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Planar metamaterials and many microwave circuits and antennas are designed by means of resonators with dimensions much smaller than the wavelength at their resonance frequency. There are many types of such electrically small resonators, and the main purpose of this paper is to compare them as building blocks for the implementation of microwave components. Aspects such as resonator size, bandwidth, their circuit models when they are coupled to transmission lines (as is usually required), as well as key applications, will be considered.

Resonators are key elements in radiofrequency (RF) and microwave engineering. Many passive and active circuits (such as filters, diplexers, oscillators, amplifiers,

To reduce device size, a possible strategy is to replace transmission line resonators with semilumped resonators. The advantage of this approach is the full compatibility with planar technology. As mentioned above, semilumped resonators are electrically small resonant elements. In such resonators, the resonance, rather than being related to a distributed Fabry-Perot type phenomenon, can be interpreted in terms of an LC equivalent circuit, with a clearly identifiable inductance,

As we have mentioned before, in semilumped resonators the inductance and capacitance can be associated to a certain element of the resonator topology (in certain cases, a single element can provide the inductive and capacitive behavior). To clarify this aspect, let us consider the implementation of a transmission line with a shunt connecting the series resonator to the ground (

(

The considered SISS is an open resonator, since it is driven by the current generated by the line. Let us now consider electrically small closed resonators, which can be electrically or magnetically driven. It is obvious that if a

Through these two examples, it is clear that electrically small resonators have a high potentiality in both microwave engineering and metamaterial synthesis. Nevertheless, microwave circuits and antennas based on metamaterial concepts and implemented by means of electrically small resonators like the SRR have been reported. Therefore, the analysis and comparison of such electrically small resonators is of highest interest and is the purpose of the next section.

(

There are many types of electrically small planar resonators. In this paper, the main focus will be on those resonators of interest for the design of metamaterials or metamaterial-based or inspired microwave circuits and antennas. The list may be too long, hence, it will be limited to the more relevant and useful implementations, according to the authors own experience.

Let us begin with the SRR already considered in the previous section [_{s}_{o}_{o}_{o}C_{pul}_{o}_{pul}_{o}_{o}

The inductance _{s}_{o}_{s}_{s}_{pul}

The model of

SRR topology and its equivalent circuit model.

The closer the rings of the SRR of

There is, however, another interesting aspect that differentiates the conventional SRR and the BC-SRR. The charge distribution in the SRR (depicted in

An alternative way to obtain inversion symmetry, thus overcoming bianisotropy is by introducing additional cuts in the edge coupled SRR, as

Topology and charge distribution of (

Another interesting particle that combines the advantages of small electrical size and uniplanar configuration is the spiral. The two-turn spiral 2-SR can be considered to be constituted of two rings with cuts in the same position and with an electric short between both rings through cross terminals, as shown in

Topology of the (

Another interesting electrically small planar particle is the stepped impedance resonator (SIR). It is well known that the length of a

The previous particles are closed particles that can be magnetically and, in some cases, electrically driven. Open particles can also be of interest in applications where the resonator must be excited by means of a voltage or current source. Inspired by the topology of the SRR of

Concerning open SIRs, the SISS reported in

Let us now consider the application of duality to some of the previous particles. If the topology of the SRR is removed from a metallic film, the resulting particle (_{c}_{o}_{o}_{o}_{pul}_{pul}

Complementary split ring resonator (CSRR) topology and its equivalent circuit model. Geometrical parameters of the CSRR are identical to those of

Complementarity can also be applied to the NB-SRR, DS-SRR, 2-SR, SIR and OSRR. Let us consider in certain detail the complementary counterparts of the SIR and OSRR, since they are of special interest for microwave and metamaterial-based circuit design. The complementary version of the open split ring resonator was called open complementary split ring resonator (OCSRR) [_{c}

The complementary counterpart of the SIR has been called dumb-bell defected ground structure (DB-DGS). The reason is that this structure is dumb-bell shaped, and it is typically etched in the ground plane of microstrip lines, creating thus a defect, or pattern, in such a ground plane (

(

Many other electrically small planar resonant particles, such as multiconductor SRRs or spirals [

