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Equivalent Circuit Models for Split-ring Resonator Arrays
Article · January 2010
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PIERS Proceedings, Cambridge, USA, July 5–8, 2010
534
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
(a)
535
(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-
PIERS Proceedings, Cambridge, USA, July 5–8, 2010
536
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
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µ
Zm =
Rdm (1 − jωRdm Cm )
2 R2
1 + ω 2 Cm
dm
537
¶
(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.
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