Revising the Equivalent Circuit Models of Resonant-Type Metamaterial Transmission Lines

Francisco Aznar, Marta Gil, Jordi Bonache and Ferran Martín

GEMMA/CIMITEC, Departament d’Enginyeria Electrònica. Universitat Autònoma de Barcelona. 08193 BELLATERRA (Barcelona), Spain. E-mail: Ferran.Martin@uab.es

Abstract — In this work, it is shown that the previously

reported and accepted equivalent circuit models of resonant type left-handed lines, either loaded with split ring resonators (SRRs), or with complementary split ring resonators (CSRRs), need a revision in order to properly account for the different elements of the structures and to accurately describe their behavior. The main relevant conclusion and novelty of this paper is that the already existing models are formally correct, but some of the reactive parameters do not have the physical interpretation given so far. However, these parameters are related to the parameters of the improved models, which are proposed and exhaustively analyzed here for the first time. A comparative analysis of SRR- and CSRR-loaded lines, to the light of these new models, is also included. The results of this work are of interest for the design of CSRR- and SRR-based microwave circuits. Index Terms — Metamaterial transmission lines, split ring resonators (SRRs), complementary split ring resonators (CSRRs), equivalent circuits.

I. INTRODUCTION

Resonant-type metamaterial transmission lines, either loaded with split ring resonators (SRRs) [1] or with complementary split ring resonators (CSRRs) [2] (Fig. 1), were proposed in 2003 and 2004, respectively, as an alternative to artificial lines loaded with series capacitances and shunt inductances [3-5] (CL-loaded approach). In all cases, the structures exhibit a composite right/left handed (CRLH) behavior [6,7], that is, backward wave propagation at low frequencies, and forward wave propagation at higher frequencies. The propagation characteristics of these artificial lines have been interpreted to the light of their lumped element equivalent circuits. For SRR- and CSRR-loaded lines, the reported models are depicted in Fig. 2 [8]. To achieve CRLH wave propagation in CSRR-loaded lines, series capacitive gaps are required, whereas shunt inductive elements are necessary in SRR-based lines. In [8], a physical interpretation was given to the different reactive parameters of both circuit models. However, in this work it is demonstrated that, by extracting such parameters, we are not actually able to link all of them to the different elements of the structures (SRRs, CSRRs, series gaps, shunt inductive strips and host line), in spite of the excellent agreement between circuit and electromagnetic simulations (or measurement) of the structures. As will be shown, the circuits of Fig. 2 are inferred through circuit transformation from other circuit models

where the reactive parameters have a direct correspondence with the different planar components and, hence, a clear physical interpretation. The relationships between the elements of the original and transformed models are obtained. From these relations, the behavior of the structures is interpreted, and interesting implications on circuit design are pointed out and discussed.

Fig. 1. Typical unit cell of a CRLH line based on SRRs (a) and CSRRs (b). In (a), the SRRs are etched in the back substrate side of a coplanar waveguide (CPW) transmission line (the upper metal level is depicted in gray). In (b), the host line is a microstrip transmission line and the CSRR is etched in the ground plane (depicted in gray).

Fig. 2. Lumped element equivalent circuit model of the unit cell of a CRLH line based on SRRs (a) and CSRRs (b).

II. IMPROVED MODEL FOR CSRR-LOADED CRLH LINES

According to previous works [8], the T-circuit of Fig. 2(b) models a CSRR-loaded CRLH line, where the CSRRs are described by the resonant tank Lc-Cc, L models the line inductance, Cg the gap capacitance and C the electric coupling between the line and the CSRRs. The parameter extraction method reported in [9] reveals that there is very

(b) (a)

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good agreement between electromagnetic simulation (or measurement) and circuit simulation. This means that the T-circuit model of Fig. 2(b) provides a good description of device behavior. It was argued that C is composed of the line capacitance and the fringing capacitance of the gap. However, the extremely high values of C previously reported [9] can not be explained to the light of this interpretation. It has been found that C increases by increasing gap distance. On the other hand, it has been found that Cg does not experience a significant decrement when gap distance is increased. The results of table I, corresponding to the extracted parameters of identical structures with different gap distance, corroborate these statements. The considered substrate is the Rogers RO3010 with thickness h=1.27mm and dielectric constant εr=10.2. Line width is 1.15mm and gap width is 4.8mm. CSRR dimensions are: internal radius ro=2.69mm, width c=0.36mm and separation d=0.24mm.