In microwave applications, rather than isolated, the resonators considered in the previous sub-section are coupled (or connected) to host transmission lines (microstrip lines, CPWs,

Let us start with the equivalent circuit models of transmission lines loaded with closed resonators. Typically, the SRR (or the 2-SR, NB-SRR, BC-SRR and DS-SRR) are useful particles to load a CPW transmission line. They can also be used by coupling them to a microstrip transmission line, but the corresponding circuit models are not so accurate. The typical topology of a CPW loaded with pairs of SRRs, as well as the lumped element equivalent circuit model of the unit cell, are depicted in _{s}_{s}_{p}

(

(

The circuit model of

with _{s}_{p}

and for a T-model, it takes the form:

It has been mentioned before that the electrical size of the SRR can be decreased through different strategies. However, as discussed in [

Layouts and transmission coefficients of several CPWs loaded with pairs of SRRs, BC-SRRs and 2-SRs (_{ext}_{ext}_{ext}_{C}_{S}_{r}

Let us now consider microstrip lines loaded with CSRRs. The typical topology and circuit model (unit cell) are depicted in _{c}_{c}_{s}_{f}

where _{par}_{f}_{L}

(

(

The circuit models depicted in

The main relevant limitation of SRR- and CSRR-loaded lines is the narrow bandwidth, which is related to the limited coupling between the line and the resonator. In band pass structures, moderate and wide bands have been achieved by merging the backward and the forward wave transmission bands [

Let us consider a CPW loaded with an OSRR (_{s}_{s}

Layout (_{s}_{s}

Layout (_{p}_{p}_{p}_{p}

To end this section, we will consider the SIRs coupled to transmission lines. These resonators are useful to implement shunt connected series resonators. In microstrip technology, the SISS topology of _{s}_{s}

Typical layout of a SIR-loaded CPW (

In this section, three illustrative applications of some of the semilumped resonators considered before are highlighted. Rather than discussing them in detail, the main aim is to point out the key characteristics of the resonators for the considered applications.

OSRRs and OCSRRs can be combined in order to implement artificial transmission lines exhibiting left handed wave propagation at low frequencies and right handed wave propagation at high frequencies. The typical layout of these structures is depicted in

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 _{r}_{ext}_{ex}_{t} = 2 mm,

(_{s}_{s}_{p}_{p}

Fabricated wideband bandpass filter and frequency response. The substrate is the _{r}_{ext}_{ext}_{p}_{p}_{s }_{s }_{hs}_{hs}

Another possible application of OSRR and OCSRR-based lines is the design of dual-band components. As compared to dual-band components based on CSRRs [

Fabricated dual-band power divider and frequency response. (_{21}_{31}_{11}_{r}_{ext}_{ext}_{s}_{s}_{p}_{p}

Elliptic-function lowpass filters are characterized by the presence of transmission zeros that improve the frequency selectivity of the filter. These transmission zeros can be achieved by using grounded series resonators. Therefore, SIRs loading a CPW transmission line are specially suited for this purpose.

Layout (_{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 [

It has been recently demonstrated that dual-band antennas with very closely spaced operating frequencies (such as those required, for instance, in UHF-RFID applications to cover different regulated bands worldwide), can be implemented by coupling electrically small resonators to mono-band antennas [

Photographs of the mono-band (_{m}_{m}_{l}_{r}_{f}_{r}_{r}_{h}_{v}_{r}

In conclusion, we have reviewed the most used electrically small resonators for the synthesis of metamaterials and metamaterial-based or inspired microwave circuits and antennas. The intrinsic circuit models of these resonators as well as the circuit models of transmission lines coupled to them have been pointed out. Such transmission lines loaded with these electrically small resonators are the building blocks for the design of many microwave components. A comparative analysis in terms of resonator’s size, bandwidth, and suitability for certain applications, has also been carried out. This comparison has been the main relevant aspect of the present paper.

This work was supported by Spain-MICIIN (project contracts TEC2010-17512 METATRANSFER and CSD2008-00066). Thanks are also given to the Catalan Government for giving support through the project 2009SGR-421.