Introduction
The performance of microwave filters directly affects the overall effectiveness of the communication systems. Waveguide filters are widely used in satellite applications because of their high unloaded quality factor (Q u) and low insertion loss (IL). Compared with rectangular waveguide resonators [Reference Latif, Macchiarella and Mukhtar1, Reference Zhang, Shang, Zhang and Xu2], spherical waveguide resonators [Reference Zhang, Gao, Li, Yu, Guo, Li and Xu3–Reference López-Oliver and Tomassoni6] have higher Q u and lower IL. Transmission zeros (TZs) can improve the out-of-band rejection. However, filters designed in [Reference Zhang, Gao, Li, Yu, Guo, Li and Xu3–Reference Zhang, Guo, Zhang, Gao, Liu, Shu and Xu5] don’t have TZs, although they use spherical resonators. In [Reference López-Oliver and Tomassoni6], spherical waveguide filters introduce TZs on both sides of the passband but have a poor shape factor since they are all far from the passband. Furthermore, this filter prototype is split, which increases the radiation and degrades the performance of the filter. In addition, all the above filters are based on single-mode resonators, which leads to a bulky volume. Dual-mode resonators can achieve more coupling paths and more TZs with a compact size [Reference Chen, Zhang, Hong, Sun, Yang, Tang and Feng7, Reference Baranowski, Balewski, Lamecki and Mrozowski8]. The filter in [Reference Chen, Zhang, Hong, Sun, Yang, Tang and Feng7] creates dual-mode resonators by adding metal perturbation posts, which are very difficult to process. The dual-mode resonators in [Reference Baranowski, Balewski, Lamecki and Mrozowski8] are deformed cavities, reducing the Q u of the cavities and increasing the complexity of the design.
This paper introduces a novel dual-mode resonator. By positioning the spherical resonators at different angles, the degenerate modes are split, without any other additional perturbation. Applying this to the classic BSCT has the desired response, which produces a TZ at the upper stopband. Then a CT coupling topology is also introduced, which produces a TZ at the lower passband. Meanwhile, slots are etched on the spherical resonant cavities to suppress the spurious response. This is similar to the method of shaping and slotting spherical resonators in [Reference Guo, Li, Yu, Zhang, Li, Attallah and Lancaster9] and slotted hemispherical resonators in [Reference Li and Yuan10]. The proposed filter in this paper has the advantages of a better shape factor and multiple TZs.
Processing the confined CT filter with the inner cross-coupling iris is a challenge for conventional machining technologies [Reference Zhang, Shang, Xu, Guo, Duan and Lu11]. Stereolithography (SLA) is a typical 3-D printing technology, which can well print complex filter structures with high precision, lightweight, and smooth surfaces [Reference Tomassoni, Peverini, Venanzoni, Addamo, Paonessa and Virone12, Reference Guo, Shang, Li, Lancaster and Xu13]. So, this paper utilizes SLA technology to fabricate the designed filter and applies metallization to the surface of the filter.
Filter analysis and design
Dual-mode spherical resonator analysis
According to the theory of spherical resonators [Reference Zhang, Li, Chang, Zhang and Li14], the magnetic field expression for fundamental mode TM101 is as follows:
\begin{equation}
\boldsymbol{H}_{{\varphi},\mathrm{TM}_{101}}=\boldsymbol{\varphi }\cdot \frac{A}{\sqrt{r}}J_{3/2}\left( kr \right) \sin \theta \,\,
\end{equation}where A and k are constants, r is the radius of the spherical cavity, and J 3/2 is the spherical Bessel function. The θ and φ are the elevation and azimuth angles. It can be seen from equation (1) that the magnetic field is related to θ. To ensure the TM101 mode has a sufficient coupling, the coupling iris position is set to θ = 90° first.
