# Electrically Small Resonators for Planar Metamaterial, Microwave Circuit and Antenna Design: A Comparative Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Electrically Small Resonators for Metamaterial and Microwave Circuit Design

**Figure 1.**(

**a**) Transmission line loaded with a grounded resonator. (

**b**) Implementation of the resonator by means of a λ/4 open stub. (

**c**) Implementation of the resonator by means of a stepped impedance shunt stub (SISS). (

**d**) Typical frequency response of the line loaded with the λ/4 open stub and SISS.

**Figure 2.**(

**a**) λ/2 ring resonator. (

**b**) Split ring resonator (SRR). (

**c**) First bulk SRR-based reported structure exhibiting backward wave propagation in a certain frequency band. Photograph courtesy of D.R. Smith.

## 3. Electrically Small Planar Resonators: A Comparative Analysis and Circuit Models

#### 3.1. Topologies and Circuit Models of the Isolated Resonators

_{s}is the SRR self-inductance and C

_{o}/2 is the capacitance associated with each SRR half. This capacitance is C

_{o}= 2πr

_{o}C

_{pul}, where r

_{o}is the mean radius of the SRR, and C

_{pul}is the per unit length capacitance along the slot between the rings. The total capacitance of this circuit is the series connection of the capacitance of both SRR halves, that is, C

_{o}/4. Therefore, the resonance frequency ω

_{o}is given by:

_{s}can be modeled as the inductance of an average ring of radius r

_{o}, i.e., the mean radius of the ring, and width c. Approximate expressions for L

_{s}, based on a variational calculation, are provided in [16]. Closed expressions for the per unit length capacitance are given in many microwave textbooks [17], and explicit expressions for L

_{s}and C

_{pul}are provided in [7].

**Figure 4.**Topology and charge distribution of (

**a**) conventional SRR, (

**b**) broadside coupled split ring resonators (BC-SRR), (

**c**) non-bianisotropic split ring resonator (NB-SRR), and (

**d**) double slit split ring resonator (DS-SRR).

**Figure 5.**Topology of the (

**a**) two-turn spiral resonator (2-SR), (

**b**) BC-SR, (

**c**) folded stepped impedance resonator (SIR) and (

**d**) open split ring resonator (OSRR) (including the circuit model).

_{c}, is the capacitance of a disk of radius r

_{o}−c/2 surrounded by a ground plane at a distance c of its edge. The inductance is given by the parallel connection of the two inductances of the metallic strips connecting the inner and outer metallic regions of the CSRR. These inductive values are given by L

_{o}/2, where L

_{o}= 2πrL

_{pul}, with L

_{pul}being the per unit length inductance of the CPWs connecting the inner disk to ground.

**Figure 6.**Complementary split ring resonator (CSRR) topology and its equivalent circuit model. Geometrical parameters of the CSRR are identical to those of Figure 3 referred to the SRR.

_{c}; the inductance is four times larger than the inductance of the CSRR. This means that the OCSRR is smaller than the CSRR by a factor of two. Indeed, the resonance frequency and electrical size of the OSRR and the OCSRR are identical (provided the same dimensions and substrate are considered).

**Figure 7.**(

**a**) Open complementary split ring resonator (OCSRR) topology and its equivalent circuit model. (

**b**) Topology of a dumb-bell defected ground structure (DB-DGS) etched in a ground plane of a microstrip transmission line.

