2009 IEEE MTT-S International Microwave Workshop Series onSignal Integrity and High-Speed Interconnects (IMWS2009-R9)

SRR- and CSRR-based Metamaterial Transmission Lines: Modelingand Comparison

Francisco Aznar, Marta Gil, Gerard Siso, Jordi Bonache and Ferran Martin

GEMMA/CIMITEC, Departament d'Enginyeria Electronica, Universitat Autonoma de Barcelona,08193 BELLATERRA (Barcelona), Spain

Abstract - This paper is focused on the comparisonbetween the two main categories of resonant-typemetamaterial transmission lines: those based on split ringresonators (SRRs) and those based on their complementarycounterparts, that is, complementary split ring resonators(CSRRs). It will be shown that both SRR- and CSRR-basedmetamaterial transmission lines exhibit a very similarbehavior, and this analogous behavior has been explained onthe basis of the equivalent circuit model of the unit cell ofthese lines and from duality arguments.

Index Terms - Metamaterial transmission lines, splitring resonators, duality.

I. INTRODUCTION

Metamaterial transmission lines are artificial linesconsisting on a host line loaded with reactive elements [1-3]. Thanks to the presence of the loading elements, theselines exhibit more degrees of freedom than conventionallines, and further control on the characteristic impedanceand dispersion of the lines can be achieved. There are twomain approaches for the synthesis of metamaterialtransmission lines: (i) the CL-loaded approach, where thehost line is periodically loaded with series capacitancesand shunt inductances (Fig. la) [4-6], and (ii) the resonanttype approach, where the host line is loaded with split ringresonators (SRRs) and shunt connected inductive elements(Fig. lb) [7] or with complementary split ring resonators(CSRRs) and series capacitances (Fig. Ic) [8]. All theselines exhibit a composite right/left handed (CRLH)behavior, namely, backward wave propagation at lowfrequencies, where the loading elements are dominant, andforward wave characteristics at higher frequencies, wherethe line parameters (line inductance and capacitance)dominate over the loading reactive elements. This CRLHbehavior has been used in many applications, including,the design of enhanced bandwidth and multibandcomponents, broadband filters, and many otherapplications (see refs. [1-3] to gain more insight on theapplications of metamaterial transmission lines tomicrowave engineering).

Actually resonant type metamaterial transmission linesare very similar to CL-loaded lines. As has beenpreviously indicated both line categories exhibit theabove-mentioned CRLH characteristics. However,

resonant-type metamaterial transmission lines exhibit atransmission zero at a finite frequency (to the left of theleft handed band) due to the coupling between the hostline and the loading resonators (magnetic coupling inSRR-loaded lines and electric coupling in CSRR-loadedlines).

It is the purpose of this paper to analyze and comparethe effects of varying the unit cell topology of these lineson the line characteristics, including the effects on theposition of the transmission zero. As will be shown thephenomenology is very similar in both SRR- and CSRR-based lines. This will be interpreted on the basis of theequivalent circuit models and duality arguments.

(a)

(b)

(c)

Fig. 1. Typical topology of a CL-loaded microstrip line (a),SRR-loaded CPW (b) and CSRR-loaded microstrip line (c). In(a), the series capacitances are implemented by means ofinterdigital capacitors and the shunt inductances throughgrounded stubs; in (b) the SRR are etched in the back substrateside, beneath the shunt connected strips; in (c), the CSRRs areetched in the ground plane, below the positions of the seriesgaps.

II. LUMPED ELEMENT EQUIVALENT CIRCUIT MODELS OFSRR- AND CSRR-LOADED LINES

The lumped element equivalent circuit models of CRLHSRR- and CSRR-based metamaterial transmission linesare depicted in Fig. 2(a) and 3(a), respectively. These

978-1-4244-2743-7/09/$25.00 C2009 IEEE 49 Guadalajara, Mexico, Feb. 19-20, 2009

2009 IEEE MTT-S International Microwave Workshop Series onSignal Integrity and High-Speed Interconnects (IMWS2009-R9)

(b)

models, which have been recently published by theauthors [9-10], are improved versions of previous modelsreported earlier by some of the authors [11].

PICs

M /21 >1.12

C/21 M/2 M2 C2j

LSCs

Fig. 2. Lumped element equivalent circuit model of the unitcell of the SRR-loaded CRLH transmission lines (a) andtransformed model (b).

L/2 C, L/2 L/2 2Cg 2Cg L/2

(a) CL Cf Cf (b) c

Fig. 3. Lumped element equivalent circuit model of the unitcell of the CSRR-loaded CRLH transmission lines (a) andtransformed model (b).

In the model of Fig. 2(a), L and C are the parameters ofthe host line, LP models the inductance of the shuntconnected strips, the SRRs are modeled by means of theresonant tank formed by LS and Cs, and the magneticcoupling between the line and the SRR is modeledthrough the mutual inductance M. This model can betransformed to that model depicted in Fig. 2(b), where thefollowing transformations apply [9].