TABLE I: EXTRACTED PARAMETERS OF IDENTICAL CSRR-

LOADED UNIT CELLS WITH DIFFERENT GAP DISTANCE Gap space (mm) Cg (pF) C (pF) L (nH) Lc (nH) Cc (pF)

1.56 0.45 21.3 3.30 1.46 3.14 0.76 0.49 8.8 4.10 1.58 2.90 0.16 0.54 3.8 5.35 1.69 2.70

To properly model the series gap and the coupling between

the line and the CSRR, we must consider the π-circuit model of the series gap. Thus, the new proposed equivalent circuit of the CSRR-loaded CRLH line is that depicted in Fig. 3, where CL is the line capacitance, Cf is the fringing capacitance of the gap and Cs is the series capacitance of the gap. Obviously, from π-T transformation, the circuit model of Fig. 2(b), which is the reported model of microstrip lines loaded with CSRRs and series gaps, is obtained, but the values of Cg and C do not actually have a physical interpretation. Indeed, Cg and C can be expressed in terms of Cs and Cpar=Cf+CL

according to:

parsg CCC += 2 (1)

s

parspar

CCCC

C)2( +

= (2)

and expressions (1) and (2) explain the behavior of C and Cg when the gap distance is increased (Cs decreased). We have considered isolated gaps, and we have estimated Cs and Cf from the S-parameters obtained through electromagnetic simulation. By using equations (1) and (2) we have obtained the values of C and Cg. Reasonable agreement with those values inferred from the parameter extraction method (shown in table I) results. These values are [Cg=0.45pF, C=17.8pF], [Cg=0.53pF, C=7.63pF] and [Cg=0.70pF, C=3.67pF], for the structures with a gap distance of 1.56mm, 0.76mm and 0.16mm, respectively. These results are of interest because they reveal that it is possible to obtain high values of C (by decreasing Cs), regardless of the substrate thickness. These high values of C are typically necessary to enhance bandwidth and to drive the transmission zero frequency

)(12

cczz CCL

f+

== πω (3)

to small values. However, this can be achieved without the penalty of a small Cg, since Cf and CL do also contribute to this capacitance (see expression 1). The increment of L in table I when the gap distance is decreased is due to the strip length variation. CSRR parameters (Lc and Cc) vary since these parameters are influenced by the geometry of the upper metallic layer (strips and series gap).

Fig. 3. Improved lumped element equivalent circuit model of the unit cell of a CRLH line based on CSRRs.

III. IMPROVED MODEL FOR SRR-LOADED CRLH LINES

In [8], it was demonstrated that the π-circuit of Fig. 2(a) is obtained by transformation of the series impedance of a primary model, intended to describe the physics of SRR-loaded CRLH transmission lines. In such primary model, the series impedance is composed of the line inductance, L, inductively coupled to the SRRs (described by a resonant tank Ls-Cs) through a mutual inductance M, whereas the shunt impedance accounts for the line capacitance, C, and the shunt inductive strips (or vias in microstrip technology), modeled by the inductance Lp (due to symmetry considerations, the magnetic wall concept was used in [8]). It has been found through parameter extraction (using a technique similar to that reported in [9] for CSRR-loaded lines) that the model of Fig. 2(a) provides a good description of device behavior. However, the parameters of the primary model, related to those of the circuit of Fig. 2(a) through

22'

ML

Co

ss ω

= (4)

sos CML 22' ω= (5)

pp LL =' (6) LL =' (7)

are not representative of the different elements of the structures. In particular, it has been found that the transmission zero frequency, ωz, depends on the characteristics of the inductive strips, something not accounted for by the model (ωz should coincide with the resonance frequency of the SRRs, ωo=(LsCs)-1/2=(Ls’Cs’)-1/2, according to the model). The simulated transmission coefficients of two identical structures (unit cells), one with shunt strips (CRLH

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line) and the other one without such elements (negative permeability line), clearly points out the variation of ωz (Fig. 4). Fig. 4. Simulated insertion losses (S21) and dispersion (φ) for the indicated SRR-loaded CPW unit cells with and without shunt inductive strips. The circuit simulation with extracted parameters is also included (symbols). The considered substrate has a thickness of h=1.524mm and a dielectric constant of εr=10.2. For the CPW, central strip width is W=7mm and slot width is G=1.48mm. The width of the shunt strips is 0.2mm. SRR dimensions are: internal radius ro=2.4mm, width c=0.6mm and separation d=0.2mm. We have actually represented the modulus of the phase.

