A Compact Second-order Chebyshev Bandpass Filter Using U-shaped Resonator and Defected Ground Structure

A compact bandpass filter using U-shaped resonators and Defected Ground Structures is proposed and designed at 5.8 GHz. The U-shaped resonators are placed around an indirectly coupled feed line while the Defected Ground Structures are positioned beneath them. The Ushaped resonator and U-shaped Defected Ground Structure are responsible for the high and low band rejection respectively. The proposed bandpass filter obeys the second order Chebyshev response which has low insertion loss of – 1.87 dB, high rejection level and a sharp roll-off performance. The design is carried out using CST Microwave studio. The design is verified by fabricating and measuring the prototype in the laboratory. A good agreement is observed between the simulated and measured results.


Introduction
Communication transceiver utilizes bandpass filters to accept/reject the signals needed by devices. Research works in improving filter performances have been a challenging task to researchers. Filters are generally designed using coupled resonators of different shapes, sizes and configuration [1]. The filter configuration is done in a manner to achieve good filter performance such as low insertion loss, good selectivity and miniaturization. Various parallel coupled bandpass filters have been reported in literature, each having unique characteristics. The order of the parallel coupled filter is usually increased to improve the selectivity as seen in [2][3][4]. However from the reported literature, increasing the filter order also increases the overall filter size and results in increase in insertion loss. In [5][6][7], the parallel coupled stub is center-tapped to form a T-shaped stub and further improve the filter selectivity. Center-tapping the parallel coupled stub creates a single transmission zero. This design achieves only a single transmission zero on either side of the passband. Two transmission zeros at both high and low edges of the passband are obtained from the T-shaped stub filter shown in [6]. The parallel coupled filter is center-tapped at both sides with different unsymmetrical lengths of the stub. From the parallel coupled bandpass filters reported, the selectivity is improved by introducing stubs on the parallel coupled resonators.
A microstrip bandpass filter using stepped impedance resonator (SIR) and defected ground structure (DGS) is presented in [8], it has a good performance however the transition band on the higher band is quite high. In [9], a U-shaped DGS and two coupled U-shaped resonators are used in the design of a bandpass filter. The selectivity of the filter is quite slow. A bandpass-bandstop filter using U-shaped DGS and U-shaped cascaded resonator is presented. The filter possesses a sharp selectivity with low insertion loss. Consequently a trade-off is observed between key filter performances such as insertion loss, size and selectivity. However, in this paper, an enhanced technique for improving the selectivity of the parallel coupled resonator is proposed. This is achieved by center offsetting and transforming the parallel coupled resonator rather than centertapping with a stub as earlier reported. This results in a U-shaped resonator formed around an indirectly coupled feedline with good rejection in the high band edge. A pair of Defected Ground Structure (DGS) is introduced beneath the U-shaped resonator to form a good rejection at the lower band edge. The presence of DGS is usually used to enhance filter behavior by suppressing out-of-band signals.
The designed second-order bandpass filters when compared with the bandpass filter in [10][11][12] are found to be compact due to the method of coupling implemented (DGS).

Theoretical Design of Filter
The U-shaped resonator used in the bandpass filter design is obtained by transforming the parallel coupled resonator. The parallel coupled resonator is transformed by center-offsetting it to form a U-shaped resonator. The general equation for parallel coupled half wavelength resonator is given in (1)-(4) where J j , j + 1 is the characteristics admittance of the J-inverters, J n , n + 1 represents the filter inverters between the resonators, Y 0 is the characteristics admittance of the terminating lines, FBW is the filter fractional bandwidth, F U and F L are the upper and lower band frequencies of the filter, and g 0 , g 1 ,…, g n represent the lowpass prototype design parameters having normalized cutoff at Ω c equal to 1.
The parallel coupled resonator is shown in Fig. 1. It has a low quality factor. The parallel coupled resonator is center-offset as shown in Fig. 2 to improve the quality factor. It is observed that as the parallel coupled resonator of half-wavelength is moved away from the center a transmission zero is obtained when the offset is 10 mm away as shown in Fig. 1

to 3.
A sharper roll-off is obtained by doubling the centeroffset parallel coupled resonator as shown in Fig. 4. This sharper roll-off results from the strong coupling of the half wavelength resonator to the indirectly coupled feedline. The two resonators are further bent to form an L-shaped resonator. The L-shaped resonator increases the S 21 level by 8 dB due to the increase in coupling strength as compared     with the offset. The two L-shaped resonators then combine to form a U-shaped resonator around the indirectly coupled feedline as shown in Fig. 5. However, it is noteworthy that the roll-off of the low band rejection remains poor. This issue will be addressed in Defected Ground Structure section.

