are also described by the two-port equivalent circuit approach. ... of an infinitely long SRR array is estimated using its equivalent circuit model and compared to.
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Equivalent Circuit Models for Split-ring Resonator Arrays P. Yasar-Orten1, 2 , E. Ekmekci1 , and G. Turhan-Sayan1 1
Department of Electrical and Electronics Engineering, Middle East Technical University, Ankara, Turkey 2 ASELSAN Inc., Macunkoy, Ankara, Turkey
Abstract— In this study, a square-shaped single-ring SRR unit cell is modeled by using a suitable two-port resonant circuit representation that accounts for the conductor loss and dielectric loss effects as well. Capacitive and inductive coupling effects between adjacent SRR unit cells are also described by the two-port equivalent circuit approach. Finally, the resonance frequency of an infinitely long SRR array is estimated using its equivalent circuit model and compared to the 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 exhibits negative 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 or square/rectangular geometry have been investigated in microwave and optical frequencies in a large number of publications [1–4]. Most of those studies have investigated the SRR behavior theoretically, experimentally or numerically by using full-wave electromagnetic solvers. Number of 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 yet practical 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 computer memory space and quite long computer processing times. Sufficiently accurate equivalent lumped circuit models, on the other hand, can be effectively used to estimate the behavior of SRR structures in a simple, fast and computationally efficient manner. This approach would even make optimization approach feasible in the design of special SRR structures. Also, when an equivalent circuit model is available, it is much easier to establish explicit relationships between the physical properties (i.e., electrical parameters, dimensions, etc.) of the SRR structure and its frequency dependent 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 in Figure 1(a) is simulated by HFSS over the frequency range from 30 GHz to 40 GHz by using PEC and PMC boundary conditions. Dimensions and electrical parameters of this SRR unit cell are as follows: Side length of SRR metal loop (L) is 2.8 mm, gap width (g) and metal strip width (w) are both 0.3 mm, substrate dimensions are Dx = Dy = 4 mm and the substrate thickness along z direction is h = 0.5 mm. Gold is assumed to be used for metal inclusions and a low loss dielectric material with relative permittivity εr = 4.6 and dielectric loss tangent tan α = 0.01 is used as the substrate. The perfect electric conductor (PEC) type boundary conditions are applied at those surfaces of the computational volume which are perpendicular to the incident electric field vector. Similarly, perfect magnetic conductor (PMC) type boundary conditions are applied at those surfaces of the computational volume which are perpendicular to the incident magnetic field vector. The remaining surfaces are the input and output planes as shown in Figure 1(b). Due to 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 dimensional infinite SRR array extending in the incident E-field and H-field directions with periodicities of Dy = L + 2DE and Dz = h + 2DH , respectively. The periodicity parameters DE and DH are indicated in Figure 2. Transmission spectrum of the SRR array is computed by HFSS for various periodicity parameters as shown in Figures 3 and 4. For the design parameters DE = 0.6 mm and DH = 1.75 mm, the SRR array resonates at f0 = 34 GHz. As the array is chosen to be sparse in z-direction, coupling effects along this direction will be neglected and hence the resulting SRR array will be assumed to be a one 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
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(b)
(a)
(b)
Figure 1: (a) Geometrical parameters of the SRR unit cell. (b) HFSS simulation setup and boundary conditions.
Figure 2: Two dimensional periodic SRR array implemented by the use of PEC and PMC boundary conditions in HFSS (a) Array in E field direction. (b) Array in H field direction.
Figure 3: Variation of resonance frequency of the SRR array for the parameter of periodicity DE = 0.6 mm (blue), 0.9 mm (red), 1.2 mm (green) in ydirection.
Figure 4: Variation of resonance frequency of the SRR array for the parameter of periodicity DH = 1.75 mm (red), 3.5 mm (blue) in z-direction.
can be justified based on the results shown in Figure 4 where the resonance frequency changes by only 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 cell is shown in Figure 5 where L is the self-inductance of the metal loop, which can be computed by the expressions given in [9, 10]. The model parameter C = Cgap /4 is the equivalent capacitance computed for four individual gap capacitances connected in series. Each gap capacitance can be computed [6, 9] as Cgap = Cpp + Ccp where Cpp and Ccp are parallel plate and coplanar capacitance contributions, respectively. The equivalent loss resistances can be approximately computed using the expression R = σ Sl eff such that l = 4(L − g) and σ = σc will be used to compute Rc , while l = 4g and σ = σd are used in 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-
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Figure 5: A feasible two-port equivalent circuit representation for the SRR unit cell including ohmic loss effects.
Figure 6: Equivalent two port circuit model for two SRR unit cells separated by 2DE = 1.2 mm along the y direction.
puted as S21 =
2Z 2Z + Z0
where Z = Rc + j2ωL +
(1)
Rd 1 + jωCRd
(2)
is the impedance of the series RLC resonant circuit in the shunt branch of this two-port circuit and Z0 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-port which consists of a parallel RC circuit in the shunt branch as shown in Figure 6. This coupling equivalent 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 inductance term 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 unit cells carry induced currents flowing in the same (clockwise or counterclockwise) direction. It should be noted that in an SRR array with n unit cells, each SRR unit cell (except those located at the ends) 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 coupling circuit block are called Z and Zm , respectively, the equivalent impedance in the shunt branch of the overall equivalent two-port circuit becomes Ztotal = nZ + (n − 1)Zm where
µ Z = Rc +
Rd 1 + ω 2 C 2 Rd2
¶
µ + j ω(Lself − 2M ) −
(3) ωRd2 C 1 + ω 2 C 2 Rd2
¶ (4)
Progress In Electromagnetics Research Symposium Proceedings, Cambridge, USA, July 5–8, 2010
and
µ Zm =
Rdm (1 − jωRdm Cm ) 2 R2 1 + ω 2 Cm dm
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¶ (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 the infinite SRR array is estimated to be f0 = 35.6 GHz from (6) by this equivalent circuit modeling approach. The resonance frequency obtained by HFSS simulations (see Figures 3 and 4) was 34 GHz. Namely, the suggested equivalent circuit model for the SRR array estimates the resonance frequency by less than 5 percent error. 4. CONCLUSIONS
In this study, an approximate equivalent two-port circuit modeling approach is suggested to describe the resonance behavior of SRR unit cells and SRR arrays. Results obtained by this approach are found in very good agreement with the results of HFSS simulations. REFERENCES
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