In recent years, there has been a growing interest in electromagnetic bandgap structures (EBG), which exhibit the properties of rejecting electromagnetic waves at certain frequencies. The most interesting applications for these structures are the filtering of frequency bands or the suppression of undesired spurious passbands and harmonics in microwave or millimeter-wave circuits. However, traditional Bragg-effected-related EBGs require several periods to provide a significant stop-band.1 These structures may therefore be relatively large for certain microwave circuit applications. Moreover, the parameters of EBGs cannot be easily controlled for a fixed stop-band.
The split ring resonator (SRR) is an alternative to traditional EBGs for tailoring certain frequency responses of microwave devices. This structure, originally proposed by Pendry, et al.,2 consists of a pair of concentric metallic rings with splits etched on opposite sides. SRRs are able to inhibit signal propagation in the vicinity of the resonant frequency, which can be interpreted as corresponding to a frequency band with a negative permeability (–μ) provided by the SRR structure. Compared with conventional ring resonators, SRRs exhibit a quasi-static resonance by virtue of the distributed capacitance between concentric rings and the rings’ over all inductance. Essentially, a unit SRR structure behaves as a LC resonant circuit that can be excited by a time-varying magnetic field applied parallel to the rings’ axis. The potential of SRRs to reduce circuit dimensions relies on the fact that the structures can be designed with dimensions much smaller than the wavelengths at their resonant frequencies. A complementary split ring resonator (CSRR) was recently proposed,3–7 which is the negative image of an SRR. According to the duality and complementary concepts, CSRR exhibits dual characteristics of an SRR. CSRRs are subwavelength structures, which are about only one-tenth of the guided wavelength.
One- and two-dimensional CSRRs, etched in the ground plane of microstrip lines, have been proposed.5–7 Because of leakage, the enclosure problems should be considered when using these kinds of microstrip lines to design practical microwave circuits. In contrast with CSRRs etched in the ground plane, one-dimensional CSRRs etched in the conductor strip are presented in this article, showing better stop-band characteristics. A simple circuit model of a unit CSRR is used to study its characteristics, based on EM simulation results, and lumped elements of this model are extracted according to circuit theory. In order to show the effectiveness of the proposed scheme, two types of microstrip band-stop filters are designed, using CSRRs to improve the stop-band characteristics. The experimental results agree well with the simulation.
Study on Complementary Split Ring Resonator
The layout of the square-shaped CSRR (also called dielectric split ring resonator) is shown in Figure 1, where the metal regions are shown in gray. It is well known that the CSRR can be obtained by replacing the metal parts of the original split ring resonator with apertures, and the apertures with a metal plate. In the case of a microstrip line, the magnetic field loops around the line (as shown in Figure 2) and the SRRs are not very well excited since the magnetic field does not penetrate through the rings. In order to best excite these rings, and considering the duality theory, the electric field of the microstrip are used. If the one- and two-dimensional CSRRs are etched in the ground plane, the electric field is found to be perpendicular to the CSRRs.5–7 If the CSRRs are etched in the center conductor, there will be no leakage through the ground plane, and this structure can be integrated more easily with other microwave circuits. Simultaneously, for the microstrip line, the electromagnetic field is concentrated around the conductor strip, so the CSRRs etched in conductor strip can be more efficiently excited by a time-varying electric field than when etched in the ground plane.
To show this fact, unit CSRRs of the same dimensions are etched in 50 Ω microstrip lines. One CSRR is etched in the center conductor and the other in the ground plane right under the strip. The dimensions of the CSRR cell are l = 3 mm, c = 0.2 mm, g = 0.2 mm and d = 0.2 mm. The 50 Ω microstrip line, with a substrate relative dielectric constant εr = 2.2, a substrate height h = 1.2 mm and a strip width w = 3.8 mm, are chosen for all the simulations. The transmission losses S21 of the two microstrip lines, with unit CSRR etched in the ground plane and the center conductor, have been simulated with an EM simulator and the results are shown in Figure 3. The attenuation pole frequency fc of the microstrip with unit CSRR etched in conductor strip (approximately 8.62 GHz) is lower than when etched in the ground plane (approximately 9.06 GHz). The –3 dB bandwidth of the CSRR etched in the strip is nearly 1.15 GHz, while the –3 dB bandwidth of CSRR etched in ground is only 0.67 GHz. The dimensions of the CSRR determine the resonant frequency. For the same resonant frequency, the dimensions of CSRR etched in conductor strip are smaller than etched in the ground plane.
