Estimation of the Resonant Frequency and Magnetic

Estimation of the Resonant Frequency and Magnetic Polarizability of an Edge Coupled Circular Split Ring Resonator with Rotated Outer Ring #

C.Saha#1, J.Y.Siddiqui*2, S.Mukherjee#3 , and R.Chaudhuri#4 Department of Electronics and Communication Engineering, Heritage Institute of Technology, Chowbaga Road, Anandapur, Kolkata-700107, India 1

[email protected], [email protected], 4 [email protected]

3

*

Royal Military College of Canada, Station Forces, Kingston, Ontario, CANADA K7K 7B4 2

[email protected]

Abstract— In this paper, a theoretical model is proposed to estimate the resonant frequency of a circular split ring resonator (C-SRR) in which the outer ring is rotated. The variation in the resonant frequency with angle of rotation of the SRR is theoretically calculated. The computed values are then compared with the simulated results using an electromagnetic simulator. Also, the magnetic polarizability for different angle of rotation is extracted from the simulated data. Keywords—circular split ring resonator, resonant frequency, magnetic polarizability, rotation, metamaterials, negative permeability.

I. INTRODUCTION The split ring resonator (SRR) along with thin metallic wires can be used to fabricate metamaterials with negative effective permeability and permittivity [1]. SRRs may vary in shape, and two such shapes, namely circular-SRR and squareSRR have been extensively studied in [1]-[4]. SRRs possess large magnetic polarizability and exhibit negative effective permeability for frequencies close to their resonant frequency [2]. Since the negative value of permeability exists over a very narrow band of frequency, accurate estimation of resonant frequency is of utmost importance. Fig. 1 shows a schematic diagram of a circular SRR having strip width c and spacing d between the rings in which the outer ring is rotated by an angle (π-φ) with respect to a conventional edge coupled-SRR in which the two gaps are diametrically opposite to each other (φ =00). The SRRs have splits or gaps g1 and g2 of identical dimensions, on the same axis within the edge of the inner and outer rings. In this paper, a simplified formulation is proposed to accurately estimate the resonant frequency of the rotational SRR where (π- φ) is the angle of rotation of the outer ring. The simulated results of a commercially available electromagnetic simulator [5] show a reasonably good agreement with the proposed theory. The simulated results are used to calculate the magnetic polarizability of the structure for a broad range of angle of rotation.

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Fig. 1. Edge coupled circular split ring resonator with rotated outer ring

II. THEORY Fig. 1 shows the structure of a circular edge coupled SRR which is placed inside a circular iris having radius R of a metallic screen located inside a rectangular waveguide [2]. When a time-varying magnetic field is applied along the zaxis, the induced electromotive force will cause loop currents to flow in the rings. The rings are coupled by a strong distributed capacitance, and have a self inductance due to the loop currents. Thus, the structure behaves as an LC circuit having resonant frequency ω0 given by [2]

ω0 =

1 LC eq

(1)

where L is the total inductance and Ceq is the equivalent capacitance of the structure. L and Ceq are calculated assuming an equivalent ring whose radius is r0 = rext − c −

d which 2

is the average radius of the outer and inner rings. L is computed using the expression in [2].

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L=

~

μ 0π 2 I

2

∞

~

∫ [ I ( k )]

2

k 2 dk

(2)

(6)

0

Where I ( k ) is the Fourier – Bessel Transform of I(k), the current function on the ring. Cpul, the per unit length capacitance between the inner and outer rings of the SRR. is calculated using the formulations presented in [2] and [6]. The equivalent circuit of the SRR is shown in Fig. 2. For a conventional or un-rotated SRR, the gaps g1 and g2 lie on the same axis. Hence, C1 and C2, the capacitances of the upper and lower half of the rings are equal. But as the outer ring is rotated by an angle φ = (π − θ ) in clockwise direction, the values of C1 and C2 becomes unequal. As the angle of rotation increases, the arc length of the rings between OA and OB decreases and that of the remaining portion increases. This results in decrease of C1 and simultaneous increase of C2. leading to a decrease of the equivalent capacitance Ceq of the structure. As a result the resonant frequency of the structure increases.

Substituting C1 and C2 from equation (3) and (4) in equation (6) and simplifying we get

⎡ cg cg ⎤⎡ g g ⎢(θr0 − ) + ⎥ ⎢((2π − θ ) r0 − ) + C pul C pul ⎥⎦ ⎢⎣ 2 2 ⎢ C eq = ⎣ 2c g ( 2πr0 − g ) + C pul

g g ⎡ ⎤⎡ ⎤ ⎢(θr0 − 2 ) + λ ⎥ ⎢((2π − θ ) r0 − 2 ) + λ ⎥ ⎦⎣ ⎦ =⎣ (2πr0 − g ) + 2λ C where. λ = g . C pul

⎤ ⎥ ⎥⎦

(7)

Though the capacitance of the SRR changes due to the effect of rotation, the inductance remains same as for unrotated SRR, since it depends only on the width of the rings and their radii, both of which are unaltered during rotation. So, the resonance frequency of the rotated SRR can be obtained from equation (1) by substituting Ceq from equation (7) and L from equation (2). The magnetic polarizability of the structure is computed as in [2] using the resonant frequency ω0 at which a peak is obtained in S21 and the frequency ωd at which a dip in transmission is observed using Fig. 2. Equivalent circuit of the rotated SRR considering the gap capacitance

