Microstrip resonators have very important applications in microwave or millimeter-wave systems. They are important components of microstrip filters,1–2 microstrip oscillators3 and microstrip antennas,4 and enable microwave equipment miniaturization with improved performance. A stepped-impedance resonator (SIR) is the fundamental resonant element that can operate from RF to millimeter-wave frequencies and can be used in many kinds of filters, oscillators and mixers. The performance of a microstrip resonator relies on electromagnetic field distribution, the resonant frequency and quality factor Q. Microstrip filters have advantages such as low cost, small volume, high selectivity and are widely used in a variety of microwave systems to transmit energy in one or more passbands and to attenuate energy in one or more stopbands.

A defected ground structure (DGS)5,6 is a novel technique for improving the performance of filters or other microwave components and is formed by etching a pattern in the ground plane. This structure can change the current and its distribution in the ground plane, and thus increase the effective capacitance and inductance of the microstrip. Periodic and non-periodic DGSs have the property to reject certain microwaves in some frequencies, permitting elimination of spurious responses due to harmonics. In this article, a novel double-T-shaped microstrip bandstop filter with DGS and a novel double-H-shaped microstrip bandpass filter with DGS are proposed and their performances simulated and optimized. The calculated results show that the performance of the filters can be effectively improved by DGS and the experimental results verify the improvement.

Analysis for characteristics of A microstrip stepped-impedance resonator

A fundamental microstrip SIR is formed by joining together two microstrip transmission lines with different characteristic impedance Z1 and Z2, as shown in **Figure 1**. Zi is input impedance, Yi is input admittance and β is phase constant. *l*1 and *l*2 are physical lengths corresponding to electric lengths θ1 and θ2, respectively. The equivalent circuit of the SIR is derived, as shown in Figure 1b, where L is inductance, C is capacitance and RL is loaded-impedance. If the discontinuous of microstrip step and fringe capacitance of open-circuit port are omitted, Zi can be expressed as

The parallel resonant condition can be obtained on the base of Yi = 0; it is

Here, K is the impedance ratio. It can be seen from Equation 2 that the resonant conditions of SIR lie on θ1, θ2 and K. The total electric length of SIR can be expressed as

It can be shown from Equation 3 that the resonator’s length reaches a minimum value of 0 < K < 1, and a maximum value for K > 1.7 For K = 1, it is a conventional quarterwave uniform-impedance resonator (UIR), as shown in Equation 4.

On the resonant condition, SIR can also be equivalent by circuit,8 as shown in Figure 1c. Here

If the susceptibility of quarter-wavelength SIR is defined as BSA and the corresponding slop parameter is defined as bSA, we have

An H-shaped resonator is a symmetrical structure that consists of two transmission lines with different characteristic impedance, as shown in ** Figure 2**. The electric length of outer step is θ

_{2}, and that of inner step is 2θ1. Its equivalent transmission line model9 is shown in Figure 2b, where

The input admittance seen from the open-circuit port can be expressed as

Based on Yi = 0, the resonant condition (impedance ratio) can be achieved as

If θ1 = θ2 = θ the resonant condition and input admittance can be simplified.

The T-shaped microstrip SIR is shown in ** Figure 3**. The heights of the patches are W and W1, and the widths are L1 and L2, respectively. The dimensions of all the dielectric substrates mentioned in this article are fixed to 50 × 50 mm2 and their relative dielectric constant is εr = 2.2 or εr = 2.6. This kind of dielectric material has properties such as low dielectric loss (tan δ ≥ 5.10–4), low cost and easy manufacturability. The relationship between the resonant frequency f0 and W is shown in

**Figure 4**. It can be seen that W variations have little influence on the resonant frequency. When W1 = 10 mm, L1 = 10 mm, L2 = 10 mm and εr = 2.2, the resonant frequency of the dominant mode is nearly fixed at 2.85 GHz and that of the second-order mode is 4.52 GHz. When εr = 2.6, the resonant frequency of the dominant mode is 2.6 GHz and that of the second-order mode is 4.15 GHz. The relationship curves of the quality factor Q and W are shown in

**. It can be seen that the quality factor decreases when both W and εr increase. The relationship curves of Q and L1, L2 are shown in**

*Figure 5***and**

*Figures 6***, respectively, with W = 30 mm and W1 = 10 mm. They show that Q decreases when L1 and L2 increase and the Q of the second-order mode is larger than the one of the dominant mode.**