#### 3.2. Transmission Lines Loaded with Electrically Small Resonators

_{s}and C

_{s}are the inductance and capacitance of the SRR (or the considered coupled particle), and M is the mutual coupling between the line and the SRRs. The structure of Figure 8 exhibits a stop band behavior that has been interpreted as consequence of the negative effective permeability of the structure (caused by the presence of the SRRs) above resonance, and by the highly positive effective permeability below SRR resonance. To implement a backward wave (or left handed) CPW transmission line, inductive strips between the central strip and the ground planes, above the positions of the SRRs, have been introduced (Figure 9(a)) [34,35]. With the presence of the shunt connected strips, the structure switches its frequency response to a bandpass type response, where wave propagation in the first allowed (narrow) band is backward (typically this structure exhibits a forward wave transmission band at higher frequencies). In this case, the circuit model is that depicted in Figure 9(b), where L

_{p}accounts for the pair of shunt connected strips. The circuit model depicted in Figure 9(b) can be transformed to the circuit depicted in Figure 9(c) [35]. The following transformations apply:

**Figure 8.**(

**a**) Typical topology of a SRR loaded CPW. (

**b**) Equivalent circuit model of the unit cell. The magnetic wall concept has been applied, so that the model is that of the one half of the structure.

**Figure 9.**(

**a**) Typical topology of a SRR- and strip-loaded CPW. (

**b**) Equivalent circuit model of the unit cell. (

**c**) Transformed circuit model.

_{s}and Z

_{p}being the series and shunt impedance, respectively, of the π-model of the considered line (for a T-model, the same expression is also valid). The characteristic impedance of a periodic structure whose unit cell can be described by a π-model is:

**Figure 10.**Layouts and transmission coefficients of several CPWs loaded with pairs of SRRs, BC-SRRs and 2-SRs (

**a**) and with shunt strips (

**b**). Relevant dimensions are: Ring width c = 0.2 mm, distance between the rings d = 1 mm, external radius r

_{ext}= 5.4mm for the SRRs; c = 0.961 mm, d = 0.19 mm, r

_{ext}= 3.36 mm for the SRs; and c = 2.12 mm, d = 0.2 mm, r

_{ext}= 3.17 mm for the BCSRRs. For the CPW the central strip width is W

_{C}= 10 mm, the width of the slots is G = 1.59 mm and the length D = 12.2 mm. Shunt strip width w

_{S}= 0.2 mm. The considered substrate characteristics are: dielectric constant ε

_{r}= 10.2 and thickness h = 2.54 mm. From [38]; copyright © 2010, John Wiley & Sons; reprinted with permission.

_{c}and the capacitance C

_{c}. The structure exhibits a stop band behavior that has been interpreted as being due to the negative effective permittivity of the structure in the vicinity of CSRR resonance. A left handed transmission band can be generated by merely cutting gaps in the microstrip line, above the positions of the CSRRs [26]. In this case the structure is a passband and can be modeled according to the circuit depicted in Figure 12(b), where the gap is modeled by a π-model consisting of the series capacitance C

_{s}plus the fringing capacitances C

_{f}. Such a model can be transformed to the model depicted in Figure 12(c), with the following transformations [39]

_{par}= C

_{f}+ C

_{L}. Like for the SRR-loaded CPW, CSRR loaded lines can be analyzed through their lumped element equivalent circuit, and expressions (6) and (8). The parameter extraction technique for these CSRR-loaded lines has been published in [40].

**Figure 11.**(

**a**) Typical topology of a CSRR loaded microstrip line. (

**b**) Equivalent circuit model of the unit cell.

**Figure 12.**(

**a**) Typical topology of a CSRR- and gap-loaded microstrip line. (

**b**) Equivalent circuit model of the unit cell. (

**c**) Transformed circuit model.

_{s}-C

_{s}, modeling the OSRR. In practice, such shifting lines can be modeled by means of a lumped capacitance C and inductance L, and the circuit model is that shown in Figure 13(b), which in turn can be simplified to the model depicted in Figure 13(c). For OCSRR loaded lines (Figure 14), a similar phenomenology applies. That is, the structure cannot be simply modeled by means of a shunt connected parallel resonator. A certain phase shift appears, and the equivalent circuit model is that depicted in Figure 14(b,c). In the models of Figure 13 and Figure 14, L and C must be considered parasitic elements. The frequency response of CPW transmission lines loaded with either OSRRs or OCSRRs reveals that the structures exhibit a band pass behavior with very wide transmission bands. As will be shown in the next section, by cascading OSRR and OCSRR-loaded CPWs, it is possible to implement transmission lines with composite backward and forward characteristics, as well as wide band filters. As compared to other structures based on closed rings, the main advantage of these OSRR and OCSRR based lines, is that the equivalent circuit of the structures models their behavior to a very good approximation within the whole transmission band.

