Vol1no5page630to633 Artificial Mag Metamat n .kn b.kb.kjby Tl

8/13/2019 Vol1no5page630to633 Artificial Mag Metamat n .kn b.kb.kjby Tl

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630 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26

SRRs Articial Magnetic Metamaterials ModelingUsing Transmission Line Theory

Qun Wu , Ming-Feng Wu , and Fan-Yi MengHarbin Institute of Technology, China

Jian WuNational Key Laboratory of Electromagnetic Environment, China

Le-Wei LiNational University of Singapore, Singapore

Abstract

This paper proposes a novel theoretical equivalent circuit model to characterize the split ring resonator(SRR), based on the analysis of the individual SRR geometry conguration and the distribution relationship of the electric eld in it. Then it becomes very desirable to analytically calculate the corresponding inductancesand capacitance of the equivalent circuit, which determine the resonance frequencies of the SRR. Numericalsimulations are used to verify the proposed equivalent circuit model. It has been proven that the results obtainedby the equivalent circuit model are in good agreement with the numerical simulation tools. Furthermore, theSRR has more than one resonance frequencies according to the equivalent circuit model, which indicates thatthere may be more than one frequency region with the negative effective permeability.

Introduction

The negative value of effective permeability was rst reported using an array of the metallic geometrycalled Split Rings Resonators (SRRs) [1], which function as electrically small resonant particles. Many researchworks have been conducted to characterize the performances of the SRRs by using numerical methods andexperimental tools [2 6]; however, there are still some aspects to be improved. Firstly, either numerical methodsor experimental tools are demanded too much, it is disadvantage to popularize in applications. Secondly, eithernumerical methods or experimental methods are not accomplished in improving performances or designing

a new geometry, although they are efficient enough to analyze a prepared geometry. Lastlypresent methodsalways demand the incident wavelength to be more larger than the SRR dimensions, which restricts the appliedfrequency region.

Equivalent model methods, especially the equivalent circuit models based on transmission line theory de-scribed here are promising to probe the micro electromagnetic effects of the individual SRR or counterpart byequivalent models, and its geometry improvement according to RF theories. This paper proposes an approachto analyze an individual SRR by a novel equivalent circuit model.

The Equivalent Circuit Model

Due to the periodicity of the SRRs arrays, we will look into the performances of an individual SRR, instead of investigating the entire SRRs arrays effective behaviors. Fig.1 shows the distribution relationship of the electriccharge and electric current in the individual SRR, while an incident plane wave magnetic eld H i is penetratingthrough it. According to the distribution of the electric charge and electric current, we develop the correspondingequivalent circuit model described in Fig.2, Ax , Ay and Az mean the dimensions along x, yand zdirections,respectively. And then the constitutive parameters can be deduced and extracted further from it, which havebeen proposed in [7]. Because of the complex nature of the SRR, which produces effective permeability throughproducing strong magnetic material-like responses, the sticking points of the equivalent circuit model are toestimate the resonance frequencies and characterize electromagnetic properties in the frequency region withnegative effective permeability. Here we will propose how to analyze and calculate the parameters in theequivalent circuit model. In the model, inductance Lx and capacitance C z respectively means the distributedinductance and capacitance per unit length of the equivalent transmission line segment, and can be expressed asLx = 0 al and C z =

0 la [8]. Here 0 and 0 are vacuum permittivity and permeability, respectively, and a and l

mean the lattice constant and the layer constant between the investigated SRR with others, respectively. There


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Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26 631

are still several inter-parameters in the above model, namely R1 , R2 , L1 , L2 , M , C 1 , C 2 . The SRR structuresconsidered here consists of two concentric rings, each with gaps, and the gaps appearing on opposite sides. Theparameters C 1 and C 2 represent the capacities with the outer gap and the inner gap, respectively. C c means thecapacity between the outer ring and the inner ring. L1 and L2 represent the self inductance of the outer splitring and the inner split ring, respectively, and the L2 could be divided into two equal parts namely L21 and L22for balance considerations. M means the mutual inductance between L1 and L2 . Their values are dependent on

the physical model of the individual circle SRR, and can be respectively estimated as followed by a quasi-staticapproach [9 11]:

Figure 1: The geometry conguration of the individ-ual SRR

Figure 2: The equivalent circuit model of the indi-vidual SRR developed from Fig. 1

R21 = R22 = 12 R2 (2a)L21 = L22 = 12 L2 (2b)

R1(2) = 2r out ( in ) /dA y (2c)

L1(2) = 0 r out ( in ) [ln(2r out ( in )

d ) + 0 .9 + r 2out ( in )

20d 2 ] (2d)

= 2d + g( r out + r in ) (2e)

M = 0( rout

+ rin

2 [ln(1 ) 0.6 + 0 .72 + (0 .2 + 112 2 ) d2

( r out + r in ) 2 ]) (2f)C 1(2) = 2.20dAy /p (2g)

C c = [0.06 + 3.5 10 5(r out + r in )] ( pF ) (2h)

Numerical Simulations and Discussions

Two numerical simulations are used for comparison from both CST Microwave Studio (MWS) and AdvancedDesign Systems (ADS) package in order to validate the equivalent circuit model. Firstly, CST Microwave Studiosimulation tool is used to consider prediction of the resonance frequency of the individual SRR. The simulationmodel consists of a two-port waveguide lled with air, and the individual SRR is centered in the waveguide.The geometry parameters of the SRR are set the same as reference [1] for convenience: r= 2.0 mm, d = 1.0mm, g = 0.1 mm, p= 0.5 mm, a=10 mm, l=2 mm. The input wave is lunched in free space toward the SRRat normal incident from each port. A variety of Reection and Transmission coefficient results are given for thescattering from the SRR showed in Fig.3, which can be interpreted in terms of the SRR having effective negativepermeability. It indicates that the effective permeability becomes negative, when the medium embedded witha SRR becomes reective at some frequency regions according to presented theories.

The special resonance frequency of the SRR which leads to negative permeability can be determined by themagnitude and phase of the S21. Because on the one hand the wave cant propagate in the air waveguide withSRR, while the effective permeability is negative; and on the other hand there will be discontinuity in the phaseof the S21 while the resonance leads to negative permeability. From Fig.3(a) one can see strong reectivityat two frequency regions, and their center frequencies are near 2.5 GHz and 13.5GHz respectively. Fig.3(b)


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632 Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26

demonstrates the phases of the S parameters predicted by MWS, and note that the phase of S 11 goes throughZero at both center frequencies, hence the SRR is responding like a PMC at the two frequencies points.

(a) MWS-predicted S parameters for the individualSRR

(b) MWS-predicted S parameters phases for the indi-vidual SRR

(c) The equivalent circuit model-predicted S param-eters for the individual SRR simulated by ADS, theline with additional marks is S 11 .

(d) The equivalent circuit model predicted S parame-ters phases for the individual SRR simulated by ADS,the dash line is phase ( S 11 ).

Figure 3: (a)-(d)

Then ADS is applied to simulate the equivalent circuit model showed in Fig.2. The series parameters inthe equivalent circuit model can be calculated through the approach demonstrated in expressions (2). Thecorresponding scattering results and phases are plotted in Fig.3(c) and Fig.3(d), one can see strong reectivityat two frequency regions too, and their center frequencies are near 1.8 GHz and 13.5GHz respectively. Andthe phase of S 11 goes through Zero at both center frequencies too, hence the SRR is responding like a PMC atabove two frequencies points too. These simulation results show that there is very good agreement between theCST Microwave Studio simulations results and the equivalent circuit model values.

Lastly, we apply the Pendry et al s model developed in reference [1], which expressed as equation (3) tocharacterize the SRR, we can get only one resonance frequency at 13.5 GHz which equal to the second resonancefrequency of above simulations, and the resonance does produce the negative effective permeability.

eff = 1 r 2

a21 +

2lr 0

i 3lc20

2 ln 2d

g r3

(3)

Above all, we can extracted from the simulation results that the individual SRR has more than one resonancefrequency, and becomes reective at more than one frequency regions, which has more information than theresults presented in reference [1]. Whether about 2 GHz is another special resonance frequency needs exper-imental tools to conrm. Nonetheless, simulations results have demonstrated the efficiency of the equivalentcircuit model.


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Vol1no5page630to633 Artificial Mag Metamat n .kn b.kb.kjby Tl

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