Thus, although the circuit of Fig. 2(a) is formally correct, the

previously proposed primary circuit [8] is only a first order approximation, and it needs a revision. However, such improved circuit must be expressible as that of Fig. 2(a). This is the main aim of this section. The proposed improved model of SRR-loaded CRLH lines is depicted in Fig. 5(a). In contrast to the work in [8], the magnetic wall concept is not used here since it is not actually necessary. In this new model, which is neither a π- nor a T-circuit, the line inductance has been divided in two parts, and the inductance of the shunt strips, Lp, is located at the intermediate position. This provides a more accurate description of the strips location and perfectly explains the variation of ωz when the shunt strips are removed. From the elements of the admittance matrix of the circuit of Fig. 5(a), we can obtain the equivalent π-circuit model. After a straightforward calculation, the series and shunt impedances of such π-circuit are found to be:

−

+−

+= 2''1

''22

2)(ω

ωωss

ss

ps CL

LLLLLjZ (8)

and

+=

22)( LLjZ pp ωω (9)

with

sp

poss

LLM

LL

CML

21

41

2' 2

2

22

+

+

= ω (10)

22

22

41

21

2'

+

+=

p

sp

o

ss

LL

LLM

ML

Cω

(11)

Fig. 5. Improved lumped element equivalent circuit model of the unit cell of a CRLH line based on SRRs (a) and transformed model (b).

From these results, it is clear that: (i) the model of Fig. 5(a) can be formally expressed as that of Fig. 2(a) (see Fig. 5(b)); (ii) expressions (4)-(7) are no longer valid; the link between the parameters of Figs. 5(a) and 5(b) is given by expressions (10), (11) and

'22

2' sp

LLLLL −

+= (12)

22' LLL pp += (13)

(iii) the transmission zero verifies ωz<ωo and it depends on Lp since it is given by ωz=(Ls’Cs’)-1/2, that is:

2/12

21

−

+=

spoz LL

Mωω (14)

We have extracted the parameters of the model of Fig. 5(b) for those structures whose insertion losses are depicted in Fig. 4. By using expressions (10)-(13) and M=L⋅f (f being the fractional area of the slots occupied by the SRRs [8]), we have obtained the parameters of the model of Fig. 5(a). Such parameters are represented in table II. It is remarkable that (i) the circuit simulation with extracted parameters provides a very good fit to the electromagnetic simulation (see Fig. 4), and (ii) the parameters of the model of Fig. 5(a) are roughly the same with and without the presence of the shunt strips. The parameter values that have been obtained from the isolated elements (SRRs, shunt strips and line elements) are also represented in table II (3rd row). The variations are

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attributed to the interaction between the CPW, the SRRs and the strips. Namely, line parameters (C and L) are affected by the presence of SRRs, and SRR parameters (Cc and Ls) are influenced by the presence of the CPW structure.

TABLE II: EXTRACTED PARAMETERS OF SRR-LOADED CPW UNIT

CELLS WITH AND WITHOUT SHUNT STRIPS Shunt strips Lp (nH) C (pF) L (nH) Ls (nH) Cs (pF)