**Figure 13.**Layout (

**a**), circuit model (

**b**) and simplified circuit model (

**c**) of a CPW transmission line loaded with a series connected open split ring resonator (L’

_{s}= L

_{s}+ 2L).

**Figure 14.**Layout (

**a**), circuit model (

**b**) and simplified circuit model (

**c**) of a CPW transmission line loaded with a pair of open complementary split ring resonators, where C’

_{p}= 2C

_{p}+ 2C and L’

_{p}= L

_{p}/2. The backside strips (in dark grey) connecting the different ground plane regions are necessary to prevent the slot mode of the CPW and the second resonance of the open complementary split ring resonators.

_{s}and C

_{s}.

## 4. Applications

#### 4.1. Wideband Bandpass Filters and Dual-band Components based on OSRRs and OCSRRs

**Figure 16.**Topology of the unit cell of a CPW composite right/left handed transmission line based on a combination of series connected open split ring resonators in the external stages and a pair of shunt connected open complementary split ring resonators in the central stage. The considered substrate is the Rogers RO3010 with thickness h = 1.27 mm and dielectric constant ε

_{r}= 10.2. Dimensions are: l = 12.3 mm, W = 5 mm, G = 1.28 mm. For the open complementary split ring resonator: r

_{ext}= 2.9 mm, c = 0.5 mm, d = 1.2 mm. For the open split ring resonators: r

_{ex}

_{t}= 2 mm, c = d = 0.2 mm.

**Figure 17.**(

**a**) Frequency response of the circuit of Figure 16. (

**b**) Characteristic impedance. (

**c**) Dispersion diagram. The element values of the circuit model are (in reference of Figure 13(c) and Figure 14(c)): L’

_{s}= 6.94 nH, C

_{s}= 0.85 pF, C = 0.28 pF, L = 0.76 nH, L’

_{p}= 1.95 nH and C’

_{p}= 2.43 pF. From [43]; copyright © 2009, IEEE; reprinted with permission.

**Figure 18.**Fabricated wideband bandpass filter and frequency response. The substrate is the Rogers RO3010 with thickness h = 0.254 mm and measured dielectric constant ε

_{r}= 11.2. Dimensions are: l = 12.32 mm, W = 5.6 mm, G = 0.574 mm. For the OCSRR: r

_{ext}=2.1 mm, c = 0.5 mm, d = 0.16 mm. For the OSRR: r

_{ext}= 1.9 mm, c = 0.3 mm, d = 0.16 mm. The element values of the circuit model are (in reference of Figure 13c and Figure 14c): C’

_{p}= 1.779 pF, L’

_{p}= 3.216 nH, C

_{s }= 1.08 pF, L

_{s }= 4.89 nH, L = 0.384 nH, C = 0.182 pF and L

_{hs}= 0.2 nH, where an additional shunt inductance L

_{hs}in series with the OCSRR tank has been considered to take into account the shunt inductive effect of the narrow strip connection between the main line and the OCSRR (this being needed to predict the broadband response of the OCSRR). From [44]; copyright ℘ 2011, Elsevier; reprinted with permission.