Furthermore, equation (1) is also associated with the φ. Figure 1(a) illustrates the eigenmode simulation at φ = 0°. The magnetic fields of the two degenerate modes are arranged as concentric rings around the X and Y axes, respectively. They are mutually orthogonal and have the same resonant frequency. So, there is no coupling between this pair of degenerate modes. In order to couple them, some perturbation to the cavity is required. Figure 1(b) depicts the eigenmode simulation at φ = 30°, where the resonant frequencies of the two degenerate modes are split. And the magnetic field distributions are almost unchanged compared with those at φ = 0°. Remarkably, this novel structure not only acts as a perturbation but also serves as a coupling iris. The effect of the perturbation is not limited to φ = 30° but also occurs in other angles. As shown in Fig. 2, the resonant frequencies of the degenerate modes are gradually split as the angle increases. So, it greatly simplifies the design and processing complexity.
The magnetic distributions with different φ (r = 13.1 mm). (a) A pair of orthogonal TM101 degenerate modes (φ = 0°). (b) Split orthogonal TM101 degenerate modes (φ = 30°).
The resonance frequencies of orthogonal degenerate modes vary with φ.
A BSCT with three rotary cavities is proposed based on the above analysis. The BSCT structure effectively couples two orthogonal TM101 degenerate modes in the middle cavity (Cavity 2) by adjusting φ, so the three cavities realize the four poles. Compared with the conventional fourth-order inline filter, the BSCT filter can effectively reduce the volume by 25%.
Filter design based on the BSCT
A fourth-order bandpass filter (BPF) with BSCT is designed, as shown in Fig. 3(a). The BPF operates at 10 GHz, a fractional bandwidth of 1.7%, and a passband return loss (RL) of 20 dB. The coupling between the cavities is achieved by using the circular iris as presented in Fig. 3(b). As can be seen from Fig. 3(c), unlike conventional spherical waveguide filters in paper [Reference Zhang, Gao, Li, Yu, Guo, Li and Xu3], Cavity 1 and Cavity 3 are rotated by the φ = 30°. So, the TM101 degenerate mode in Cavity 2 (dual-mode cavity) can be coupled. In addition, due to the proximity of feed ports, a source-load coupling is also created. Moreover, there are also numerous holes in the spherical resonators of Filter A. These holes can further lighten the weight of the filter and facilitate the electroplating process. These holes have been carefully designed so as not to interfere with the filter.
The rotary structure waveguide bandpass filter (Filter A). (a) Main view. (b) Side view and section. (c) Top view. The crucial dimensions in millimeters are: a = 22.86, b = 10.16, r 1 = r 3 = 12.65, r 2 = 12.94, r 12 = 4.56, t = 3, w s = 11.02.
The topology of Filter A is depicted in Fig. 4(a), which is a fourth-order BSCT with source-load coupling. Furthermore, the solid line is the positive coupling, while the dashed line means the negative coupling. The calculated N+ 2 coupling coefficients are [Reference Mendoza, Martinez, Rebenaque and Alvarez-Melcon15, Reference Cameron, Harish and Radcliffe16]:
\begin{align}\left[ {\begin{array}{*{20}{c}}
0&{1.0300}&0&0&0&{0.00002} \\
{1.0300}&{0.3323}&{0.6600}&{ - 0.4000}&0&0 \\
0&{0.6600}&{0.6479}&{0.0150}&{0.6600\,}&0 \\
0&{ - 0.4000}&{0.0150\,}&{ - 0.4032}&{0.4000}&0 \\
0&0&{0.6600}&{0.4000\,}&{0.3066}&{1.0300} \\
{0.0002}&0&0&0&{1.0300}&0
\end{array}} \right].\nonumber\\ \end{align}As shown in Fig. 4(b), the matrix and EM simulations are in good agreement.
(a) Dual-mode box section coupling topology. (b) S-parameters of extracted coupling matrix and simulation.
