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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,

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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

PIC S&T

'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

'2020

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Kharkiv, Ukraine

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

Problems of Infocommunications. Science and Technology

PIC S&T

'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).

PIC S&T

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October 6-9, 2020

Kharkiv, Ukraine

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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