Yes 0.36 2.16 2.27 14.81 0.35 No ---- 2.16 2.29 14.69 0.35

Estimated 0.57 1.14 2.60 12.55 0.50

IV. COMPARATIVE ANALYSIS AND DISCUSSION

It is interesting to mention that the transmission zero frequency can be controlled with the shunt strip (equation 14), in a similar form that ωz depends on the characteristics of the gap (through C) in CSRR-loaded lines (expression 3). Specifically, widening the inductive strip and the capacitive gap (enhancing gap separation) has the same effect: lowering the transmission zero frequency. Notice that this effect is due to the fact that the strips/gaps are located above the position of the SRRs/CSRRs. Actually, this similarity is not surprising, if one takes into account that CSRR-loaded microstrip lines and SRR-loaded CPWs roughly exhibit a dual behavior. Another interesting aspect of SRR-loaded CPW lines is that the frequency where the phase nulls, ωs, does not depend on Lp. In a π- or T-circuit, such phase is given by:

)()(

1cosωωφ

p

s

ZZ

+= (15)

Thus, by forcing Zs(ω)=0, we obtain such frequency, that is:

−

==

LMLC

f

ss

zs 2

2

12πω (16)

and it does not depend on Lp, as anticipated. It has been verified from electromagnetic simulation (Fig. 4) that identical value of ωs is obtained with and without the presence of shunt strips. Analogously, ωs does not depend on the presence of the series gap in CSRR-loaded microstrip lines. According to the model of Fig. 3, this frequency is obtained when Zp(ω)=∞, and this occurs at the resonance frequency of CSRRs, ωo=(LcCc)-1/2 (such frequency is the same for the 3 considered cases of table I). Again, we obtain a similar behavior concerning the frequency where the phase nulls in both the SRR- and CSRR-loaded lines. This further supports the dual behavior of the considered structures.

V. CONCLUSIONS

In conclusion, the previously existing and accepted models of SRR- and CSRR-loaded metamaterial transmission lines have been revised. The new proposed models accurately account

for the different elements of the structures. It has been demonstrated that although the new models can be transformed to π- or T-circuits formally identical to the previous ones, the elements of these new transformed circuits are very different, and they are able to perfectly explain the electromagnetic behavior of the structures. The roughly dual behavior of the two considered structures has also been discussed.

ACKNOWLEDGEMENT

This work has been supported by MEC (Spain) by project contract TEC2007-68013-C02-02 METAINNOVA. Special thanks are also given to CIDEM (Generalitat de Catalunya) for funding CIMITEC. MEC has given an FPU Grant to Marta Gil (Reference AP2005-4523).

REFERENCES

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[2] F. Falcone, T. Lopetegi, M.A.G. Laso, J.D. Baena, J. Bonache, R. Marqués, F. Martín, M. Sorolla, “Babinet principle applied to the design of metasurfaces and metamaterials”, Phys. Rev. Lett., vol. 93, p 197401, November 2004.

[3] A. K. Iyer and G. V. Eleftheriades. “Negative refractive index metamaterials supporting 2-D waves,” in IEEE-MTT Int’l Microwave Symp., vol. 2, Seattle, WA, pp. 412– 415, June 2002.

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

[5] C. Caloz and T. Itoh. “Application of the transmission line theory of left-handed (LH) materials to the realization of a microstrip LH transmission line,” in Proc.IEEE-AP-S USNC/URSI National Radio Science Meeting, vol. 2, San Antonio, TX, pp. 412–415, June 2002.

[6] C. Caloz and T. Itoh, “Novel microwave devices and structures based on the transmission line approach of metamaterials”, in IEEE-MTT Int’l Microwave Symp, vol. 1 Philadelphia, PA, pp. 195-198, June 2003.

[7] M. Gil, J. Bonache, J. Selga, J. García-García, F. Martín, “Broadband resonant type metamaterial transmission lines”, IEEE Microwave and Wireless Components Letters, vol. 17, pp. 97-99, February 2007.

[8] J.D. Baena, J. Bonache, F. Martín, R. Marqués, F. Falcone, T. Lopetegi, M.A.G. Laso, J. García, I Gil and M. Sorolla, “Equivalent circuit models for split ring resonators and complementary split rings resonators coupled to planar transmission lines”, IEEE Transactions on Microwave Theory and Techniques, vol. 53, pp. 1451-1461, April 2005.

[9] J. Bonache, M. Gil, I. Gil, J. Garcia-García and F. Martín, “On the electrical characteristics of complementary metamaterial resonators”, IEEE Microwave and Wireless Components Letters, vol. 16, pp. 543.545, October 2006.

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