Theoretical Model for Estimation of Resonance

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Aug 1, 2012 - for frequency tuning of the magnetic resonance of the split-ring resonator. ... size of these structures is much smaller than the free-space ...
Electromagnetics

ISSN: 0272-6343 (Print) 1532-527X (Online) Journal homepage: http://www.tandfonline.com/loi/uemg20

Theoretical Model for Estimation of Resonance Frequency of Rotational Circular Split-Ring Resonators Chinmoy Saha & Jawad Y. Siddiqui To cite this article: Chinmoy Saha & Jawad Y. Siddiqui (2012) Theoretical Model for Estimation of Resonance Frequency of Rotational Circular Split-Ring Resonators, Electromagnetics, 32:6, 345-355, DOI: 10.1080/02726343.2012.701540 To link to this article: http://dx.doi.org/10.1080/02726343.2012.701540

Published online: 01 Aug 2012.

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Electromagnetics, 32:345–355, 2012 Copyright © Taylor & Francis Group, LLC ISSN: 0272-6343 print/1532-527X online DOI: 10.1080/02726343.2012.701540

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Theoretical Model for Estimation of Resonance Frequency of Rotational Circular Split-Ring Resonators CHINMOY SAHA 1 and JAWAD Y. SIDDIQUI 2 1 2

Swami Vivekananda Institute of Science and Technology, Kolkata, India Institute of Radio Physics and Electronics, University of Calcutta, Kolkata, India Abstract In this article, a split-ring resonator with a rotated inner ring is analyzed for frequency tuning of the magnetic resonance of the split-ring resonator. Simulated results reveal an increase in the resonance frequency with an increasing angle between the splits. A simple theoretical model is proposed for accurate estimation of this resonance frequency for angular separation between the splits using simplified analytical formulations. The computed results using the proposed model are verified with simulated results. A close comparison between the present formulation and previously measured results for conventional circular split-ring resonators is also revealed. Keywords split-ring resonator, circular split-ring resonator, magnetic resonance

1. Introduction Double-negative (DNG) media or negative index materials (NIMs) can be realized using periodic alignment of structures exhibiting negative permittivity and negative permeability. Pendry et al. (1999) showed that while negative permittivity is realized using conducting wires, negative permeability can be realized using split-ring resonators (SRRs). The size of these structures is much smaller than the free-space wavelength at resonance with dimensions only a tenth of the corresponding wavelength at the frequency of interest (Gay-Balmaz & Martin, 2002). Apart from being an integral component for realizing metamaterial, the potential of SRRs in passive microwave planar circuit design in the form of a narrow and wide band filter (Martín et al., 2003; Falcone et al., 2004), a phase shifter (Saadoun & Engheta, 1992), and a power divider (Antoniades & Eleftheriades, 2005) has recently been explored by different researchers. However, the negative permeability in an SRR occurs within a narrow frequency band of its resonance. Therefore, this resonance determines the range over which the material will simultaneously have negative . Thus, precise estimation of the resonance frequency of the SRR thus becomes imperative for such narrowband structures. The circular SRR with a rotated inner ring and its equivalent circuit is shown in Figure 1. The figure shows a schematic view of a circular SRR, printed on the surface Received 17 January 2012; accepted 21 April 2012. Address correspondence to Jawad Y. Siddiqui, Institute of Radio Physics and Electronics, University of Calcutta, 92 Acharya Prafulla Chandra Road, Kolkata 700 009, India. E-mail: [email protected]

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Figure 1. (a) Schematic view of a circular SRR with rotated inner ring formed with metallic rings of width c, having radii r0 and rext , with inter-ring spacing d and split gap dimensions g D g1 D g2 , printed on a dielectric substrate having height h and dielectric constant "r ; (b) equivalent circuit of SRR with rotated inner ring.