In the filter realization, the external quality factor (Q e) is controlled by the width of the rectangular coupling apertures at the source and load, while the coupling coefficient (k) between the cavities can be controlled by varying the sizes of the coupling iris. The Q e and the k of the filter can be extracted by the following equations [Reference Hong and Lancaster17]:
\begin{equation}Q_e\;=\;\frac{f_0}{BW_{3dB}}\end{equation}
\begin{equation}k = \frac{{f_2^2 - f_1^2}}{{f_2^2 + f_1^2}}\end{equation}where f 0 is the resonant frequency of the filter and BW 3dB is the 3-dB bandwidth, f 1, and f 2 are the first and second resonant frequencies of the filter, respectively.
Figures 5(a) and (b) display the relationship between Q e and k with corresponding physical dimensions. It can be observed that Q e decreases as w s increases, while k increases as r 12 increases. Then, the dimensions of the filters are optimized based on the ideal models by EM simulation.
(a) The external quality factor (Q e). (b) The coupling coefficient (k cavity12 = k cavity23) between cavities.
In Fig. 6(a) and (b), the current distributions in Cavity 2 change with the different phases. This is caused by two split degenerate modes resonating at different frequencies. The current distributions in Cavity 1 and Cavity 3 are kept unchanged with φ.
(a) The current distributions at phase = 0°. (b) The current distributions at phase = 90°. (c) S-parameters of Filter A and an inline coupling filter.
In Fig. 6(c), due to the dual-mode resonator, Filter A exhibits four poles. And compared with the inline coupling filter, Filter A has TZs at 10.12 GHz and 12.34 GHz. The TZ at 10.12 GHz results from a 180° phase difference between the two main coupling paths, further improving the filter’s shape factor. Besides, the TZ at 12.34 GHz is caused by source-load coupling, which enhances out-of-band rejection and broadens the stopband bandwidth.
Filter design based on the hybrid BSCT and CT topology
To further optimize the performance of Filter A, the BSCT combined with the CT topology is proposed, which can also introduce a TZ at the lower stopband. Figures 7(a–c) show a new cross-coupling between Cavity 1 and Cavity 3. The calculated N + 2 coupling matrix is as follows:
\begin{align}\left[ {\begin{array}{*{20}{c}}
0&{0.9800}&0&0&0&{0.0002} \\
{0.9800}&{0.5020}&{0.6646}&{ - 0.3793}&{ - 0.1277}&0 \\
0&{0.6646}&{0.9218}&{0.0180}&{0.6646}&0 \\
0&{ - 0.3793}&{0.0180}&{ - 0.3223}&{0.3870}&0 \\
0&{ - 0.1227}&{0.6646}&{0.3870}&{0.4835}&{0.9800} \\
{0.0002}&0&0&0&{0.9800}&0
\end{array}} \right].\nonumber\\ \end{align}The optimized filter based on the hybrid BSCT and CT topology (Filter B). (a) Top view. (b) Sectional view. (c) The hybrid BSCT and CT topology. The crucial dimensions in millimeters are: a = 22.86, b = 10.16, r 1 = r 3 = 12.68, r 2 = 12.98, r 12 = 4.61, h = 5.75, w = 12, t = 3, w s = 11.02, φ = 60°.
As shown in Fig. 8(a), TZ2 remains unchanged when the height h of the iris is changed. And the movement of TZ3 towards the passband makes the shape factor better. Nonetheless, in Fig. 8(b), Filter B has a spurious response at 10.33 GHz, which severely deteriorates its performance. Figures 9(a) and (b) present the current distributions of Filter B at 10 GHz and 10.33 GHz, respectively. The current in Cavity 1 and Cavity 3 at 10.33 GHz are perpendicular and higher than those at 10 GHz. As shown in Fig. 9(c), the slots that are perpendicular to the direction of the spurious response current are etched. They are parallel to the current at 10 GHz simultaneously, so the current in the passband is not interrupted. However, the spurious modes can be suppressed effectively, which is further confirmed in Fig. 9(d).
(a) The positions of TZ change with different h values. (b) Simulated S-parameters of Filter A and Filter B.
(a) The current distributions of Filter B at 10 GHz. (b) The current distributions of the spurious response (10.33 GHz). (c) The hybrid BSCT and CT topology filter with slots (Filter C). (d) Simulated S-parameters of Filter C and Filter B. The crucial dimensions in millimeters are: l = 8.70, s = 2.00.
