534 PIERS Proceedings, Cambridge, USA, July 5–8, 2010

Equivalent Circuit Models for Split-ring Resonator Arrays

P. Yasar-Orten1, 2, E. Ekmekci1, and G. Turhan-Sayan1

1Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey2ASELSAN Inc., Macunkoy, Ankara, Turkey

Abstract— In this study, a square-shaped single-ring SRR unit cell is modeled by using a suit-able two-port resonant circuit representation that accounts for the conductor loss and dielectricloss effects as well. Capacitive and inductive coupling effects between adjacent SRR unit cellsare also described by the two-port equivalent circuit approach. Finally, the resonance frequencyof an infinitely long SRR array is estimated using its equivalent circuit model and compared tothe value of the resonance frequency obtained by HFSS simulations for the same array topology.

1. INTRODUCTION

Split Ring Resonator (SRR) is a well known sub-wavelength metamaterial structure that exhibitsnegative values of permeability (µ) over a narrow frequency band around it resonance frequency.Theory and applications of single ring, double ring or multiple ring SRR cells with circular orsquare/rectangular geometry have been investigated in microwave and optical frequencies in alarge number of publications [1–4]. Most of those studies have investigated the SRR behaviortheoretically, experimentally or numerically by using full-wave electromagnetic solvers. Numberof publications concentrated on the analysis of SRR structures using equivalent circuit models,however, has been relatively few [2, 5–8].

Describing SRR unit cells and SRR arrays by accurate circuit models offers an approximate yetpractical alternative to the use of full-wave electromagnetic solvers based on numerical methods(such as the Ansoft’s HFSS or CST Microwave Studio), which usually need very large computermemory space and quite long computer processing times. Sufficiently accurate equivalent lumpedcircuit models, on the other hand, can be effectively used to estimate the behavior of SRR struc-tures in a simple, fast and computationally efficient manner. This approach would even makeoptimization approach feasible in the design of special SRR structures. Also, when an equivalentcircuit model is available, it is much easier to establish explicit relationships between the physicalproperties (i.e., electrical parameters, dimensions, etc.) of the SRR structure and its frequencydependent transmission/reflection behavior.

2. DESIGN AND NUMERICAL SIMULATIONS FOR SRR ARRAYS

In this paper, transmission spectrum of the fully symmetrical four-split SRR unit cell shown inFigure 1(a) is simulated by HFSS over the frequency range from 30GHz to 40 GHz by using PECand PMC boundary conditions. Dimensions and electrical parameters of this SRR unit cell areas follows: Side length of SRR metal loop (L) is 2.8 mm, gap width (g) and metal strip width(w) are both 0.3mm, substrate dimensions are Dx = Dy = 4 mm and the substrate thicknessalong z direction is h = 0.5mm. Gold is assumed to be used for metal inclusions and a low lossdielectric material with relative permittivity εr = 4.6 and dielectric loss tangent tanα = 0.01 isused as the substrate. The perfect electric conductor (PEC) type boundary conditions are appliedat those surfaces of the computational volume which are perpendicular to the incident electricfield vector. Similarly, perfect magnetic conductor (PMC) type boundary conditions are appliedat those surfaces of the computational volume which are perpendicular to the incident magneticfield vector. The remaining surfaces are the input and output planes as shown in Figure 1(b). Dueto the imaging effects of PEC and PMC boundaries, the computed transmission spectrum (i.e.,magnitude of the S21 parameter computed as a function of frequency) represents a two dimensionalinfinite SRR array extending in the incident E-field and H-field directions with periodicities ofDy = L + 2DE and Dz = h + 2DH , respectively. The periodicity parameters DE and DH areindicated in Figure 2.

Transmission spectrum of the SRR array is computed by HFSS for various periodicity parametersas shown in Figures 3 and 4. For the design parameters DE = 0.6mm and DH = 1.75 mm, the SRRarray resonates at f0 = 34 GHz. As the array is chosen to be sparse in z-direction, coupling effectsalong this direction will be neglected and hence the resulting SRR array will be assumed to be aone dimensional infinite array extending in y-direction in the rest of the paper. This assumption

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010 535

(a) (b)

Figure 1: (a) Geometrical parameters of the SRRunit cell. (b) HFSS simulation setup and boundaryconditions.

(a) (b)

Figure 2: Two dimensional periodic SRR array im-plemented by the use of PEC and PMC boundaryconditions in HFSS (a) Array in E field direction.(b) Array in H field direction.

Figure 3: Variation of resonance frequency of theSRR array for the parameter of periodicity DE =0.6mm (blue), 0.9mm (red), 1.2 mm (green) in y-direction.

Figure 4: Variation of resonance frequency of theSRR array for the parameter of periodicity DH =1.75mm (red), 3.5 mm (blue) in z-direction.

can be justified based on the results shown in Figure 4 where the resonance frequency changes byonly 0.3 GHz (less than one percent) when DH is doubled.

3. TWO-PORT EQUIVALENT CIRCUIT MODEL FOR SRR STRUCTURES

The two-port equivalent circuit representation suggested for this fully symmetrical SRR unit cellis shown in Figure 5 where L is the self-inductance of the metal loop, which can be computed bythe expressions given in [9, 10]. The model parameter C = Cgap/4 is the equivalent capacitancecomputed for four individual gap capacitances connected in series. Each gap capacitance can becomputed [6, 9] as Cgap = Cpp + Ccp where Cpp and Ccp are parallel plate and coplanar capacitancecontributions, respectively.

