Dual-Narrowband/Wideband Filters Using Modified Coaxial Cavities With Large Frequency Ratios

This article presents two types of dual-narrowband/ wideband filters using modified coaxial cavities with large frequency ratios. The dual-narrowband/ wideband filters mean two dual-band bandpass filters, one of which is a dual-band wideband filter, and the other is a dual-band narrowband filter. The proposed coaxial resonator in the dual-narrowband filter consists of a pair of disk-loaded coaxial stepped impedance resonator (DLCSIR), which provides a large frequency ratio. The frequency ratio of the first two resonant modes of the DLCSIR can be precisely controlled by adjusting the impedance ratios and dimensions of the loading disk. By contrast, the proposed coaxial resonator in the dual-wideband filter is a rectangular coaxial stepped impedance resonator (RCSIR), which has the merits of miniaturized size, large frequency ratio, and convenience for strong coupling. Finally, two second-order dual-narrowband/ wideband filters with large frequency ratios were designed, fabricated, and measured. The exhibited dual-passband performance of the two filters verifies the proposed concept.


I. INTRODUCTION
Now, modern wireless communications systems are heading for high-speed data transmission. However, the frequency spectrum is getting crowded below 6 GHz, and the fifthgeneration (5G) wireless systems have been moved up to the millimeter-wave spectrum, which triggers the demand for multiband filters with large frequency ratios, especially multi-wideband filters. Recently, dual-band planar filters with a large frequency ratio based on the complementary frequency response characteristics of the stepped-impedance resonator bandpass filter and low-pass filter [1] and mode composite coplanar waveguide [2] have been proposed. In addition, a tunable dual-band filter with a large frequency ratio, combining a microstrip bandpass filter and a dielectricloaded bandpass filter, has been proposed in [3]. However, these structures are difficult to apply to the base transceiver stations. By contrast, multiband cavity filters can be broadly classified as dielectric-loaded filters [4], [5], [6], [7], helical filters [8], waveguide filters [9], [10], [11], [12], [13], [14], [15], and coaxial filters [16], [17], [18], [19], [20], [21], [22]. Despite the size advantage of dielectric-loaded filters, the closer harmonics make it more difficult to achieve large frequency ratios. Compared to significant insertion loss helical cavity filters and large-size waveguide cavity filters, coaxial cavity filters are more suitable for achieving large frequency ratio characteristics due to their use of quarter wavelength resonators, which are naturally far harmonics.
In this paper, DLCSIR is used to realize a dual-narrowband filter with a large frequency ratio, and RCSIR is employed to achieve a dual-wideband filter with a large frequency ratio for the first time. There are two innovations in this paper. The first innovation is to achieve controllable resonant frequency by adding a disk on the coaxial stepped impedance resonator and achieving a controllable coupling coefficient through different magnetic fields of the fundamental and second modes. The second innovation is to achieve a strong coupling through rectangular step transmission lines across the coupling window, thus realizing a dual-wideband filter. The dual-narrowband filter has resonators with a large frequency ratio and an adaptive feeding structure and coupling structure, which provides independently controlled frequencies and bandwidths of the two bands in a specific range. Moreover, the dual-wideband filter adopts a resonant structure which is easy to realize strong coupling. Meanwhile, miniaturized size is also one of its characteristics. A detailed analysis of the two novel filters is introduced, including resonator analysis, feeding analysis, coupling analysis, and filter design. Finally, two examples of the proposed second-order dual-narrowband/ wideband filters are demonstrated. This paper is organized as follows. A detailed design analysis of the dual-narrowband filter and dual-wideband filter is provided in Section II and Section III, respectively. Finally, Section IV is the conclusion of this paper, and the whole design is summarized.