Fabrication and measurement
Since the cross-coupling iris between Cavity 1 and Cavity 3 is concealed, a prototype of Filter C is fabricated using SLA printing technology. A photosensitive resin is used for printing, with a printing resolution of 50 μm. After printing, a 10 μm copper layer is deposited on the surface of the filter to metalize it, as shown in Fig. 10(a). Before printing, the dimensions of the filter are increased by 10 μm to compensate for the thickness of the metal layer. Filter C is measured using a ZVB 20 vector network analyzer. In Fig. 10(b), the measured results are in good agreement with the simulated results. The average IL is 0.64 dB, and the RL is better than 17 dB in the passband. The measured results shift toward lower frequency about 35 MHz. The losses are slightly larger than the simulated ones due to the unavoidable tolerance in fabrication and measurement.
(a) Fabricated prototype of Filter C. (b) Measured and simulated results.
Finally, some comparisons between Filter C and some previous filters are summarized in Table 1. From the comparisons, Filter C is not dominant in size. However, it can generate a dual-mode resonator without adding metal perturbation posts or changing the shape of the cavity. Furthermore, the advantages of Filter C are obvious, offering not only a better shape factor but also more TZs.
Comparison with previously reported 3-D printed BPFs
SF: shape factor (BW40dB/BW3dB). N/A: not available.
Conclusion
Waveguide filters have attracted attention in satellite communications because of their excellent characteristics, such as higher Q u and lower IL. This paper designs an X-band spherical waveguide filter based on the hybrid BSCT and CT Topology. When the cavities are rotated, the coupling irises can serve as the perturbations to split the degenerate modes directly. So, three rotary resonator cavities can realize a fourth-order filter without additional design. Compared with an inline filter, this filter has TZs on both sides of the passband, which greatly improves the filter’s performance. A good agreement between the measurement and simulation results demonstrates the feasibility of the design.
Acknowledgement
This work was supported by the National Key R&D Program of China (2021YFC2902701).
Competing interest
The authors report no conflict of interest.
Sheng Zhang received the Ph.D. degree in electronics science and technology from Shanghai University, Shanghai, China, in 2007.
He is currently an Associate Professor at the School of Information and Control Engineering, China University of Mining and Technology (CUMT), Xuzhou, China. He has authored or coauthored over 40 journal and conference papers in this field. His current research interests include RF/microwave integrated circuits and antennas, especially in 3-D metal printing and substrate-integrated waveguides.
Hongliang Qiao was born in 1999 and is pursuing a master’s degree at the School of Information and Control Engineering, China University of Mining and Technology. His main research interests include high-Q waveguide filters based on 3-D printing and microstrip technologies.
Jiayu Rao (Member, IEEE) received the M.Sc. degree (Hons.) in electronics science and technology from China University of Mining and Technology (CUMT), Xuzhou, China, in 2018, and the Ph.D. degree in engineering science from Heriot-Watt University (HWU), Edinburgh, U.K., in 2022.
He is currently an Associate Professor at the School of Electrical and Information Engineering, Jiangsu University of Technology (JSUT), Changzhou, China. He has authored and coauthored over 20 journal articles and conference papers in this field. His research interests include RF/microwave passive devices miniaturization and integration, such as high-Q coaxial/waveguide filters, filtering antennas, and tunable lossy filters based on 3-D metal printing, substrate integrated waveguide (SIW), and microstrip technologies.
Dr. Rao is a guest Research Fellow with the High-Tech Research Institute, Nanjing University, Changzhou.
Xilong Wu was born in 2000 and is pursuing a master’s degree at the School of Information and Control Engineering, China University of Mining and Technology. His main research interests include substrate-integrated waveguide (SIW) and microstrip technologies.
Enhui Zhao was born in 2000 and is pursuing a master’s degree at the School of Information and Control Engineering, China University of Mining and Technology. Her main research interests include filtering antennas and microstrip technologies.