of a dielectric slab having dielectric constant "r , formed with metallic strips of width c, and radii r0 and rext forming the inner and outer rings, respectively, with inter-ring spacing d . The splits on the inner and outer rings have identical gap dimensions g1 and g2 , respectively, lying diametrically opposite on the same axis. The alignment of the split gaps g1 and g2 can be altered by rotating gap g2 of the inner ring shown in the figure. The angular variation of the gaps g1 and g2 can be quantified as an angle  with respect to their conventional alignment. The variation in this angle  would determine the increasing or decreasing angular variation between the split gaps g1 and g2 . When an external magnetic field is applied along the z-axis of the SRR, an electromotive force appears around the SRR and couples the two metallic rings with the induced current passing from one ring to the other ring through a distributed capacitance formed due to the inter-ring spacing. Analytical formulations to predict this resonance frequency for circular SRRs were proposed in Marquez et al. (2003) and Saha and Siddiqui (2011). A method to tune the resonance frequency by rotating the inner ring of the circular SRR was recently proposed in Wang et al. (2008). This tunability of the resonance frequency helps to achieve the overlapping of the magnetic resonance of the SRRs with the electrical resonance of the conducting wires and exhibit left-handed properties. A simple formulation is proposed in this article to estimate the resonance frequencies of an SRR, with different angular rotation of the SRR rings, using an equivalent circuit approach. Several parametric studies have been performed by varying the external radius (rext ), ring width (c), inter-ring spacing (d ), split gap (g), and dielectric constant ("r ) of the substrate for different angular variations between the split gaps. One of the most optimum and convenient ways to characterize these resonant structures is printing them on the back side of a coplanar waveguide (CPW) (Martín et al., 2003; Falcone et al., 2004) and measuring the transmission parameter (S21 ). Due to resonance, the transmission coefficient S21 will exhibit a sharp dip, which can also be exploited to design notch filters (Martín et al., 2003; Falcone et.al., 2004). By staggering multiple SRRs on the back side of the CPW medium with gradually shifted angular orientations, multiple resonance is also achieved, and these characteristics have been reported in this article. The computed resonant frequency using the present formulation is verified with the simulated results, showing excellent agreement between them. The computed results are also verified with

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the measured and computed results reported in Marquez et al. (2003) for conventional unrotated circular SRRs. The change in frequency due to variation of the gap width of a circular SRR has also been incorporated into the formulation, and the results are verified with the simulated data. However, the change in gap alignment plays a significant role in determining the resonance frequency and significantly changes the resonance frequency (Wang et al., 2008). This shift in resonance frequency has been quantitatively analyzed in this article for different angular variations between the split gaps. The closed-form expressions of Smith et al. (2005) have been used to extract the constitutive parameter (negative permeability) around the SRR’s resonance frequency as a function of frequency, and this is verified using the simulation results obtained for different angle of rotations of the SRR.

2. Theoretical Formulation The SRR and its equivalent circuit model is shown in Figure 1. When a magnetic field is applied along the the z-axis, an electromotive force appears around the SRR and induces currents that pass from one ring to the other through the inter-ring spacing d , and the structure behaves as an LC circuit. As shown in the equivalent circuit, the metallic rings contribute a total inductance LT and distributed capacitances C1 and C2 forming at the two halves of the SRR structure above and below the split gaps. This new equivalent circuit also incorporates the gap capacitances Cg1 and Cg2 , formed due to the split within the inner and outer rings, respectively. The resonance frequency !0 of the circular SRR is thus given as s 1 !0 D ; (1) LT Ceq where Ceq is the total equivalent capacitance of the structure. For a conventional SRR, the split gaps g1 and g2 are on the same axis. Hence, C1 and C2 , the capacitances of the upper and lower half of the rings, are equal. For a rotational SRR, where the inner ring is rotated by an angle , the values of C1 and C2 become unequal. For an angle of rotation  of the inner ring, the capacitance of the portion within OA and OB of the SRR, as shown in Figure 1(a), are given as C1 D .

/ravg Cpul ;

(2)

where Cpul is the per unit length capacitance between the inner and outer rings of the SRR, and the uniform average dimension of the inner and outer rings of the SRR, ravg , is calculated as d ravg D rext c : (3) 2 The capacitance of the remaining portion of the SRR is given by C2 D . C /ravg Cpul :

(4)

The per unit length capacitance Cpul between the inner and outer rings of the SRR is calculated as p "e Cpul D ; (5) c 0 Z0

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where c0 D 3108 m/s is the velocity of light in free space, "e is the effective permittivity of the medium, and Z0 is the characteristic impedance of the line, which is a function of SRR dimensions c and d . The effective permittivity "e can be calculated as (Bahl & Bhartia, 1998) "e D 1 C

"r

1 K.k 0 /K.k1 / ; 2 K.k/K.k10 /

(6)

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where kD

c=2 ; c=2 C d

aD

k1 D

c ; 2

bD

c C d; 2

sinh. a=2h/ ; sinh. b=2h/

and k0 D

p

1

(7) (8)

k2I

(9)

K is a complete elliptic function of the first kind, and K 0 is its complimentary function. An approximate expression for K=K 0 is given as (Bahl & Bhartia, 1998) " p !# 1 1 1 C k0 K.k/ D ln 2 p for 0  k  0:7; (10) K.k 0 /  1 k0 p ! K.k/ 1 1C k D ln 2 p K.k 0 /  1 k

for 0:7  k  1:

(11)

The characteristic impedance Z0 is given as 120 K.k/ Z0 D p : "e K.k 0 /

(12)

