EQUIVALENT CIRCUIT OFCOMPLEMENTARY SPLIT-RINGRESONATOR LOADED TRANSMISSIONLINE

Xin Hu,1,2 Qiaoli Zhang,2 Zhili Lin,3 and Sailing He1,2

1 Division of Electromagnetic Engineering, School of ElectricalEngineering, Royal Institute of Technology, S-100 44 Stockholm,Sweden2 Centre for Optical and Electromagnetic Research, ZhejiangUniversity, 310058 Hangzhou, China3 Department of Microelectronics and Applied Physics, Royal Instituteof Technology, SE-164 40 Kista, Sweden; Corresponding author:sailing@kth.se

Received 26 January 2009

ABSTRACT: In this article, a circuit model is proposed for the com-plementary split-ring resonators (CSRRs) loaded transmission line, andcomparisons between the results derived from the equivalent circuitmodel and the experimental results are given and a good agreementbetween them over a wide frequency band supports the effectiveness ofthe proposed modeling methodology. Both the results show the negativepermittivity at the vicinity of the resonance frequency of CSRR. By com-paring the results of different periodic length (i.e., the capacitance per-unit-cell of the host microstrip line is changed), the validity of theequivalent circuit is further confirmed. © 2009 Wiley Periodicals, Inc.Microwave Opt Technol Lett 51: 2432–2434, 2009; Published online inWiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24625

Key words: circuit model; CSRR; microstrip line; metamaterial

1. INTRODUCTION

The possibility to obtain media with simultaneously negative per-meability and permittivity (also called left-handed metamaterials(LHMs)) was firstly hypothesized by Veselago in the late 60s [1].Inspite of the interesting properties in such media, it was not until2000 that the first experimental evidence of such a medium wasdemonstrated [2]. The original medium proposed in Ref. 2 consistsof a bulky combination of metal wires and split ring resonators(SRRs) [3], and the frequency-selective property (band rejection)of SRRs [3–5] was interpreted as due to the negative effectivepermeability of the structure. By virtue of the distributed capaci-tance between concentric rings and overall rings inductance, SRRbehaves as an LC resonant tank that can be excited by an externalmagnetic flux. Thereafter, another key particle has been proposedfor metamaterial design, namely, the complementary split-ringresonator (CSRR), which is the negative image of an SRR and isshown in Figure 1 [6, 7]. The authors have demonstrated thatCSRRs etched in the ground plane of planar transmission mediaprovide a negative effective permittivity to the structure [8]. SinceSRRs and CSRRs are both planar configurations, SRRs and CS-RRs (properly combined with shunt metallic wires or series gaps)have been successfully applied to the design of novel planarmicrowave circuit and devices [9–11] based on the equivalentcircuit model [7].

In this article, a new circuit model for the CSRR loadedmicrostrip line is proposed and analyzed. In section 2, the topologyof the CSRR loaded microstrip line and the equivalent circuitmodels are presented. The validity of the equivalent circuit isfurther confirmed by comparing the results when the periodicvaries. Finally, the main conclusions are highlighted.

2. THEORY

2.1. Equivalent Circuit ModelThe unit cell of the CSRR loaded transmission lines with CSRRsetched in the ground plane is shown in Figure 2(a). Due to thesmall electrical dimensions of CSRRs at resonance, the unit cellcan be described by means of lumped-element equivalent circuits.As indicated in Ref. 6, CSRRs are mainly excited by the electricfield induced by the line, so we can derive its equivalent-circuitmodel shown in Figure 2(b), where L and C are the per-unit-cellinductance and capacitance of the host transmission line, whereasCSRRs are modeled as a resonant tank (with inductance LC andcapacitance CC) electrically coupled to the line through CM.

The equivalent impedance of the parallel branch can be sim-plified to the circuit in Figure 3 [12, 13], where

L�c �Cc

�2C2M;C�c � Lc�

2C2M; (1)

From the circuit of Figure 3, we can have the transmission matrixfor a unit cell,

�A BC D� � �1 � ZSYP ZS�2 � ZSYP�

YP 1 � ZSYP� (1)

The dispersion relation of CSRRs loaded transmission lines can beeasily obtained as follows:

cos��l� �A � D

2� 1 � ZSYP � 1 � �2L�C � C�C�

1 � �2/�S2

1 � �2/�C2

(2)

Figure 1 Topology of the CSRR. Gray area represents the metallization

2432 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 10, October 2009 DOI 10.1002/mop

where �C � 2�fC � 1�L�CC�C is the angular resonance frequencyof the CSRR, and �S � 2�fS � 1/�LCCC�C/�C � C�C� is the angularresonant frequency of the whole parallel impedance.