Filter Synthesis
The filter synthesis technique is used to determine the external quality factor and coupling coefficient of the filter. Q e1 and Q en representing the external quality factor of the  input and output of the filter. M i,i + 1 is the coupling coefficient between any two resonators. Equations (5)-(7) are used to determine Q e1 , Q en and M i,i + 1 .
A second-order filter with a passband ripple requirement of 0.1 dB and FBW of 0.03 is used in the filter design. The low-pass prototype parameter at cut-off frequency Ω c = 1, for the passband ripple is given as g 0 = 1, g 1 = 0.8431, g 2 = 0.6220 and g 3 = 1.3554. The calculated Q e1 and Q e2 is given as 18.735 while the coupling coefficient is given as 0.053. The external quality factor is used to determine the gap, g between the feedline and the first resonator. On the other hand, the coupling coefficient is used to determine the gap, s between the resonators.
The arrangement in Fig. 6a is used to extract the external quality factor of the filter. The arrangement helps to determine the length between the resonator and the feedline with the Port 2 weakly coupled which implies that Q e2 = ∞. Figure 6b shows the transmission coefficient magnitude response S 21 from which the external quality factor Q e is determined from simulation. The external quality factor is given as shown in (8), where f 0 is the resonant frequency and Δf is the 3 dB bandwidth as obtained from the simulation. The design curve represents the plot of Q e against the gap g as shown in Fig. 6c.
The mutual coupling coefficient is determined by the arrangement shown in Fig. 7a to obtain the gap s. The ports 1    Fig. 7b. The first and second peaks of S 21 are used to calculate the mutual coupling M as seen in (9). The design curve in Fig. 7c shows the plot of M versus length s. Fig. 6c and 7c are used to determine the filter dimensions. The filter dimensions are determined to be g = 0.4 mm and s = 0.5 mm. The resulting structure is shown in Fig. 8a. The performance of the filter and parallel coupled resonator line is shown in Fig. 8b.The filter is observed to have a good rejection in the high band however a poor rejection in the lower band is experienced. In order to overcome the poor rejection in the lower band a DGS is introduced to the filter structure as discussed in Defected Ground Structure section.

Integration of Filter with Defected Ground Structure
A pair of U-shaped DGS is used to introduce a transmission zero in the lower band of the filter. The U-shaped DGS is known for creating a single stop band [13]. The U-shaped DGS is placed directly underneath the U-shaped resonator. The structure and performance of the U-shaped DGS is shown in Fig. 9a and Fig. 9b, respectively. The parametric sweep obtained from varying the length of the DGS from 11.8 mm, 12.4 mm and 13.4 mm to yield a frequency of 5.5 GHz, 5 GHz and 4.5 GHz respectively is shown in Fig. 9b. The DGS slot functions at half-wavelength. The DGS slot is directly proportional to the effective inductance and inversely proportional to the effective capacitance. This infers that a decrease in the DGS slot results in a corresponding decrease in the effective capacitance hence an increase in the cut-off frequency vice-versa.
Therefore a pair of U-shaped DGS is placed beneath the U-shaped resonator as shown in Fig. 10a to give a better rejection at the lower frequency band. However the matching of the filter S 11 is disturbed and noticed to be poorly matched at -9 dB as shown in Fig. 10b   there is the need to optimize and obtain a suitable matching. This optimization is achieved through running parametric sweep. The spacing between the two U-shaped resonators is optimized from 0.5 mm to 0.8 mm to give a good matching. Figure 10c shows the comparison of the filter performance with and without the inclusion of DGS after optimization. The filter performance with DGS gives an enhanced selectivity on both the lower and higher band of the frequency as can be seen from the S 21 result. The filter has a good rejection level of which is greater than -49 dB with fast roll-off at both edges. The transmission zero on the lower passband edge and upper passband edge can be altered by modifying the length of the DGS and U-shaped resonators respectively.

Discussion of Results
The bandpass filter is designed on a TLY5 Taconic substrate with a dielectric constant of 2.2 and substrate height of 0.8 mm. The filter is designed to operate at 5.8 GHz. It is observed to have a measured low insertion loss of -1.87 dB, good rejection and fast roll-off at both sides of the passband. The measured passband of the filter is located between 5.65 GHz and 5.89 GHz. The filter has two poles located at 5.66 GHz and 5.83 GHz as seen from the S 21 performance. The transmission zeros are located at 5.2 GHz and 6.3 GHz respectively. The higher band rejection results from the filter synthesis of two U-shaped resonators while the lower band rejection results from the Ushaped DGS pair. The introduction of DGS helps to improve the selectivity without increasing the filter order thereby keeping a compact filter size. The final designed filter with its parameters are shown in Fig. 11 and Tab. 1. The filter is designed and simulated using CST Microwave Studio. The simulated results are validated through fabrication of the prototype and measurements in the laboratory with the aid of a Rohde and Schwarz ZVL Network Analyzer. The fabricated prototype is shown in Fig. 12. The comparison between the simulated and measured S 11 and S 21 results are shown in Fig. 13, it is observed that they agree to a large extent. The little discrepancy between the simulated and measured is caused by fabrication defects and the tolerance level of the substrate used.  The comparison between the proposed filter and other filters employing similar methods of design are presented in Tab. 2. The proposed filter has the advantage of a sharp rejection on both sides of the passband as compared with others who show good rejection on a single side of the passband.

Conclusion
A compact microstrip bandpass filter is proposed in this paper. The bandpass filter operates as a second order Chebyshev response at 5.8 GHz with a low insertion loss, good rejection and sharp roll off performance. The bandpass filter results from center offsetting a parallel coupled feedline to form a U-shaped resonator around an indirectly coupled feedline. The U-shaped resonator is responsible for the higher band rejection while the U-shaped DGS results in the lower band rejection. The bandpass filter is designed from a combination of U-shaped resonators and U-shaped DGS. The designed filter is suitable for WLAN applications.