
0.2
Fig. 7. Estimation of the DNG frequency domain for the Rogers RO3003
substrate material: a) – Re(εr); b) – Re(µ).
Fig. 8. Estimation of the DNG frequency domain for the Rogers RO3006
substrate material: a) – Re(εr); b) – Re(µ).
Considering the presence of an SRR modification
without a substrate (see Fig. 6), it is advisable to analyze
the influence of the angle of inclination of the SRR rings
around the Ox axis on the frequency properties of the
metamaterial unit cell model (Fig. 11) for the same
variables: t_copper = Wide= 0.2 mm. It is significant that
an increase in these values may lead to contact between the
SRR rings, which is unacceptable. The research results
were systematized and summarized in Table IV. Thus, the
inclination of the SRR rings around the Ox axis did not
give a significant improvement in the properties of the
metamaterial cell according to the maximum criterion
δ fDNG.
Fig. 9. Estimation of the DNG frequency domain for the Rogers RO3010
substrate material: a) – Re(εr); b) – Re(µ).
Fig. 10. Estimation of the DNG frequency domain for the model shown
in Fig. 6: a) – Re(εr); b) – Re(µ).
Fig. 11. Model No. 2, with AN option of independent selection of the
angle of inclination of the outer and inner rings of the SRR.
TABLE IV. RESULTS OF CALCULATIONS OF THE FREQUENCY AREA
DNG WHEN CHANGING AN ANGLE OF THE RINGS SRR AROUND THE OX
AXIS
Angles of
inclination,
deg
Variable
structures, mm
Boundary
frequencies,
GHz
Bandwidth
A1
A2
t_copper
Wide
f1
f2
∆f
δ fDNG
0
0
0.2
0.2
13.92
18.06
4.14
0.259
4
0
0.2
0.2
13.47
17.52
4.05
0.261
5
0
0.2
0.2
13.96
18.19
4.23
0.263
10
0
0.2
0.2
14.07
18.15
4.08
0.253
0
-5
0.2
0.2
14.03
17.99
3.96
0.247
0
5
0.2
0.2
14.03
17.99
3.96
0.247
5
5
0.2
0.2
14.16
17.99
3.83
0.238
15
-15
0.2
0.2
15.31
17.61
2.3
0.14
-15
15
0.2
0.2
15.51
17.07
1.56
0.096
5
-5
0.5
0.2
13.86
18.12
4.26
0.267
A further area of research was the determination of the
influence of the orientation of the conductor relative to the
SRR rings. At the same time, it was located in the xOy
plane, and it rotated around the Oz axis by 90 degrees
(Fig. 12). This approach has led to an expansion in
bandwidth in the DNG region (Fig. 13), which indicates its
effectiveness. In this case, the value ∆f = 8.64 GHz
(δ fDNG = 1.13) was achieved. For a given SRR arrangement
in Fig. 14 shows the frequency dependence of Re(n).
2020 International Scientific-Practical Conference
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'2020
Fig. 12. Model SRR No. 3.
Fig. 13. Assessment of the DNG frequency domain for model No. 3:
a) – Re(εr); b) – Re(µ).
Fig. 14. Evaluation of the real component of the refractive index Re(n)
for model No. 3.
Hereinafter, we analyzed the influence of changes in
the dimensions of the conductor, its placement in relation
to the center of the SRR, as well as the previously
considered versions of the SRR models, but taking into
account the changed spatial orientation of the conductor.
In addition, the SRR model was investigated with rings
rotated by 90 degrees around the Oy axis (Fig. 15), with
sections of the SRR frames located on the Oz axis. This
allowed us to obtain ∆f = 10.78 GHz and δ fDNG = 1.9
(Fig. 16).
Fig. 15. Model SRR No. 4.
Fig. 16. Assessment of the frequency domain DNG for model No. 4: a) –
Re(εr); b) – Re(µ).
Considering an obtained effect, the SRR layout option
was studied in the most detail, which differed by the
deviation of the rings section in relation to the Oz axis by
angles of up to ± 30 degrees (Fig. 17). In this case,
fluctuations in the value of δ fDNG occurred. As a result, the
maximum effect was observed at the deviation angle of 20
degrees (Fig. 18).
Fig. 17. Model SRR No. 4 (with variation of the inclination angle of the
rings rupture in relation to the axis Oz).
Fig. 18. Assessment of the DNG frequency domain for model No. 4 with
an inclination angle of the cut of rings 20 degrees: a) – Re(εr); b) – Re(µ).
To increase ∆f, it is possible to introduce mutual
rotation of SRR rings around the Oz axis. For instance, for
the model shown in Fig. 19, this made it possible to
provide an additional increment of ∆f at 0.5 GHz.
Fig. 19. SRR with mutual rotation of the rings around the axis Oz (top
view).
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In general, the estimates of the synthesized SRR
models allow us to conclude that for the expansion of the
DNG frequency regions and their simultaneous shift down
in frequency, it is advisable to perform mutual
displacement between the SRR rings. However, it should
be considered that such a shift is acceptable only within
certain limits.
V. PERSPECTIVES OF FURTHER RESEACH
It is advisable to focus further research on studying of
the electromagnetic parameters of SRR structures, in which
the constituent elements are modeled on the basis of
fractals [18] or antifractals.
Another direction of designing such SRRs may be the
combination of square frames and round rings, as well as
the introduction of various types of dielectric inserts into
the SRR, which are based on dielectric resonator antenna
technologies [16].
VI. CONCLUSIONS
The study of the electromagnetic properties of the
synthesized SRR variants was carried out taking into
account the influence of geometric parameters (orientation
and size of the conductor; axial rotation of the rings,
substrate material).
Compared with the prototype [7], the synthesized SRR
models allow one to obtain a 10-fold increase in δ fDNG с
0.19 to ≈1.9.
SRR with the conductor placed in the xOy plane and its
axis oriented perpendicular to xOz is an effective solution
that has led to the expansion of the DNG bandwidth in the
low frequency region.
The use of a conductor with a rectangular cross section
is advisable if its larger side is oriented perpendicular to the
Floquet Ports.
When determining the optimal angle of inclination of
the section plane of the SRR frames in relation to the yOz
plane, it is advisable to choose it within the range of 0 ÷ 90
degrees.
If we consider the synthesized SRR model as a
template for creating a metastructure, then the cut width of
the frames should be equal to the double thickness of the
rings.
To expand the frequency domain of the DNG and
simultaneously shift it down in frequency, it is advisable to
perform a mutual offset between the SRR rings. However,
it should considered that it only makes sense to do this
within certain limits.
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