Equation (2) indicates the presence of a frequency gap in thevicinity of fC, delimited by fC (lower limit) and fS (upper limit),where the total parallel impedance is inductive. In the frequencyinterval, the structure behaves as a one-dimensional effective me-dium with negative permittivity. Therefore, propagating modes areexcluded in this frequency band.

The two characteristic frequencies fC and fS can be identified asbelow: the frequency that nulls the shunt admittance (transmissionzero frequencies, fC) and the frequency that null the shunt imped-ance (fS). fS can be determined by the intersection between the S11

curve and the unit normalized resistance circle as the shunt path tothe ground is opened at fS, and the real part of the input resistanceseen from the ports is that of the opposite port (50 �). Thus, wecan obtain L�C and C�C from the two characteristic frequencies,which can be experimentally determined or obtained through fullwave electromagnetic simulation.

To verify the equivalent circuit model, a two-unit prototypedevice has been designed and simulated. The substrate employedis Taconic CER-10 substrate (Epson � 10, thickness � 1.27 mm).The upper plane conductor strip has a width of 1.2 mm, which iscorresponding to a characteristic impedance of 50 �. The resonantfrequency of the CSRRs has been set to be the same as [8], whichis around 2.7 GHz. The physical dimensions in our case are d �s � 0.3 mm, rext � 3 mm (Fig. 1). The width of the slit at each ringhas been taken equal to 0.3 mm, and the period of the CSRR array

Figure 2 (a) Layout of the unit cell of CSRR loaded transmission lineand (b) its equivalent circuit

Figure 3 Simplified equivalent circuit for the unit cell of the CSRRloaded microstrip line without capacitive gaps

Figure 4 The experimental result and simulation result from the equiv-alent circuit model of the CSRR loaded transmission line

Figure 5 Experimental result of CSRR loaded transmission line withdifferent periodic length (a) S12; (b) S11

DOI 10.1002/mop MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 10, October 2009 2433

P is 9 mm. The total length of the region occupied by CSRRs is 18mm. The reference planes of the input and the output ports arechosen to lie at the edges of the two units of CSRRs.

In the equivalent circuit model of the unit cell of the CSRRloaded transmission line the per-unit-cell inductance and capaci-tance of the host transmission line, L and C in Figure 3 are 4.23 nHand 1.69 pF, respectively, which equals the per-unit-cell induc-tance and capacitance of the host transmission line, whereas L�C andC�C to model CSRRs in Figure 3 are (3.2e12/�2) H and (�2/9.3e32)F, respectively.

The experimental result and simulation result from the equiv-alent circuit model are shown in Figure 4. Very good agreementover a large frequency range is obtained between them. A deeprejection band is obtained around the design frequency, with sharpcutoffs, maximum rejection of �40 dB, and low return losses.Below the rejection frequency band, a flat and perfect matchedpassband is present with very low-insertion losses less than �0.3dB and nearly linear phase variation (the same as [7] not shown).

2.2. CSRR Loaded Transmission Lines With Different PeriodicLengthTo further verify the validness of the equivalent circuit model inFigure 3, we simulated the transmission response of the CSRRloaded transmission line with different periodic length, in whichthe per-unit-cell inductance and capacitance of the host transmis-sion line in the equivalent circuit model are different. Two unitsare applied and the periodic length is varied from 7 to 11 mm. Theexperimental results are given in Figure 5, which show that withdifferent periodic length, the rejection band is not changed. Thefrequency of the transmission dip depends only on the resonancefrequency of the CSRR, which is consistent with the equivalentcircuit model. The experimental [Fig. 5(a)] and simulated results(Fig. 6) of the reflection response (S11) also show a good agree-ment.

3. CONCLUSIONS

In this article, a novel circuit model is proposed for the CSRRsloaded transmission line, and comparisons between the resultsderived from the equivalent circuit model and the experimentalresults are given and verify the correctness of the proposed circuitmodel. The validity of the equivalent circuit is further confirmed

by comparing different results when the periodic length varies (i.e.,the capacitance per-unit-cell of the host microstrip line arechanged). Meanwhile, the experimental results show the transmis-sion dip does not depend on the periodic length, but only on theresonance frequency of the CSRR, which is consistent with theequivalent circuit model.

ACKNOWLEDGMENTS

This work was supported by the National Basic Research Program(973) of China (NO.2004CB719802) and the Swedish ResearchCouncil (VR) under Project No. 2006-4048.

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© 2009 Wiley Periodicals, Inc.

Figure 6 Simulated result of the equivalent circuit model of the CSRRloaded transmission line with different periodic length

2434 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 51, No. 10, October 2009 DOI 10.1002/mop