International Journal of Antennas and Propagation

Volume 2015, Article ID 792750, 13 pages

http://dx.doi.org/10.1155/2015/792750

## Recent Advances in the Modeling of Transmission Lines Loaded with Split Ring Resonators

GEMMA/CIMITEC, Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain

Received 19 December 2014; Accepted 23 April 2015

Academic Editor: Ivan D. Rukhlenko

Copyright © 2015 Jordi Naqui et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper reviews the recent advances in the modeling of transmission lines loaded with split ring resonators (SRRs). It is well known that these artificial lines can exhibit a negative effective permeability in a narrow band above the SRR fundamental resonance, providing stopband functionality. By introducing shunt inductive elements to the line, the stopband can be switched to a pass band with left-handed (LH) wave propagation. For the design of microwave circuits based on these artificial lines, accurate circuit models are necessary. The former circuit model of SRR-loaded lines was presented more than one decade ago and is valid under restrictive conditions. This paper presents the progress achieved in the modeling of these artificial lines during the last years. The analysis, restricted to coplanar waveguide (CPW) transmission lines loaded only with SRRs (negative permeability transmission lines), includes the effects of SRR orientation, the coupling between adjacent resonators, and the coupling between the two SRRs constituting the unit cell. The proposed circuit models are validated through electromagnetic simulation and experimental data. It is also pointed out that the analysis can be easily extended to negative permittivity transmission lines based on complementary split ring resonators (CSRRs).

#### 1. Introduction

The topic of metamaterials has attracted the interest of researchers since 2000 when the first left-handed (LH) metamaterial structure was synthesized [1]. These artificial media exhibit unusual electromagnetic properties, derived from the simultaneous negative effective permeability and permittivity, as predicted by Veselago in 1968 [2]. After the seminal work [1], various papers have experimentally confirmed the properties of LH media. Among them, the experimental verification of negative refraction [3] and the demonstration of backward leaky wave radiation in one-dimensional LH media, that is, the analogous of backward Cherenkov radiation in bulk LH media, are worth mentioning [4]. There are many books devoted to the topic of metamaterials among the available literature [5–13]. The unusual (or exotic) properties of metamaterials can be applied to the design of microwave circuits and antennas. In particular, by replacing the ordinary transmission lines in distributed circuits by artificial lines based on metamaterial concepts, it is possible to implement microwave devices with reduced size, enhanced performance, and novel functionalities on the basis of impedance and dispersion engineering [5, 6, 8]. These artificial lines based on (or inspired by) metamaterials have been designated as metamaterial transmission lines, consisting of a host line loaded with reactive elements (inductors, capacitors, and/or resonators), and can be considered to be one-dimensional metamaterials (sometimes these lines are also referred to as transmission line metamaterials [14–16]). By loading appropriately the host lines, it is possible to achieve negative effective permeability , permittivity , or both negative parameters simultaneously . (Despite the fact that the permittivity and permeability are parameters of bulk media, the effective permeability and permittivity in transmission lines can be defined from the equivalence between plane wave propagation in source-free, isotropic, linear, and homogenous media and TEM wave propagation in transmission lines. The wave equations are identical if the permeability and permittivity are given by and , respectively, where and are the per-unit-length impedance (series branch) and admittance (shunt branch), respectively, of the equivalent T- or -circuit model (unit cell) of the considered line.) In the last case, wave propagation is allowed, and it is backward (i.e., the phase and group velocities are antiparallel). If only one of the constitutive parameters is negative, wave propagation is not allowed. One of the most interesting properties of metamaterials and metamaterial transmission lines is the controllability (of course within certain limits) of the effective permeability and permittivity. In metamaterial transmission lines, this controllability is equivalent to the controllability of the characteristic impedance and dispersion (or phase constant), which are the parameters of interest in circuit or antenna design.

From the unit cell model of the considered metamaterial transmission line, the characteristic impedance and dispersion are given by [17]where, , , and are elements of the matrix of the unit cell, is the unit cell length, and is the complex propagation constant; that is, , where and are the attenuation and the phase constant, respectively. If the unit cell is symmetric with regard to the ports and it is described by a T-model, the previous expressions can be written aswhereas if the considered model is the -model, the dispersion is given by the same expression, and the characteristic impedance iswhere and are the impedance of the series and shunt branch, respectively, of the T- or -circuit models, with and being the series reactance and shunt susceptance, respectively (it is assumed that losses are negligible).