L5= 2MC 2 X P (1)

1+M2LpLS2 2

2M 2Ls 2L'LS (2)

Ls=K 2+L2 34Lp

L 2 + L2 (3)2L )2 s

Lp '=2Lp +-L (4)

With regard to the model of Fig. 3(a), CL is the linecapacitance, Cf is the fringing capacitance of the gap andC5 is the series capacitance of the gap. L models theinductance of the host line and finally, the CSRR isaccounted for the resonant tank formed by LC and Cc. Thismodel can by transformed to the model depicted in Fig.3(b) with:

Cg = 2Cs + Cpar

= Cpar (2Cs + Cpar )

Cs

(5)

(6)

where Cpar=CL+Cf.Indeed the models of Figs. 2(b) and 3(b) are formally

identical to the older models proposed by the authors.However, the interpretation of the parameters of thesemodels is different than previously reported. Specifically,for the circuit of Fig. 3, Cg is no longer the capacitance ofthe series gap, C is no longer the fringing capacitance ofthe gap plus the line capacitance. From the circuit of Fig.2, the resonance of the tank formed by C's and L's is nolonger the resonance of the SRRs, and L 'p is no longer theinductance of the shunt connected strips. Let us see in thenext section that with these modifications, thephenomenology associated to these metamaterialtransmission lines can be perfectly explained.

III. PHENOMENOLOGY AND INTERPRETATION OF THERESULTS

Obviously, the frequency response of SRR- and CSRR-loaded metamaterial transmission lines depends on thegeometry of the resonators and the line, as well as on thedimensions of the series gap and shunt strips. By varyingthe gap and shunt strip dimensions we obtain interestingresults that can be interpreted to the light of the newmodels.The effects of varying the gap width in CSRR-loaded

microstrip lines is depicted in Fig. 4, whereas the effectsof varying the shunt strip width in SRR-loaded lines isdepicted in Fig. 5 (in both cases we report the simulationof a single unit cell structure). A significant variation inthe position of the transmission zero is obtained in bothcases, whereas in the phase response we observe that thefrequency at which the phase shift is null is preserved. Inthe SRR-loaded lines, the transmission zero is given by:

co=ooi m2L J/ (7)0),0)0 2L,L,where w0 is the intrinsic resonance frequency of the SRRs.Clearly, this transmission zero depends on LP. Thisexplains the variation with the shunt strip width.

978-1-4244-2743-7/09/$25.00 C2009 IEEE 50 Guadalajara, Mexico, Feb. 19-20, 2009

2009 IEEE MTT-S International Microwave Workshop Series onSignal Integrity and High-Speed Interconnects (IMWS2009-R9)

Concerning the frequency that nulls the phase shift of thecell, it is given by:

CS Ls2L

and it does not depend on Lp.

0

-10

-20

co -30m)

-40N

U)

0

-10(8)

-20m

30

U) -40

-50

-601.5

CY)-o

U)C,(U-

n~

Frequency (GHz)0

-30

m -60a)

-o-gUa) -90en

CL - 120

-150

-180 L_-1.9 2.0 2.1 2.2

Frequency (GHz)2.3 2.4

Fig. 4. Transmission coefficient (a) and phase (b) of differentCSRR-loaded structures with different gap separation (indicatedin the figure). The phase nulls at identical frequency in all theconsidered cases. Dimensions are: the strip line widthW,,=1.15mm, the length D=8mm; for the CSRRs: outer ringwidth c01t=0.364mm, inner ring width ci,=0.366mm, distancebetween the rings d=0.24mm, internal radius r=2.691mm. Theconsidered substrate is Rogers R03010 with dielectric constant£r= 10.2 and thickness h=1.27mm.

With regard to CSRR-loaded lines, the variation of thetransmission zero is due to the variation of C that resultswhen gap dimensions are modified. On the other hand, asis given by the intrinsic resonance frequency of theCSRRs, and it does not depend on gap dimensions. Thus,the phenomenology can be explained to the light of thenew models.

1801501209060300

-30--60--90-

-120-150-180

1.8

2.0 2.5Frequency (GHz)

2.0 2.2 2.4Frequency (GHz)

3.0

2.6 2.8

Fig. 5. Transmission coefficient (a) and phase (b) of differentSRR-loaded structures with different shunt strip width (indicatedin the figure). The phase nulls at identical frequency in all theconsidered cases. Dimensions are: the central strip widthW,=8mm, the width of the slots G=1.43mm, length of the lineD=8mm. For the SRRs the dimensions are those of Fig. 4, andwe have considered identical substrate.

