Embed Size (px)
Citation preview
Square Split Ring Resonator: A New Approach in Estimation of Resonant Frequency
Swagato Mukherjee(1)
, Romita Chaudhuri(1)
, and Chinmoy Saha(1)
(1) Heritage Institute of Technology, Chowbaga Road, Anandapur, Kolkata-700107, India,
Email: [email protected], [email protected], [email protected]
Abstract: In this paper, the resonant frequency of a square split ring resonator (S-SRR) is calculated using a new approach. The model is studied by a wide range of parametric variations of the SRR geometry, namely edge length, width of the rings, gap within the edge, and spacing between the rings. The calculated results have been verified using an electromagnetic simulator.
Introduction: The split ring resonator (SRR), along with thin metallic wires can be used to
fabricate metamaterials with negative magnetic permeability and electric permittivity as
proposed in [1]-[4]. Two common shaped SRRs namely circular-SRR and square-SRR have
been extensively studied in [1]-[4]. SRRs possess large magnetic polarizability and show
negative permeability over a very narrow band of frequency close to its resonance frequency
[2]. In fig.1, a schematic view of a square SRR is shown having strip width c, spacing d
between the rings and gaps g1 and g2 in the edge of the rings. The rings are printed on a
dielectric substrate with dielectric constant and thickness h. Since the negative value of
permeability lies over a very narrow band of frequency accurate estimation of resonant
frequency is of utmost importance. In [2] resonant frequency is calculated for a circular SRR
but involves a complex calculation of inductance and the gap capacitance in the equivalent
circuit of the SRR is not considered. The present model calculates the resonance frequency of
the S-SRR using the inductance computation of [6]. Computed results for different design
parameters are compared with the results of a commercially available electromagnetic
simulator [5].
Theory: When a time varying magnetic field is applied along the z-axis, it induces an
electromotive force in the plane of the SRR. Since the size of the SRR structure is much
smaller than the free space wavelength at resonance a quasistatic analysis is plausible. The
two split rings of the SRR are coupled by a strong distributed capacitance between the rings
as well as by the gap capacitances. The induced current flows from one ring to the other
through the capacitive gaps in the form of displacement current. Hence the entire structure
behaves like a LC circuit whose resonant frequency ω0 is given by,
(1)
where, L and C are the total inductance and total capacitance of the S-SRR structure
respectively.
Calculation of Inductance: To calculate the inductance of the SRR a single ring is
considered having effective edge length a0 calculated from the center of the SRR and broken
up into rectangular segments as shown in fig. 2(a).
Let the length of a segment be l. Defining the normalized widths W = c/l and T = t/l and the
normalized distances as
r = , αw = , αt = , αr = the inductance L of each segment can be calculated using the closed form expression given in
[6] as
(2)
where, ) (3)
and ) (4)
From fig. 2(b), the total inductance of the structure is
L = 2L1 + L2 + 2L3 (5)
where L1, L2 and L3 are the inductances of the segments of lengths l1, l2 and l3, respectively.
Calculation of Capacitance: Calculation of total capacitance of the SRR involves
calculation of distributed ring capacitances C1 and C2 and calculation of gap capacitance
and as done in [4].
(6)
The gaps’ capacitance because of identical gap dimensions g1=g2=g is given by,
and (7)
where, K1 = 60, K2 = 400 are the correction factors due to brink effects, calculated from the
ratio of the gap capacitance, found using the electric field magnitude and the surface charge
distribution from the HFSS simulation, in a region very close to the gap, to the theoretical gap
capacitance. Therefore,
(8)
substituting L and Ceq from (5) and (8) in equation (1) we can determine resonant frequency
of the SRR.