II. DUAL-NARROWBAND FILTER DESIGN A. DLCSIR ANALYSIS
The geometry of DLCSIR is depicted in Fig. 1. This resonator is formed on a cuboid cavity with the length and width both are parameter w, and the height is parameter h. A disk-loaded stepped impedance coaxial line is embedded vertically in the center of the bottom sidewall of the cavity. The specific parameters are shown in Fig. 1(b). The fundamental and second modes are used to achieve the two narrow passbands of the filter.
CST Studio (EM simulation software) simulated the two proposed filters in this study. Field distribution of the fundamental mode and second mode is displayed in Fig. 2.  Fig. 2(a)-(d) respectively depict the electric field (E-field) distribution of the fundamental mode, the E-field distribution of the second mode, the magnetic (B-field) distribution of the fundamental mode, and the B-field distribution of the second mode. We can find that with the loading of a disk, the position of the strongest E-field moves down from the top to the loading position at the fundamental mode resonance. On the contrary, the electric field at the loading position is the weakest at the second mode resonance. For B-field, its strength is just the opposite of the electric field. The stronger the electric field is, the weaker the magnetic field is, and vice versa.
The first two resonant frequencies of DLCSIR are employed as the first and second passband center frequencies, namely, f res1 and f res2 . The resonant frequencies of the first three resonant modes against the dimensions of Ø 2 and Ø 3 are shown in Fig. 3(a) and (b). In Fig. 3(a), Ø 3 = Ø 1 is a testing condition, and the aim is to observe the resonance characteristics of coaxial SIR without disk loading. Fig. 3(a) illustrates that the resonant frequencies of the first two modes (f res1 and f res2 ) keep away from each other as Ø 2 increases. Meanwhile, the frequency of the third resonant mode (f res3 ) decreases as Ø 2 increases, especially when Ø 2 is greater than 8, and f res3 decreases sharply. Fig. 3(b) illustrates that the  resonant frequencies of the first and the third modes (f res1 and f res3 ) decrease as Ø 3 increases, while the resonant frequencies of the second mode are nearly unchanged. The reason is that the electric field at the loading position is the weakest at the second mode resonance, which leads to the loading disk having little effect on DLCSIR at the second mode [23]. Therefore, according to Fig. 3(a) and (b), the conclusion is that the frequency ratio (f res2 /f res1 ) can be further increased based on coaxial SIR by loading a disk.
As shown in Fig. 4, DLCSIR is equivalent to the transmission line model of the stub-loaded quarter-wavelength step impedance resonator [24]. The input admittance of DLCSIR can be obtained by Analytic expressions of resonance condition can be derived by Where f 1 and f 2 are the first two (fundamental mode and second mode) resonant frequencies of the resonator, respectively. θ 1(2) and θ 2(2) denote the electrical lengths at f 2 . θ 1(1) , θ 2(1) , and θ op(1) are the electrical lengths at f 1 . Y 1 and Y 2 represent the admittances of the resonator. According to Eqs. (2) and (4), the conclusion is that θ 1 = 2θ 2 and the loading stub doesn't affect the resonator at the second resonant mode but have an effect on the first resonant mode. The parameters of DLCSIR H 1 , H 2 , H 3 , Ø 1 , Ø 2 , Ø 3 , f res1 , and f res2 are related to the parameters of the resonator θ 1 , θ 2 , Y op , Y 1 , Y 2 , θ op , f 1 , and f 2 , respectively. Therefore, the previous conclusion can be changed to H 1 = 2H 2 , and the loading disk has little effect on DLCSIR at the second resonant mode, but it affects the resonant frequencies of the fundamental mode.

B. FEEDING STRUCTURE AND COUPLING STRUCTURE
As shown in Fig. 5(a)-(c), a simple feeding structure using a rectangular plate is proposed. Fig. 6(a) and (b) show the external quality factors of the first two resonant modes against the dimensions of the rectangular plate (H 5 and H 6 ). In Fig. 6(a), the external quality factors of the first two resonant modes (Q e1 and Q e2 ) decrease as H 5 increases from 0 mm to 6 mm.  When H 5 increases from 6 mm to 8 mm, the Q e1 remains decreasing, while the Q e2 increases. The Q e1 and Q e2 decrease with the increase of H 6 in Fig. 6(b). Fig. 7(a) and (b) show the coupling structure with three apertures. The top aperture and the bottom aperture are used to provide electric and magnetic coupling for two coupling coefficients of the first two modes (K 1 and K 2 ), respectively, and the middle aperture, whose height is facing the loading point is used to control K 2 independent. As shown in Fig. 8(a), K 1 and K 2 decrease when H 7 varies from 1 to 6 mm, and K 1 decreases faster than K 2 . By contrast, in Fig. 7(b), when W 3 varies from 6 to 11 mm, K 2 increases, and K 1 is nearly unchanged. The reason is the difference in B-field distributions of the first two modes. The B-field at the loading position is the weakest at the fundamental mode resonance and strongest at the second mode. Therefore, K 1 and K 2 can be achieved by selecting different combinations of H 7 and W 3 .