The model also incorporates the effect of gap capacitances Cg1 and Cg2 in the equivalent circuit of the SRR. Because of identical gap dimensions g1 D g2 D g, Cg1 D Cg2 D Cg D

"0 A "0 ct D : g g

(13)

From the equivalent circuit of Figure 2, the equivalent capacitance is given as Ceq D

.C1 C Cg /.C2 C Cg / : .C1 C Cg / C .C1 C Cg /

(14)

Substituting C1 and C2 from Eqs. (2) and (4) into Eq. (14) gives Ceq D

Œ. /ravg Cpul C Cg Œ. C /ravg Cpul C Cg  Œ. /ravg Cpul C Cg  C Œ. C /ravg Cpul C Cg 

(15)

. C q/2  2 ravg Cpul ; 2. C q/

(16)

or Ceq D

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Figure 2. Simulated jS21j of a circular SRR as a function of angle of rotation  of the inner ring; rext D 2:2 mm, "r D 2:43, c D 0:5 mm, d D 0:2 mm, g D 0:1 mm, h D 0:49 mm, and t D 35 m. (color figure available online)

where qD

Cg : ravg Cpul

(17)

As the angle of rotation increases and the angular separation between the split gap decreases, the area of the portion of rings between OA and OB decreases, and the area of the remaining portion increases. This leads to a decrease in C1 and an increase in C2 , which results in a decrease of the equivalent capacitance Ceq of the structure, hence resulting in an increase of the resonance frequency of the SRR structure. So, the resonance frequency of the rotated SRR can be obtained from Eq. (1) as 1

f00 D 2

:

s

(18)

. C q/2  2 LT ravg Cpul 2. C q/

For conventional SRRs, resonance frequency is given by 1

f0 D 2

s

LT



(19)

 ravg Cpul "0 ch C 2 2g



or 1

f0 D 2

s

LT



 Cg C 2 2ravg Cpul

: 

ravg Cpul

(20)

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Simplifying Eq. (20) gives 1

f0 D

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2

s

LT



 Cq 2

: 

(21)

ravg Cpul

A simplified formulation for the evaluation for the total equivalent inductance LT for a wire of rectangular cross-section having finite length l and thickness c is proposed as in Terman (1943) and Saha and Siddiqui (2011) as   4l LT D 0:0002l 2:303 log10

micro H ; (22) c where the constant D 2:451 is for a wire loop of circular geometry. Here, length l and thickness c are in mm. The evaluation of the wire length is straightforward as l D 2 rext

g:

(23)

From Eqs. (18) and (21), f00  Cq Dp f0 . C q/2

2

:

(24)

If the gap dimension (g < 0:2 mm) is very small, q may be neglected, thus  f00 p ; 2 f0  2

(25)

where  and  are in radian. Equation (25) is applicable for the rotation of both inner and outer rings.

3. Results and Discussion The proposed theoretical formulations for the evaluation of the resonance frequency was validated using simulated results obtained using an electromagnetic simulator highfrequency structure simulator (HFSS v11; ANSYS, Inc., Canonsburg, Pennsylvania, USA). and measurements reported in Marquez et al. (2003) and are presented in this section for different parametric variations. Different methods to excite the magnetic resonance of a single-element SRRs were proposed in Martín et al. (2003), Falcone et al., (2004), Marquez et al., (2003), and Saha and Siddiqui (2011). When excited by an electromagnetic wave, propagating along the x-direction with a z-oriented magnetic field results in a transmission coefficient jS21j, as shown in Figure 2 for different angles of rotation of the inner ring of the SRR. The dip in transmission coefficient S21 corresponds to the resonance frequency for three discrete angular separations of 0ı , 60ı , and 90ı. The computed resonance frequencies for a wide range of angular rotation  between the rings are compared with the simulation results, as shown in Figure 3. The plot explicitly shows the increase in the resonance frequency with the increase in the rotational angle . This can be attributed to the decreasing equivalent capacitance (Ceq ) within the rings of the SRR rings with an increasing angle of rotation. The computed plot shows excellent

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Figure 3. Computed, simulated, and measured resonance frequency of a circular SRR as a function of angle of rotation of the inner ring for different rext ; "r D 2:43, c D 0:5 mm, d D 0:2 mm, g D 0:1 mm, and h D 0:49 mm.

agreement with the simulation data for a rotational angle of the inner ring up to  ˙ 95ı. Further increase in  diminishes the resonance characteristics of the SRR with less current flowing through the rings. Computed results are also compared with the measured datum (Marquez et al., 2003) reported for  D 0ı (conventional SRR). A similar comparison of the computed resonance frequencies with simulated values for different rext is shown in Figure 4.