Results and Discussions: To verify the present model a set of SRRs with different geometric
parameters were printed on a substrate with r = 2.43, h = 0.035 mm and were simulated
using HFSS. Fig.3. shows the HFSS simulated transmission parameter when the SRR is
placed in a waveguide and held by a square metallic aperture. As explained in [2]-[4] the
peak in transmission correspond to the resonant frequency and dip frequency can be exploited
to estimate the magnetic polarizability of the structure. Fig. 4 shows the plot of resonant
frequency as a function of different geometrical parameters of the SRR. The computed results
show very close agreement with the simulated results. Fig.5. shows the plot of surface current
density on the two rings of the SRR for three different aext all taken in same phase and same
scale. The plot gives us the detailed idea on the current distribution which is extremely
important for estimation of magnetic polarizability of the structure. The charge distribution in
the two rings is shown in fig. 6 which clearly confirms the equivalent circuit of the SRR.
Conclusions: A simple and new model is proposed for the estimation of resonant frequency
of a square SRR with various design parameters. The accuracy of the present model is
established using a commercially available electromagnetic simulator. Several variations with
the edge length, width of ring, spacing between rings and gap between the edges are modeled
using the simulator and the obtained results compared with the proposed theory show
excellent agreement. Plot of current distribution is extremely important and can be exploited
for extraction of constitutive parameter of the SRR. The resonant frequency for various
design parameters of S-SRR varies from about 4 GHz to 6 GHz which is lesser than its
variation in the circular SRR for similar design parameters as given in [2]. As the resonant
frequency of the square SRR is lesser than its circular counterpart, its electrical size also
decreases and hence, it can be used as one of the strategies of miniaturization of split ring
resonators.
Reference:
[1] J.B. Pendry, A.J. Holden, D.J. Ribbins, and W.J. Stewart, “Magnetism from conductors
and enhanced nonlinear phenomenon” IEEE Trans. Microwave Theory Tech., vol. 47, pp.
2075-2084, Nov. 1999.
[2] R.Marquez, F. Mesa, J. Martel, F. Medina, “Comparative analysis of edge- and
broadside- coupled split ring resonators for metamaterial design-theory and experiments,”
IEEE Trans. Antennas Propagation., vol. 51, pp. 2572-2581, Oct. 2003.
[3] C. Saha, J.Y. Siddiqui, Y.M.M. Antar, “Theoretical investigation of the square split ring
resonator” Proc. URSI NA Radio Science Meet, Ottawa, Canada, July, 2007.
[4] C. Saha, J.Y. Siddiqui, D. Guha, Y.M.M. Antar, “Square Split Ring Resonators:
Modeling of Resonance Frequency and Polarizability” Proc. IEEE AEMC, India, Dec.
2007.
[5] HFSS: High Frequency Structure Simulator, Ansoft.
[6] Ruey-Beei Wu,Chien-Nan Kuo, and Kwei K.Chang, “Inductance and Resistance
Computations for Three-Dimensional Multiconductor Interconnection Structures” IEEE
Trans. on Microwave Theory and Techniques, Vol. 40, No. 2, February 1992.
[7] I. Bahl and P. Bhartia, Microwave Solid State Circuit Design, Ch.2,Wiley, New York,
1998.
Fig. 1 Square split ring resonator
(a) (b) (c)
Fig. 2 (a) Equivalent ring of the SRR for inductance calculation (b) Equivalent circuit
diagram of (a) Circuit modeling of S-SRR
(a) (b)
Fig.3(a) A S-SRR placed inside a square iris of length, L within a waveguide (b). HFSS
Simulated transmission co-efficient of the square SRR placed within a rectangular waveguide
c=0.6 mm, d=0.2 mm, aext=2.6 mm, c=0.6 mm,
L
Fig. 4. Computed and Simulated resonant frequency as function of various design parameters
r = 2.43, h = 0.035 mm.
aext=2.6mm aext=2.7mm aext=2.8mm
Fig.5. HFSS simulated surface current density on the two rings of the SRR. c=0.6mm,
d=0.2mm,g1g2=0.4mm,r = 2.43 , h = 0.035 mm.
Fig. 6. HFSS simulated surface charge density on the two rings of the SRR.aext=2.6mm
c=0.6mm, d=0.2mm,g1g2=0.4mm,r = 2.43 , h = 0.035 mm.