As previously mentioned, the electrical characteristics of the stop-band for a CSRR can be simply represented as a parallel LC resonant circuit, as shown in Figure 4. The lumped L-C parameters of the CRSS have been investigated in recent years, but their calculation methods are complicated for practical microwave circuit designs.3–4 However, the equivalent parameters of the parallel LC circuit of a CSRR can be extracted from EM simulation results, and the equivalent circuit of the CSRR is matched to the one-pole Butterworth low pass filter (LPF) response. The reactance of CSRR can be expressed as
where
ω0 = angular resonance frequency of the parallel LC resonator
The series inductance of the one-pole Butterworth LPF can be written as
where
ω' = normalized angular frequency
Z0 = characteristic impedance
g1 = normalized parameter of a
one-pole Butterworth LPF
According to circuit theory, the two reactance values must be equal at ωc
where
ωc = cut-off angular frequency of the parallel LC resonator
From Equations 1 to 3, the inductance L and capacitance C can be obtained as
where
f0 = resonant frequency, which can be obtained from EM simulation results
The equivalent-circuit parameters can be calculated from Equations 4 and 5. The simulated frequency responses using the extracted equivalent-circuit parameters for a 50 Ω microstrip line with unit CSRR are compared with the EM simulation results and show excellent agreement. The equivalent C and L of the unit CSRR in the conductor strip are calculated as 1.6479 pF and 0.2066 nH, respectively, while the equivalent C and L of the unit CSRR in ground plane are 2.5424 pF and 0.1212 nH. A small discrepancy between the EM and circuit simulations still exists because the LC parallel circuit is an ideal simple model, and the resistive loss of the radiation are not considered.
Simulations and Measurements
In order to demonstrate the stop-band validity of the CSRR, microstrip lines with double cells and quadruple cells etched in conductor strips are designed, respectively. The dimensions of the CSRRs are shown in Table 1. The layout of the double CSRR cells in the conductor strip is shown in Figure 5. The lengths of the CSRRs are l2 = 3.00 and l3 = 2.9 mm, respectively. The distance between the two CSRR centers is p = 6 mm. The simulated and measured results are shown in Figure 6. For the stop-band characteristics, the maximum insertion losses S21 are better than –20 dB, and the –3 dB relative bandwidth is approximately 16.8 percent. In order to improve the insertion loss performance and widen the bandwidth of the stop-band, a quadruple cells filter, etched in the conductor strip, is proposed, as shown in Figure 7. The dimensions of the CSRRs are shown in Table 1, and the period between adjacent CSRRs is ω = 6 mm. Figure 8 shows the simulated and measured S11 and S21 responses of this filter. The maximum S21 is greater than –23 dB and low return loss ripples are obtained in the stop-band; the –3 dB bandwidth of the filter is approximately 2.15 GHz (the relative bandwidth is nearly 25 percent). The measured S-parameters compare well with the ones obtained from EM simulators in the two cases. The small discrepancies may be attributed to fabrication tolerances and radiation losses. The measurements of the band-stop filters were obtained with a HP8510 vector network analyzer.
Conclusion
In this article, a CSRR etched in the conductor strip of a microstrip line and its equivalent circuit are studied. Compared with the CSRR etched in the ground plane, it can be excited more efficiently by a time-varying electric field and there are no enclosure problems. Two compact microstrip band-stop filters based on CSRRs etched in conductor strips are proposed and their S-parameters are calculated and measured. Deep and wide bandgap characteristics are shown in the proposed filters.
Acknowledgments
This work is supported by the Shanghai Leading Academic Discipline Project (No. T0102), the Foundation of Antenna and Microwave Key Laboratory in China (No. 51437080204QT0601) and the National Natural Science Foundation of China (No. 60571054).
References
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Sheng Zhang obtained his BS degree from Anhui Normal University, China, in 2000, and his MS degree from Hefei University of Technology, China, in 2004, respectively. He is currently working toward his PhD degree from Shanghai University, China. His research interests include electromagnetic bandgaps, substrate integrated waveguides and passive microwave components.
Jian-Kang Xiao obtained his BS degree in electronics and his MS degree in radio physics from Lanzhou University, China, in 1996 and 2004, respectively. Since then, he has been an electrical engineer at the Institute of Modern Physics, Chinese Academy of Sciences. He is currently working toward his PhD degree in radio physics, Shanghai University, China. His research interests include microwave and millimeter-wave theory and technologies.
Ying Li is currently a professor at Shanghai University. His research interests include electromagnetic theory, microwave and millimeter-wave techniques and their applications.