C. FILTER ANALYSIS AND RESULT
A second-order dual-narrowband bandpass filter (Filter I) using DLCSIR is designed and implemented. The two center frequencies of the two passbands are1.2 and 5.8 GHz, and two ripple 0.0432 dB FBWs of 1.2% and 0.6% are selected. The lumped circuit element values of the Chebyshev lowpass prototype filter with ripple 0.0432dB are g 0 = 1, g 1 = 0.6648, and g 2 = 0.5445. The external quality factors (Q e1 and Q e2 )  and coupling coefficients (K 1 and K 2 ) of the two passbands can be calculated from [25] According to Fig. 6, Q e1 and Q e2 can be realized by H 5 and H 6 . By contrast, according to Fig. 8, K 1 and K 2 can be achieved by H 7 and W 3 , respectively. The configuration of the proposed second-order dual-narrowband filter, named Filter I, is shown in Fig. 9. The circuit is symmetric about the central plane of the coupling window. The fabricated filter by adopting a silver-plated copper cavity is provided in Fig. 10. The physical dimensions of the fabricated prototype of Filter I are shown in Table 1. The internal dimensions of this filter I are 12 mm × 25 mm × 40 mm, and in the two passbands, the unloaded Q values are respectively 1900 and 3470. Fig. 11 shows the filter along with the simulated and test results. The measured inband insertion losses are approximate 0.43 and 0.40 dB at 1.2 and 5.8 GHz, respectively. The measured inband return losses are below 14 and 28 dB, and the simulated inband return losses are below 21 and 24 dB. The slight discrepancy is primarily caused by an extra soldering width between the Small-A-Type (SMA) connectors and the  feeding lines. Besides, the S-parameters of test results and simulation results are consistent regarding the center frequencies, the transmission zeros, and fractional bandwidth.

III. DUAL-WIDEBAND FILTER DESIGN A. RCSIR ANALYSIS
To achieve dual-wideband characteristics with a large frequency ratio, RCSIR is proposed. Compared with the DLC-SIR mentioned above, to achieve strong coupling, the cylindrical structure is changed to a rectangular structure, and the loaded disk is removed. The configuration of RCSIR is shown in Fig. 12. This resonator is formed on a cuboid cavity with the length and width both are parameters w 1 and w 2 , and the height is parameter h. The bottom sidewall of the cavity is embedded with a rectangular stepped impedance line, which is not placed in the center of the cavity for realizing independent control of resonant frequencies and strong coupling. AA' and BB' are the symmetry planes of Fig. 12(a) and (b), and the corresponding offsets are p and q, respectively. The value of gap S 1 is minimal for equivalent strong coupling between resonators. The Field distribution of the fundamental mode and second mode of RCSIR is displayed in Fig. 13. Fig. 13(a)-(d) respectively show the E-field distribution of the fundamental mode, E-field distribution of the second mode, B-field distribution of the fundamental mode, and B-field distribution of the second mode. Compared with the E-field of the two modes, the E-field of the fundamental mode is stronger than that of the second mode in the upper left region of Fig. 13(a) and (b), and even the reverse electric field appears at the second mode when the height decreases. Compared with the B-field of the two modes, the reverse magnetic field appears at the second mode in the bottom region of Fig. 13(d).
The first two resonant frequencies of RCSIR are used as the first and second passband frequencies, namely, f res1 and f res2 . The resonant frequencies of the first three resonant modes against the dimensions of H 2 and W 1 are shown in Fig. 14(a) and (b). In Fig. 14(a), the resonant frequencies of the first two modes (f res1 and f res2 ) keep away as H 2 increases due to the reverse electric field at the second mode. Fig. 14(b) illustrates that the resonant frequencies of the first and the third modes (f res1 and f res3 ) decrease as W 1 increases, while the resonant frequencies of the second mode are nearly unchanged. The reason is that the E-field of the fundamental mode is much stronger than that of the second mode in the upper left region of the Fig. 13(a) and (c). Therefore, according to Fig. 14(a)  and (b), the desired f res1 and f res2 can be achieved.