In the propagation regions, and , and the dispersion relation in those regions can be expressed asAccording to the dispersion relation, a necessary (although not sufficient) condition for wave propagation is an identical sign for the series reactance () and shunt susceptance (). Moreover, if both signs are positive, wave propagation is forward, whereas it is backward (LH) if the signs of and are negative. To demonstrate this, we calculate the product using expression (4). The following result is obtained:Since, according to the Foster reactance theorem, the slope of the reactance/susceptance of a lossless network must be always positive, it follows that if , , then , corresponding to forward wave propagation. Conversely, if , , then , and wave propagation is backward (note that ).

Depending on the complexity of the loading elements of the host line, it is possible to achieve not only either forward or backward wave transmission but also a composite right-/left-handed (CRLH) behavior, namely, LH wave propagation at certain frequency bands and forward (or right-handed (RH)) wave propagation in other frequency regions. There are two main approaches for the implementation of metamaterial transmission lines: (i) the CL-loaded approach [14–16] and (ii) the resonant-type approach [18, 19]. In the first case, the host line is loaded with series capacitors and shunt inductors. At low frequencies, the loading elements are dominant and wave propagation is backward; at high frequencies, the host line dominates over the loading elements, and wave propagation is forward. Thus CL-loaded lines actually exhibit a CRLH behavior. In the resonant-type approach, a host line is loaded with electrically small resonators, such as split ring resonators (SRRs) [20], or complementary split ring resonators (CSRRs) [21] and other reactive elements. CPWs loaded with pairs of SRRs and shunt inductive elements exhibit a CRLH behavior [18]. Similarly, microstrip lines loaded with CSRRs and series capacitive gaps support the propagation of backward (low frequencies) and forward (high frequencies) waves [22]. Obviously, CRLH lines can also be implemented by combining the CL-loaded and the resonant-type approach (hybrid approach [23]) or by using multiple reactive elements and/or resonators in many different configurations loading the host line. By increasing the degrees of freedom, it is possible to obtain circuit functionalities not easily achievable by using ordinary lines, for example, the multiband functionality attainable by means of generalized (or extended) metamaterial transmission lines [24–27].

In this paper, the focus is on the modeling of metamaterial transmission lines based on SRRs. The aim is to analyze in detail some effects related to the resonant elements (SRRs) which are not usually taken into account but that may be important under certain conditions, namely, the coupling between SRRs and the SRR orientation with regard to the transmission line. For this reason, the analysis is restricted to SRR-loaded lines without the presence of shunt inductive elements or other elements loading the line. Thus, the considered structures exhibit stopband functionality and provided that the number of unit cells is high enough, such structures are one-dimensional negative permeability metamaterials. These structures have found applications as bandstop filters [28], multiband devices, including dual-band matching networks [29] and dual-band printed dipole antennas [30] and sensors [31].

#### 2. Structure under Study and Former Circuit Model

The typical topology of an SRR-loaded metamaterial transmission line, based on CPW technology, is depicted in Figure 1(a). The line is loaded with an array of SRR pairs etched in the back substrate side, with their centers roughly aligned with the slots of the line and their symmetry planes orthogonal to the line axis. With this configuration, the magnetic field generated by the line is able to excite the SRRs, and the structure exhibits a stopband in the vicinity of the SRR fundamental resonance frequency. This stopband has been interpreted as due to the negative effective permeability of the line just above the SRR resonance and to the high positive permeability below it (causing a strong mismatch); that is, the effective permeability is resonant and is described by the Lorentz model [1]. However, the stopband can also be explained from the lumped element equivalent circuit model of the unit cell of the structure, depicted in Figure 1(b). According to this model, firstly reported in [18], the SRRs are described by the resonant tank ; the CPW line section is described by the series inductance and the shunt capacitance , and accounts for the magnetic coupling between the line and the SRRs. This model can be transformed to the model depicted in Figure 1(c), where the following transformations applywithIt should be noted that interresonator coupling (i.e., the coupling between resonators of different cells as well as between the pair of resonators forming the unit cell) is neglected in that model.