Miniaturization is important for many high-frequency circuits, but often limited by the use of certain transmission-line technologies and circuit structures. Fortunately, compact composite right/left-handed transmission lines (CRLH-TLs) can be used to form resonant circuits without additional lumped circuit elements, helping to dramatically reduce the size of numerous printed-circuit-board (PCB) designs. In the present work, the CRLH-TLs are implemented by loading a host line with complementary split ring resonators (CSRRs) in combination with series gaps. Two designs will be presented, with the first employing equilateral triangular CSRRs and the second a novel CSRR; the latter will be realized by loading a pair of narrow slots in the split ring of the CSRR. In both approaches, the resonant frequencies are reduced by significant amounts. Equivalent circuit models will be offered for both cases.

A great deal of interest has been shown by the design community recently in metamaterials: Since they exhibit simultaneous negative values of effective permittivity and permeability, they are intriguing candidates for various high-frequency circuit structures. As far back as 1968, the existence of such materials was predicted, and the electrodynamic behavior of such materials was theorized.^{1} In 2000, Smith fabricated the first left-handed structure, consisting of metallic posts and split-ring resonators (SRRs).^{2} Metamaterials with negative permittivity and permeability can be implemented by SRR-based left-handed transmission lines, although another possible strategy employs a new structure based on the Babinet principle—i.e., complementary split-ring resonators (CSRRs).^{3}

These represent the negative image of SRRs, producing a sharp bandgap and negative permittivity effect around their resonant frequencies. The equivalent-circuit model of a CSRR structure behaves like a parallel inductive-capacitive (LC) resonant circuit.^{4} CSRRs can be effectively excited by electric fields and are more suitable for implementation in microstrip circuits than SRRs in view of the electromagnetic (EM) field distributions of microstrip lines.^{5} Composite right/left-handed transmission lines (CRLH-TL) can be synthesized in microstrip circuits, which is achieved by etching CSRRs in the ground plane and series capacitive gaps in the conductor strip.^{6}

In modern communication systems, various characteristics are favored for passive circuit design. Therefore, compact CRLH-TLs are important structures for consideration in passive circuit designs.^{7,8} Fractal technologies are often used to reduce the size of antennas and other components because of the unique space-filling properties of the fractal curves. These properties offers great potential for miniaturizing passive microwave circuits.^{9} As an example, ref. 10 details novel CSRRs geometries that use square second- and third-order Sierpinski fractal curves.

To better understand the potential for CRLH-TLs in microstrip circuits, two compact CRLH-TLs will be examined in this report. Figure 1(a) shows equilateral triangular CSRRs etched in a circuit ground plane. The dark grey area denotes the conductor strip and the light grey area is the ground plane. In Fig. 1(b) and Fig. 1(c), first- and second-order Koch fractal curves are used to form equilateral triangular CSRRs (of zero order). The original curve is an equilateral triangle with side length of l. All the iterations are circumscribed inside a circumference of radius r = (3l)^{0.5}/3 and overall perimeter of C_{k} = 3l(4/3)^{k}.^{11}

Figure 2 presents an equivalent-circuit model of these structures. In the model, parameter C_{s} represents the slot capacitance under the conductor strip, L_{s} is the line inductance, C_{g} is the gap capacitance, and C is the coupling capacitance between the line and the part surrounded by the slot. The equilateral triangular CSRRs are described by means of a tandem tank, L_{p} and C_{p} being the reactive elements and R accounting for losses. Two sections of microstrip line are used to compensate the phase response in the proposed structure.

CRLH-TL circuits were fabricated on F4B-2 woven-glass polytetrafluoroethylene (PTFE) copper-clad printed-circuit-board (PCB) material with relative dielectric constant of 2.65 and thickness of 1.5 mm. The dimensions of the proposed structure were optimized as follows: l = 11 mm; d_{p} = 0.6 mm; d_{s} = 0.7 mm; and d_{g} = 0.4 mm. The width of the fractal CSRRs is 0.3 mm. The electrical parameters were extracted by means of the Ansoft Serenade circuit simulation software from ANSYS, with details shown in Table 1.

Figure 3 shows images of the top and bottom layers of these structures, with circuit model, simulated S-parameters, and measured S-parameters shown in Fig. 4. The consistency of the simulated and measured results is apparent from Fig. 4. The resonant frequencies for the CRLH-TL structure with zero-order, first-order, and second-order Koch fractal CSRRs are 2.73, 2.12, and 1.80 GHz, respectively. As can be seen, reductions of 22% and 34% in the resonant frequencies are achieved, respectively, when first- and second-order Koch fractal CSRRs are substituted for conventional circuit structures.

Figure 5(a) shows a conventional C SRR etched into a ground plane. By loading a pair of narrow slots in the split of the ring, the second CRLH-TL approach was achieved, as shown in Fig. 5(b). Figure 6 shows simulated S-parameters for a CRLH-TL circuit based on the conventional CSRR and one based on the loaded-slot approach. The approach with the loaded slots reveals a lower resonant frequency. This proposed CRLH-TL design can be modeled by means of the equivalent circuit of Fig. 7. In this model, L is the line inductance and C_{g} is the gap capacitance. The CSRR is modeled by the parallel resonant circuit (with inductance L_{c} and capacitance C_{c}), while its coupling to the host line is modeled by capacitance C. To reduce the size of the proposed structure, the slots are altered as the meander-shaped lines shown in Fig. 8. The total length of the slots is fixed.