B. FEEDING STRUCTURE AND COUPLING STRUCTURE
As shown in Fig. 15(a), a simple tapping feeding structure is employed. Fig. 15(b) shows the external quality factors of the first two resonant modes against the dimensions of the tapping position, H t . The external quality factors of the first two resonant modes (Q e1 and Q e2 ) decrease as H t increases from 5 mm to 25 mm. When H t increases from 25 mm to 40 mm, the Q e1 remains decreasing, while the Q e2 increases.     The ranges of Q e1 and Q e2 are relatively large due to the wide range of H t . Fig. 16(a)-(c) show the coupling structure, which consists of two apertures. The top aperture provides strong electric coupling for two coupling coefficients of the first two modes (K 1 and K 2 ). In order to realize strong coupling, the two rectangular stepped impedance lines pass through the top aperture, and the strong coupling coefficients are mainly decided by S 2 and W 6 . The bottom aperture is used to fine-tune K 1 and K 2 . As shown in Fig. 17(a), K 1 and K 2 decrease when S 2 increases from 0.2 to 1.8 mm, and K 1 decreases faster than K 2 . The large values of K 1 and K 2 also verify the strong coupling characteristic. In Fig. 17(b), K 1 and K 2 decrease as H 5 increases from 5 mm to 20 mm. When H 5 increases from 20 mm to 35 mm, K 1 decreases, while the K 2 increases due to the reverse E-field at the second mode. Therefore, K 1 and K 2 can be realized by selecting different combinations of S 2 and H 5 .

C. FILTER ANALYSIS AND RESULT
A second-order dual-wideband bandpass filter (Filter II) using RCSIR is designed and implemented. The two center frequencies of the two passbands are 1 and 4 GHz, and two ripple 0.0432 dB FBWs of 9% and 7% are selected. The lumped circuit element values of the Chebyshev lowpass prototype filter with ripple 0.0432 dB are g 0 = 1, g 1 = 0.6648, and g 2 = 0.5445. The external quality factors (Q e1 and Q e2 ) and coupling coefficients (K 1 and K 2 ) of the two passbands can be calculated as: According to Fig. 15, Q e1 and Q e2 are decided by H t . By contrast, according to Fig. 17, K 1 and K 2 can be achieved by S 2 and H 5 . The configuration of the proposed second-order dual-wideband filter, named Filter II, is shown in Fig. 18. The circuit is symmetric about the central line of the coupling windows. The fabricated filter by adopting a silver-plated copper cavity is provided in Fig. 19. The physical dimensions of the fabricated prototype of Filter II are given in Table 2. The internal dimensions of Filter II are 22 mm × 16 mm × 68 mm, and in the two passbands, the unloaded Q values are respectively 740 and 1190.
The measured S-parameters are compared with the simulated ones in Fig. 20. The measured minimal insertion losses are approximate 0.39 and 0.16 dB at 0.98 and 4.06 GHz, respectively. In contrast, the measured return losses are below   14 and 24 dB. Finally, Table 3 compares the two proposed filters with several reported cavity dual-band filters, which show the advantages of this work in terms of large frequency ratios and miniaturized size.

IV. CONCLUSION
Two types of dual-narrowband/wideband filters have been proposed using modified coaxial cavities with large frequency ratios. DLCSIRs are adopted to realize the dual-narrowband filter. The frequency ratios of DLCSIR can be controlled by the impedance ratio and loading disk. In contrast, RCSIRs are utilized to achieve the dual-wideband filter. Two rectangular stepped impedance lines pass through the top aperture, which provides a strong coupling. Finally, the two proposed filters are designed, fabricated, and tested with good results. The proposed techniques are potentially applicable to multi-band systems with large frequency ratios. ZHI-CHONG ZHANG (Member, IEEE) was born in Ji'an, Jiangxi, China. He received the B.S. degree in communication engineering from Nanchang University, Nanchang, China, in 2008, the M.E. degree in communication and information system from the East China of Jiaotong University, Nanchang, China, in 2012, and the Ph.D. degree in electromagnetic fields and microwave technology from the South China University of Technology, Guangzhou, China, in 2015. Since 2020, he has been an Associate Professor with the School of Electronic and Information Engineering, Jinggangshan University, Ji'an, China. His research interests include the design of microwave filters and associated RF modules for microwave and millimeter-wave applications.