**Figure 19.**Fabricated dual-band power divider and frequency response. (

**a**) Top view. (

**b**) Bottom view. (

**c**) Power division (S

_{21}, S

_{31}) and matching (S

_{11}). The substrate is the Rogers RO3010 with thickness h = 0.635 mm and dielectric constant ε

_{r}= 10.2. Dimensions are: l = 9 mm, W = 4 mm, G = 0.74 mm. For the open complementary split ring resonator: r

_{ext}= 0.9 mm, c = 0.2 mm, d = 0.2 mm. For the open split ring resonators: r

_{ext}= 1.5 mm, c = 0.3 mm, d = 0.2 mm. The wide metallic strip in the back substrate side has been added in order to enhance the shunt capacitance of the open complementary split ring resonator stage, as required to achieve the electrical characteristics of the device. The circuit parameters, referred to the circuits of Figure 13(c) and Figure 14(c) are: C = 0.2 pF, L = 0.25 nH, C

_{s}= 0.66 pF, L’

_{s}= 3.74 nH, C’

_{p}= 2.99 pF and L’

_{p}= 0.83 nH. From [43]; copyright © 2009, IEEE; reprinted with permission.

#### 4.2. Ultra Compact Elliptic-function Lowpass Filters Based on SIRs

**Figure 20.**Layout (

**a**), photograph (

**b**), and frequency response (

**c**) of the fabricated elliptic lowpass filter. Dimensions are: W = 5 mm, G = 0.55 mm, l = 6.6 mm, b = 2.42 mm, c = 2.7 mm. The strips of the meander lines are of 0.2 mm. The filter has been fabricated on the Rogers RO3010 substrate with measured dielectric constant ε

_{r}= 11.2 and thickness h = 254 μm. The circuit simulation corresponds to an ideal elliptic filter with the following characteristics: order-3 elliptic-function LPF with a passband ripple of L

_{Ar}= 0.1 dB, a cutoff frequency of f

_{c}= 1 GHz and a stopband attenuation of L

_{As}= 30.52 dB with the equal-ripple stopband starting normalized frequency Ω

_{s}= 2.5. From [46], copyright © 2010, IEEE; reprinted with permission.

#### 4.3. Dual-band Antennas Based on SRs

**Figure 21.**Photographs of the mono-band (

**a**) and dual-band (

**b**) fabricated UHF-RFID tags. (

**c**) Measured and simulated read ranges. The dimensions of the tag are 48 mm × 48 mm, and the strip width is 1.4 mm. The other relevant dimensions are l

_{m}= 16.3 mm, w

_{m}= 4.8 mm, d

_{l}= 7.3 mm, d

_{r}= 33.9 mm and d

_{f}= 14.2 mm. The final dimensions concerning the 2-SR and its relative position inside the antenna (shown in Figure 21b) are l

_{r}= 16.1 mm, w

_{r}= 6.8 mm, s

_{h}= 8.3 mm and s

_{v}= 5.4mm. The strip width of the 2-SR is 0.5 mm and the separation between spirals is 0.3 mm. The considered substrate is the Rogers RO3010 with the dielectric constant ε

_{r}= 10.2 and thickness h = 0.127 mm. From [47], copyright © 2011, IEEE; reprinted with permission.

## 5. Conclusions

## Acknowledgments

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**MDPI and ACS Style**

Durán-Sindreu, M.; Naqui, J.; Paredes, F.; Bonache, J.; Martín, F. Electrically Small Resonators for Planar Metamaterial, Microwave Circuit and Antenna Design: A Comparative Analysis. *Appl. Sci.* **2012**, *2*, 375-395.
https://doi.org/10.3390/app2020375

**AMA Style**

Durán-Sindreu M, Naqui J, Paredes F, Bonache J, Martín F. Electrically Small Resonators for Planar Metamaterial, Microwave Circuit and Antenna Design: A Comparative Analysis. *Applied Sciences*. 2012; 2(2):375-395.
https://doi.org/10.3390/app2020375

**Chicago/Turabian Style**

Durán-Sindreu, Miguel, Jordi Naqui, Ferran Paredes, Jordi Bonache, and Ferran Martín. 2012. "Electrically Small Resonators for Planar Metamaterial, Microwave Circuit and Antenna Design: A Comparative Analysis" *Applied Sciences* 2, no. 2: 375-395.
https://doi.org/10.3390/app2020375