The equivalent loss resistances can be approximately computed using the expression R = lσ Seff

such that l = 4(L − g) and σ = σc will be used to compute Rc, while l = 4g and σ = σd are usedin Rd computation. The effective current carrying cross-sectional area Seff terms can be computed,in general, in terms of the geometrical parameters w, h, t and the skin depth δ.

Using the conversion formulas between Z-matrix and S-parameter matrix representations [11],the scattering parameter S21 of the two-port circuit representation shown in Figure 5 can be com-

536 PIERS Proceedings, Cambridge, USA, July 5–8, 2010

Figure 5: A feasible two-port equivalent circuit rep-resentation for the SRR unit cell including ohmicloss effects.

Figure 6: Equivalent two port circuit model for twoSRR unit cells separated by 2DE = 1.2mm alongthe y direction.

puted as

S21 =2Z

2Z + Z0(1)

where

Z = Rc + j2ωL +Rd

1 + jωCRd(2)

is the impedance of the series RLC resonant circuit in the shunt branch of this two-port circuit andZ0 is the terminal impedance which is given as an output by HFSS simulations.

The coupling effects between two SRR unit cells can also be described by a coupling two-portwhich consists of a parallel RC circuit in the shunt branch as shown in Figure 6. This couplingequivalent circuit is connected in series between two SRR blocks. The parameters Cm and Rdm

can be computed by similar approaches used for the computation of the parameters Cgap and Rd

mentioned above. The inductive coupling effect is taken into account by the effective inductanceterm L = Lself − M for each SRR unit cell where M is the mutual inductance computed [9]between two SRR unit cells and it acts in a subtractive manner as the metal loops of both unitcells carry induced currents flowing in the same (clockwise or counterclockwise) direction. It shouldbe noted that in an SRR array with n unit cells, each SRR unit cell (except those located at theends) experiences subtractive inductive coupling stemming from both of its neighbors. Therefore,L = Lself − 2M should be used in their equivalent circuit models.

When the equivalent impedances in the shunt branches of each SRR block and of each couplingcircuit block are called Z and Zm, respectively, the equivalent impedance in the shunt branch ofthe overall equivalent two-port circuit becomes

Ztotal = nZ + (n− 1)Zm (3)

where

Z =(

Rc +Rd

1 + ω2C2R2d

)+ j

(ω(Lself − 2M)− ωR2

dC

1 + ω2C2R2d

)(4)

Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010 537

and

Zm =(

Rdm(1− jωRdmCm)1 + ω2C2

mR2dm

)(5)

with n being the number of SRR unit cells in the array.The resonance frequency of this array can be estimated by equating the imaginary part of Ztotal

to zero, namely by solving the following equation:

Im{Ztotal} = 0 (6)

Using the geometrical and physical parameters specified earlier, the resonance frequency of theinfinite SRR array is estimated to be f0 = 35.6GHz from (6) by this equivalent circuit modelingapproach. The resonance frequency obtained by HFSS simulations (see Figures 3 and 4) was34GHz. Namely, the suggested equivalent circuit model for the SRR array estimates the resonancefrequency by less than 5 percent error.

4. CONCLUSIONS

In this study, an approximate equivalent two-port circuit modeling approach is suggested to describethe resonance behavior of SRR unit cells and SRR arrays. Results obtained by this approach arefound in very good agreement with the results of HFSS simulations.

REFERENCES

1. Pendry, J. B., A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductorsand enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech., Vol. 47, No. 11,2075–2084, 1999.

2. Marques, R., F. Mesa, J. Martel, and F. Medina, “Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design-theory and experiments,” IEEE Trans.Antennas Propag., Vol. 51, No. 10, 2572–2581, 2003.

3. Sauviac, B., C. R. Simovski, and S. A. Tretyakov, “Double split-ring resonators: Analyticalmodeling and numerical simulations,” Electromagnetics, Vol. 24, 317–338, 2004.

4. Sydoruk, O., E. Tatartschuk, E. Shamonina, and L. Solymar, “Analytical formulation for theresonant frequency of split rings,” Journal of Applied Physics, Vol. 105, 014903, 2009.

5. Bilotti, F., A. Toscano, L. Vegni, K. Aydin, K. B. Alici, and E. Ozbay, “Equivalent-circuitmodels for the design of metamaterials based on artificial magnetic inclusions,” IEEE Trans.Microwave Theory Tech., Vol. 55, No. 12, December 2007.

6. Chen, H., L. Ran, J. Huangfu, T. M. Grzegorczyk, and J. Kong, “Equivalent circuit model forleft-handed metamaterials,” Journal of Applied Physics, Vol. 100, 024915, 2006.

7. Wu, M.-F., F.-Y. Meng, Q. Wu, J. Wu, and L.-W. Li, “SRRs’ artificial magnetic metamaterialsmodeling using transmission line theory,” PIERS Online, Vol. 1, No. 5, 630–633, 2005.

8. Ekmekci, E. and G. Turhan-Sayan, “Comparative investigation of resonance characteristics andelectrical size of the double-sided SRR, BC-SRR and conventional SRR type metamaterialsfor varying substrate parameters,” Progress In Electromagnetics Research B, Vol. 12, 35–62,2009.

9. Johnson, N. P., A. Z. Khokhar, H. M. H. Chong, R. M. de La Rue, and S. McMeekin, “Char-acterisation at infrared wavelengths of metamaterials formed by thin-film metallic split-ringresonator arrays on silicon,” Electronics Letters, Vol. 42, No. 19, September 14, 2006.

10. Thompson, M. T., Inductance Calculation Techniques, Part II: Approximations and Handbook,1999.

11. Pozar, D. M., Microwave Engineering, 2005.