This second CRLH-TL circuit was fabricated on a TP-2 circuit substrate with relative dielectric constant of 6.0 and thickness of 1 mm. The dimensions of the proposed structure were: a = 8.8 mm; c = 0.4 mm; g = 0.25 mm; h = 0.4 mm; l = 14 mm; and t = 0.25 mm. In order to demonstrate the correctness of the equivalent circuit model and analyze the reason why the resonant frequency is lower than the conventional cell, the electrical parameters are extracted as follows in Table 2.

Two specific frequencies are used in the process: the resonant frequency, f_{r}, and the transmission zero frequency, f_{z}, where f_{z} = 1/2π[L_{c}(C + C_{c})]^{0.5} at which the impedance of the shunt branch is equal to zero. The shunt branch presents inductive impedance between 1/[L_{c}(C + C_{c})]^{0.5} and 1/(L_{c}C_{c})^{0.5}, which denotes equivalent negative permittivity. The series branch presents capacitive impedance when the frequency is less than 1/(LC_{g})^{0.5}, which denotes equivalent negative permeability.

Therefore, the left-handed band (including f_{r}) will appear if the frequency of the equivalent negative permittivity and negative permeability overlap. By comparing the results presented in Table 2, the frequencies f_{r} and f_{z} of the proposed structures can be lowered with respect to the resonant frequencies of a conventional CSRR,while proposed structures 1, 2, and 3 remain physically almost the same. Capacitances C and C_{c} are increased largely due to the significant increase of the coupling between the CSRR and the host line, essentially explaining why the resonant frequencies are lower in these modified approaches compared to conventional CSRR circuits.

The resonant and transmission-zero frequencies, f_{r} and f_{z}, of CSRR structures 1, 2, and 3 using the second CRLH-TL approach are almost the same. Only one of the structures, structure 3, was fabricated (Fig. 9) and then simulated to compare its modeled and measured behavior (Fig. 10). The simulated and measured results were consistent, revealing that the resonant frequency in this second CRLH-TL approach can be reduced by 52% when the CSRR structure 3 is substituted for a conventional CSRR.

In summary, both of these compact CRLH-TL approaches were developed and evaluated by implementing them as CSRR circuit structures. Both of these approaches are ideal for designing compact antennas.

**Acknowledgment**

This research has been supported by National Natural Science Foundation of China under Grant 60971118.

**References**

- V.G. Veselago, "The electrodynamics of substances with simultaneously negative values of e and ," Soviet Physics Uspeaki, Vol. 10, 1968, pp. 509-514.
- D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat-Nasser, and S. Schultz, "Composite medium with simultaneous negative permeability and permittivity," Physical Review Letters, Vol. 84, May 2000, pp. 4184-4187.
- F. Falcone, T. Lopetegi, J. D. Baena, R. Marqus, F. Martn, and M. Sorolla, "Effective negative-e stopband microstrip lines based on complementary split ring resonators," IEEE Microwave and Wireless Component Letters, Vol. 14, June 2004, pp. 280-282.
- J.D. Baena, J. Bonache, F. Martn, R. Marqus, F. Falcone, T. Lopetegi, M.A.G. Laso, J. Garca, I Gil, and M. Sorolla, "Equivalent circuit models for split ring resonators and complementary split rings resonators coupled to planar transmission lines," IEEE Transactions on Microwave Theory & Techniques, Vol. 53, April 2005, pp. 1451-1461.
- Sheng Zhang, Jian-kang Xiao, and Ying Li, "Novel microstrip band-stop filters based on complementary split rings resonator (CSRRs)," Microwave Journal, Vol. 49, November 2006, pp. 100-112.
- F. Falcone, T. Lopetegi, M. A.G. Laso, J.D. Baena, J. Bonache, R. Marqus, F. Martn, and M. Sorolla, "Babinet principle applied to the design of metasurfaces and metamaterials," Physical Review Letters, Vol. 93, November 2004, pp. 197401-4.
- H.Y. Zeng, G.M. Wang, J.G. Liang, and X.J. Gao, "Complementary Split Ring Resonators Using Equilateral Triangular Koch Fractal Curves," Proceedings of the IEEE International Symposium on Microwave Antennas & Propagation, EMC Technology on Wireless Communications (MAPE), October 2009, pp. 911-913.
- H.Y. Zeng, G.M. Wang, Z.W. Yu, and C.X. Zhang, "Novel Compact Single Complementary Split Ring Resonator," Proceedings of the IEEE International Symposium on Microwave Antennas & Propagation, EMC Technology on Wireless Communications (MAPE), October 2009, pp. 914-916.
- F. Falcone, T. Lopetegi, J.D. Baena, R. Marqus, F. Martn, and M. Sorolla, "Effective negative-e stopband microstrip lines based on complementary split ring resonators," IEEE Microwave and Wireless Component Letters, Vol. 14, June 2004, pp. 280-282.
- Vesna Crnojevic-Bengin, Vasa Radonic, and Branka Jokanovic, "Fractal Geometries of Complementary Split-Ring Resonators," IEEE Transactions on Microwave Theory & Techniques, Vol. 56, October 2008, pp. 2312-2321.
- V. Crnojevic-Bengin, "Novel compact microstrip resonators with multiple 2D Hilbert fractal curves," Microwave and Optical Technology Letters, Vol. 48, February 2006, pp. 270-273.