Figure 4. Computed and simulated resonance frequency of a circular SRR as a function of rext for different angles of rotation of the inner ring; "r D 2:43, c D 0:5 mm, d D 0:2 mm, g D 0:1 mm, and h D 0:49 mm.

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Figure 5. Computed and simulated resonance frequency of a circular SRR as a function of c for different angles of rotation of the inner ring; "r D 2:43, rext D 2:6 mm, d D 0:2 mm, g D 0:1 mm, and h D 0:49 mm.

Results for two different angle of rotation,  D 0ı and 60ı , are reported. The computed values are compared with the measured datum (Marquez et al., 2003) for  D 0ı. The present formulations show better coherence with the measured and simulated data. With an increase in rext , the total inductance (LT ) and equivalent capacitance (Ceq ) of the SRR increases, thereby decreasing the resonance frequency. A similar comparison of the measured, simulated, and computed data for two different angular rotations for varying widths of the conducting strip c is shown in Figure 5. The present computation shows an excellent match with the simulated and measured values. The variation of the

Figure 6. Computed and simulated resonance frequency of a circular SRR as a function of d for different angles of rotation of the inner ring; rext D 2:2 mm, "r D 2:43, c D 0:5 mm, g D 0:1 mm, and h D 0:49 mm.

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Figure 7. Computed and simulated resonance frequency of a circular SRR as a function of g for different angles of rotation of the inner ring; rext D 2:2 mm, "r D 2:43, c D 0:5 mm, d D 0:2 mm, and h D 0:49 mm.

inter-ring spacing d and its effect on the resonance frequency is presented in Figure 6. The present computation shows excellent agreement with the measured and simulated data. The present formulation has also been verified for prediction of the resonance frequency due to a change in split gap dimension g of the circular SRR. Figure 7 shows the comparison of the computed resonance frequency compared with the simulated values as a function of the split gap dimension g for a circular SRR with two different angular rotations of the inner ring. The effect of varying the dielectric constant on the resonance frequency for different angular rotations  is depicted in Figure 8. Staggering multiple SRRs in a linear array format on the back side of the CPW medium with gradually shifted angular orientations

Figure 8. Computed and simulated resonance frequency of a circular SRR as a function of dielectric constant of substrate "r for different angles of rotation of the inner ring; rext D 2:2 mm, c D 0:5 mm, d D 0:2 mm, g D 0:1 mm, h D 0:49 mm.

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Figure 9. Simulated S21 of a circular SRR one-dimensional array as a function of frequency; rext D 2:4 mm, ro D 1:7 mm, "r D 2:43, c D 0:5 mm, d D 0:2 mm, h D 0:49 mm, and g D 0:1 mm;  D 0ı , 60ı, and 90ı. (color figure available online)

helps in achieving multiple resonances, and a simulated result is plotted in Figure 9. The extraction of negative permeability is computed using Smith et al. (2005), and the extracted values as a function of frequency is presented in Figure 10. An one-dimensional array of SRRs is designed and simulated using HFSS v.11, yielding S21 (magnitude and phase) and S11 (magnitude and phase) values. The effective permeability of the medium is extracted using the values of S21 and S11 in the closed-form expressions of Smith et al. (2005). This is done for different angular orientations of the inner ring  D 0ı, 60ı, and

Figure 10. Extracted magnetic permeability (real) of SRR for different angles of rotation between the rings as a function of frequency; rext D 2:4 mm, c D 0:5 mm, d D 0:2 mm, and g D 0:1 mm;  D 0ı , 60ı , and 90ı.

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90ı . The resultant permeability curve follows the Lorentzian profile, exhibiting negative values around the resonance frequency of the SRR.

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4. Conclusions A simple model is proposed to accurately estimate the resonance frequency of a rotational circular SRR. Simple empirical relations for determining the ring inductance and capacitance for different angular separation of the split gaps are proposed and incorporated in the present formulations. Change in the resonance frequency due to a change in the split gap dimension is also verified using the proposed formulation. The comparison of the computed and simulated results shows excellent correspondence for all parametric variations of the SRR. The computed results are also compared with the measured data for an unrotated ring, showing close correspondence. The formulation can be implemented easily and is mathematically less complicated and is computationally inexpensive. Multiple resonances are achieved using an array of SRRs with different angular separations between the gaps and can be used for frequency tunability. The sharp rejection around the resonance frequency of each element of the array can successfully be exploited in designing a notch filter with multiple rejections. The extracted constitutive parameter shows negative values of permeability around the resonance of the SRR. The present formulation can be implemented easily for designing tunable SRR structures useful for NIM applications.

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