For an angle of rotation φ= (π − θ ) of the outer ring in clockwise direction, capacitance of the portion within OA and OB of the SRR is given by,

g⎞ ⎛ C 1 = ⎜ θ r0 − ⎟ C pul 2⎠ ⎝

(3)

The capacitance of the remaining portion of the SRR is given by,

g⎤ ⎡ C 2 = ⎢ ( 2 π − θ ) r0 − ⎥ C pul 2⎦ ⎣

(4)

The present model also incorporates the effect of gap capacitances Cg1 and Cg2 as in [8], enabling an accurate estimation of the resonant frequency of the SRR. Cg1 and Cg2 are calculated using parallel plate capacitance approximation. As the dimensions of gap are identical,

C g = C g1 = C g 2 =

ε0A g1

=

ε 0 ch g1

From the equivalent circuit of Fig. 2 the equivalent Capacitance is given as


(5)

α0 =

8R 3 3μ 0

⎛ ω 02 ⎞ ⎜ 2 − 1⎟ ⎜ω ⎟ ⎝ d ⎠

(8)

Theoretically magnetic polarizability of the structure should remain nearly constant as dimension of the SRR is unaltered. However, a small deviation from the un-rotated SRR is expected because of asymmetric gap orientation. III. RESULTS AND DISCUSSION The computed resonant frequencies for different angle of rotation are compared with the simulated results using [5] as shown in Fig.5. Fig.3 shows a snapshot of simulated S21 in dB, indicating the two frequencies of interest ω0 and ωd. The simulated S21 for φ = 00,300,600 and 900 are shown in fig. 4. The HFSS generated charge density plots (fig.7-9) for φ = 00,300,-500 (negative sign indicating anticlockwise rotation) verify the equivalent circuit shown in fig. 2. The surface current density on the rings have also been shown for φ = 00 in fig.10. The extracted magnetic polarizability from the simulated S21, for rext =2.4mm, 2.6mm, and 2.8mm against different angle of rotation is shown in Fig 6. The plot suggests that the magnetic polarizability is almost invariant with the angle of rotation of the outer ring.

CONCLUSION

A simple and new circuit model is hereby proposed to estimate the resonant frequency of a circular SRR with rotated outer ring. The accuracy of the present model is established using the commercially available simulation software [5] whose results show excellent agreement with the theoretical calculations for a wide range of angular rotation of the outer ring. However, for angles greater than 900, the deviation of the simulated values of resonant frequency increases. The value of resonant frequency is same for both clockwise and anticlockwise rotation of the outer ring, which is evident from the symmetry of the curve in fig. 5 about 00. Thus, rotation of outer ring can be used to fine-tune the operating frequency of the circular SRR to the desired value without having to change the design parameters.

8.0

Theory Simulated

7.5

7.0

f0 (GHz)

IV.

6.5

6.0

5.5

5.0 -100

-80

-60

-40

-20

0

20

40

60

80

100

φ (deg.) Fig. 5. Computed and simulated resonant frequency of the rotated SRR for different angle of rotations. rext=2.6mm, εr = 2.43, c=0.5mm, d = 0.2mm, h = 0.49 mm, g1=g2=0.4mm.

Fig. 3 . A snap-shot from HFSS simulated S21 of the circular SRR. rext=2.6mm, εr = 2.43, c=0.5mm, d = 0.2mm, h = 0.49 mm, g1=g2=0.4mm.

0

0 deg. 30 deg. 60 deg. 90 deg.

-10 -20

S21(db)

-30 -40 -50 -60 -70 -80 5.5

6.0

6.5

7.0

Resonant Frequency (GHz) Fig 6. Simulated magnetic polarizabilty of the rotated SRR for different angle of rotations. c=0.5mm, εr = 2.43, d = 0.2mm, h = 0.49 mm. Fig. 4. Simulated S21 for different angle of rotation of the EC-SRR. rext=2.6mm, εr = 2.43, c=0.5mm, d = 0.2mm, h = 0.49 mm, g1=g2=0.4mm.


Fig. 7 HFSS Simulated profile for surface charge density on an unrotated C-. SRR rext=2.6mm, εr = 2.43, c=0.5mm, d = 0.2mm, h = 0.49 mm. g1=g2=0.4mm.

Fig. 10 HFSS Simulated profile for surface current density on an unrotated C-. SRR rext=2.6mm, εr = 2.43, c=0.5mm, d = 0.2mm, h = 0.49 mm. g1=g2=0.4mm.

REFERENCES [1] [2]

[3] [4] Fig. 8 HFSS Simulated profile for surface charge density on a C-SRR with rotation -300. SRR rext=2.6mm, εr = 2.43, c=0.5mm, d = 0.2mm, h = 0.49 mm. g1=g2=0.4mm.

Fig. 9 HFSS Simulated profile for surface charge density on a C-SRR with rotation -500. rext=2.6mm, εr = 2.43, c=0.5mm, d = 0.2mm, h = 0.49 mm. g1=g2=0.4mm.


[5] [6]

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. 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. C. Saha and J.Y. Siddiqui, “Estimation of resonant frequency of Conventional and rotational Split Ring Resonator” Proc. IEEE AEMC, India, Dec. 2009. C. Saha, J.Y. Siddiqui, D. Guha, Y.M.M. Antar, “Square Split Ring Resonators: Modelling of Resonance Frequency and Polarizability” Proc. IEEE AEMC, India, Dec. 2007. HFSS: High Frequency Structure Simulator, Ansoft. I. Bahl and P. Bhartia, Microwave Solid State Circuit Design, Ch.2,Wiley,.New York, 1998