*7*Design of Novel SIR Coupling Microstrip Filters

A novel double-T-shaped bandstop filter with DGS is shown in ** Figure 8**. The dimensions of each T-shaped patch are W = 30 mm, W1 = 10 mm, L1 = 5 mm and L2 = 4 mm. The overall width of the coupled patch is L = 26 mm. The double-T-shaped resonators form a bandstop filter when coupled to each other, but the filter performance is not acceptable because of its narrow passbands near the stopband; consequently, DGS was used. The performance of the bandstop filter was calculated with Ansoft HFSS software. The dumbbell-shaped DGS on the ground plane is shown in

**Figure 9**, where a = 10 mm, b = 4 mm, g = 3 mm and l = 2 mm.

The simulated S-parameters of the filter are shown in Figure 10. If the relative dielectric coefficient of dielectric substrate is εr = 2.2, it can be seen that there is a wide stopband at a center frequency of 8.71 GHz, with a 0.49 GHz bandwidth (5.62 percent). If εr = 2.6, there is a stopband at a center frequency of 8.004 GHz, with a 0.444 GHz bandwidth (5.55 percent). It can be seen that the attenuation within the stopband is greater than 20 dB, the stopband center frequency will shift with different dielectric substrates and the 3 dB bandwidth of the passband near the operational stopband is more than 200 MHz. The experimental results, measured with a HP8510 vector network analyzer, are shown in Figure 11 with εr = 2.2. They agree with the simulated ones.

The layout of a novel double-H-shaped microstrip bandpass filter is shown in **Figure 12**. The dimension of every H-shaped patch is w = 15 mm, w1 = 5 mm, L = 40 mm, L1 = 10 mm. The coupling coefficient k is bandpass filter can be expressed as

In order to improve the performance of the filter effectively, a pair of DGSs is applied, and for each DGS, a = 9 mm, b = 5 mm, l = 4.2 mm and g = 3 mm. The simulated S-parameters of the double-H-shaped microstrip bandpass filter are shown in **Figure 13**, and the experimental results are shown in ** Figure 14**. For εr = 2.2, it can be seen that the center frequency is 3.275 GHz and there is an attenuation pole at 4.5 GHz. The 3 dB bandwidth is 0.75 GHz, the maximum insertion loss is 0.55 dB at the center frequency and the coupling coefficient is k = 0.229. For εr = 2.6, the center frequency is 3 GHz, the 3 dB bandwidth is 0.676 GHz, the maximum insertion loss is 0.37 dB at the center frequency and the coupling coefficient is k = 0.225. There is an attenuation pole at 4.18 GHz, and the maximum attenuation is greater than 32 dB. It can be seen from the figures that the simulated and measured frequency responses are in good agreement.

Conclusion

In this article, the performance of stepped-impedance resonators are computed and analyzed. A novel double-T-shaped microstrip bandstop filter with DGS and a double-H-shaped microstrip bandpass filter with DGS are designed and their performance is calculated, measured and analyzed. The simulated and measured frequency responses show good agreement. The proposed filters have advantages such as compact and simple structures, they are easily integrated and manufactured, and they show wide bandwidths, high attenuations and also demonstrate that the filters’ performance can be effectively improved by using DGS.

Acknowledgment

This work was supported by the Shanghai Leading Academic Discipline Project (No. T0103) and the National Natural Science Foundation of China (No. 60571054).

References

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**Jian-kang Xiao** received his BS degree in electronics from Lanzhou University, China, in 1996, and his MS degree in radio physics from Lanzhou University in 2004. He is currently working toward his PhD degree. Since 1996 he has been an electrical engineer at the Institute of Modern Physics, Chinese Academy of Sciences. His research interests include microwave and millimeter-wave theory and their applications.

**Shi-wei Ma** received his BS, MS and PhD degrees in radio physics, electronics, control theory and control engineering from Lanzhou University, Shanghai University, China, in 1986, 1991 and 2000, respectively. After two years as an STA research fellow at the Japan National Institute of Industrial Safety, he is now an associate professor in the automation department of Shanghai University. His current research interests include signal processing, time-frequency analysis theory, and applications and microsystems integration.

**Ying Li** is currently a professor at Shanghai University. His research interests include electromagnetic theory, microwave and millimeter-wave techniques and their applications.