IV. COMPARISON

This similar phenomenology between the SRR- and theCSRR-loaded lines can be interpreted to the light ofduality. SRR-loaded CPW transmission lines and CSRR-loaded microstrip lines are not dual structures, in the sense

of the Babinet Principle. However, they are roughly dual,and thus it is not surprising that they exhibit a roughlydual behavior, and hence similar phenomenology. In Fig.6, it can be appreciated the different transformationsbetween the CSRR-loaded microstrip line and the SRR-loaded CPW, where it is indicated where duality isapplied. Thus, according to these transformations the twostructures are roughly dual. Indeed, the equivalent circuitsof Figs. 2b and 3b are circuit duals. In a recent paper byGetsinger [12], it is argued that an impedance Z1representing an element in a planar circuit is inverselyproportional to the impedance Z2 of the correspondingelement in its dual according to:

978-1-4244-2743-7/09/$25.00 C2009 IEEE

(b)

-w = 1.8 mms

----- w = 0.2 mms

-- w = 0.0 mm

I

I

51 Guadalajara, Mexico, Feb. 19-20, 2009

2009 IEEE MTT-S International Microwave Workshop Series onSignal Integrity and High-Speed Interconnects (IMWS2009-R9)

2

Z1Z2= 4 (9)1 2 4

(where 77i is the vacuum impedance) provided that thestructures can be described reasonably as quasi-TEMstructures. This is for instance the case of shunt inductivestrips in a CPW, and series gaps in a twin strip (TS)structure (Fig. 7). The corresponding impedances are oneinverse of the other, and the elements are related by [12]:

2

4(10)

In the circuits of Figs. 2b and 3b, the impedance of theseries/shunt branch is formally proportional to the inverseof the impedance of the shunt/series branch, thisconfirming that the circuits of Fig. 2b and 3b are formallycircuit duals. Transformation of the circuit elements is notsimple because the conditions for duality do not applystrictly, but this discussion explains the similar behaviorof SRR- and CSRR-loaded lines.

(d)

(a) (b) (c)

Fig. 6. CSRR-loaded microstrip line (a), application of dualityin the ground plane (b), application of duality in the upper metallayer (c) and duplication of the structure to obtain the SRR-loaded CPW.

(a) (b)

lines are dual structures.

V. CONCLUSION

It has been shown in this paper that SRR-loaded CPWtransmission lines and CSRR-loaded microstrip linesexhibit a very similar behavior and this behavior has beeninterpreted to the light of duality arguments. Thestructures are roughly dual and they are described byformally circuit duals. These circuits are improved

versions of previous circuit models proposed by theauthors.

ACKNOWLEDGEMENT

This work has been partially supported by MCI-Spain(TEC2007-68013-C02-02 and CSD2008-00066) andGeneralitat de Catalunya under project 2005SGR-00624.

REFERENCES

[1] C. Caloz and T. Itoh, Electromagnetic Metamaterials:Transmission Line Theory and Microwave Applications,John Wiley & Sons, New Jersey, 2006.

[2] G.V. Eleftheriades and K.G. Balmain, Negative refractionmetamaterials: fundamental principles and applications,John Wiley & Sons, Inc, New Jersey 2005.

[3] R. Marques, F. Martin and M. Sorolla, Metamaterials withnegative parameters: theory, design and microwaveapplications, John Wiley & Sons, New Jersey, 2007.

[4] A. K. Iyer and G. V. Eleftheriades. "Negative refractiveindex metamaterials supporting 2-D waves," IEEE-MTTInt'l Microwave Symp., vol. 2, Seattle, WA, pp. 412- 415,June 2002.

[5] A. A. Oliner. "A periodic-structure negative-refractive-index medium without resonant elements," in URSI Digest,IEEE-AP-S USNCIURSI National Radio Science Meeting,San Antonio, TX, pp. 41, June 2002.

[6] C. Caloz and T. Itoh. "Application of the transmission linetheory of left-handed (LH) materials to the realization of amicrostrip LH transmission line," in Proc.IEEE-AP-SUSNCIURSI National Radio Science Meeting, San Antonio,TX, vol. 2, pp. 412-415, June 2002.

[7] F. Martin, F. Falcone, J. Bonache, R. Marques and M.Sorolla, "Split ring resonator based left handed coplanarwaveguide", Appl. Phys. Lett., vol. 83, pp. 4652-4654,December 2003.

[8] F. Falcone, T. Lopetegi, M.A.G. Laso, J.D. Baena, J.Bonache, R. Marques, F. Martin, M. Sorolla, "Babinetprinciple applied to the design of metasurfaces andmetamaterials", Phys. Rev. Lett., vol. 93, p 197401, Nov.2004.

[9] F. Aznar, J. Bonache and F. Martin, "An improved circuitmodel for left handed lines loaded with split ringresonators", Appl. Phys. Lett. Vol. 92, paper 043512, Feb.2008.

[10] J. Bonache, M. Gil, 0. Garcia-Abad and F. Martin,"Parametric analysis of microstrip lines loaded withcomplementary split ring resonators", Microwave andOptical Tech. Lett., vol. 50, pp. 2093-2096, August 2008.

[11] J.D. Baena, J. Bonache, F. Martin, R. Marques, F. Falcone,T. Lopetegi, M.A.G. Laso, J. Garcia, I Gil and M. Sorolla,"Equivalent circuit models for split ring resonators andcomplementary split rings resonators coupled to planartransmission lines", IEEE Transactions on MicrowaveTheory and Tech., vol. 53, pp. 1451-1461, April 2005.

[12] W.J. Getsinger, "Circuit duals in planar transmissionmedia", IEEE MTT-S Int'l Microw. Symp. Dig., pp. 154-156, 1983.

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