Compact 3-D Printed Bandpass Filters Using Folded Mixed Hemispherical Resonators

A new class of high quality factor and highly miniaturized hemispherical resonator-based bandpass filters (BPFs) with asymmetric transfer functions is presented. The proposed BPF concept is based on a monolithic integration scheme using 3-D printed hemispherical resonators. Size compactness is achieved by: 1) effectively using the available 3-D volume through resonator folding and capacitive loading; 2) introducing transmission zeros (TZs) by cross coupling; and 3) by monolithic integration enabled by a stereolithography apparatus (SLA) 3-D printing manufacturing method. A design methodology starting from coupled-resonator-based synthesized examples and translating them to a physical geometry is presented. The concept has been validated at <inline-formula> <tex-math notation="LaTeX">$X$ </tex-math></inline-formula>-band through the manufacturing and testing of a three-pole/one-TZ BPF prototype with: center frequency: 8 GHz, 3-dB BW of 0.37 GHz fractional bandwidth (FBW): 4.6%, and minimum insertion loss (IL): 0.287 dB (effective <inline-formula> <tex-math notation="LaTeX">$Q$ </tex-math></inline-formula> of 1400).


I. INTRODUCTION
Emerging Internet-of-Space (IoS) and 5G+/6G communications are aiming to provide seamless and high data-rate connectivity.Furthermore, they will be exploited for a variety of governmental and crisis management applications.However, their practical development is really challenging due to the lack of multifunctional, highperforming, and compact RF front ends/transceivers.In particular, RF transceivers supporting IoS and medium-to-high-power terrestrial base stations will require highly miniaturized high unloaded quality factor (Q u ) filters to select the band of interest while rejecting interference.This is also the case in radar and electronic warfare systems that are challenged by strong interferers.
Bandpass filters (BPFs) using high Q u 3-D resonators are the only RF filtering solution for these systems due to the requirements for low insertion loss (IL) and high RF power handling [1].However, existing 3-D metallic RF filters using rectangular resonators are really large.Miniaturization techniques using multimode resonators [1] or folded waveguide arrangements are explored for size compactness [2], [3].Furthermore, combline or capacitively loaded coaxial resonators are also considered for 3-D BPF realization [1], however with reduced power handling capabilities.In addition to the size constraints imposed by the 3-D geometry itself, the integration of these systems using split blocks further increases their weight and size due to the need for auxiliary parts and screws for the assembly process.Additive manufacturing (AM) (or 3-D printing) techniques, such as stereolithography apparatus (SLA) and direct laser metal sintering (DLMS), have been creating new avenues for the realization of compact 3-D RF BPF [4].This is due to their free-form manufacturing capabilities that have facilitated the realization of new types of filter geometries with unconventional shapes that cannot be manufactured with conventional computer numerical control (CNC) machining.Typical examples of this trend include spherical [5], [6] and hemispherical [7], [8] resonator geometries, elliptical resonator-based BPFs [9], [10], and conical [11] and coaxial-resonator-based configurations [12], [13].Another advantage of AM is that it has enabled the realization of monolithic RF components eliminating the need for screws and additional volume for the assembly process.Whereas monolithic integration has been mostly shown for rectangular waveguides [1], [4] and using DLMS; SLA has also enabled the realization of monolithic fully enclosed configurations such as the coaxial BPFs in [12] and [13] and the spherical BPFs in [5], [6], [7], and [8].
In this brief, an SLA-based manufacturing and integration scheme is investigated for the realization of compact high Q u hemispherical BPFs with highly selective transfer functions.Size compactness is achieved through a folded architecture and monolithic integration.The organization of this manuscript is given as follows.In Section II, the operating principles of the folded-hemispherical filter concept are presented alongside its practical implementation using unloaded and capacitively loaded hemispherical resonators.In Section III, the concept is validated through the manufacturing and RF testing of a three-pole/one-TZ BPF.Finally, Section IV outlines the major contributions of this work.

II. THEORETICAL FOUNDATIONS
The details of the folded hemispherical resonator-based BPF are shown in Fig. 1 in terms of its 3-D geometry and its corresponding coupling routing diagram (CRD).It facilitates transfer functions comprising three poles and one transmission zero (TZ) to enhance selectivity in a compact form factor as opposed to conventional in-line coupled resonator configurations.It consists of three (unloaded and capacitively loaded) hemispherical resonators that are represented by resonating nodes (gray circles) 1-3 in the CRD in Fig. 1(a) that are arranged in a folded configuration so that the CRD can be implemented with a compact structure that optimally occupies the available 3-D volume.Both unloaded and capacitively loaded hemispherical resonators are utilized in the filtering geometry for size compactness as well as to facilitate effective coupling between the resonators.The geometrical and E-field distribution details for the fundamental mode for the two resonators are, respectively, shown in Fig. 2(a) and (b).As shown in [7], the hemispherical resonator resonates at the same frequency as a spherical, however occupying half of the available volume.To further reduce its size, capacitive loading can be applied by incorporating a metallic post on the upper wall as shown in Fig. 2(b), at the expense of Q u , and however, their Q u /Vol is comparable.The tradeoffs between f cen and Q u for the unloaded and the capacitively loaded resonator are shown in Fig. 3.By increasing the height of the post H P , the resonant frequency of the resonator is lowered and so is its Q u .
To examine the applicability of the mixed hemispherical BPF concept to the realization of compact and highly selective filtering geometrics, the example case of a three-pole/one-TZ transfer function is considered using the CRD in Fig. 1(a).Its theoretically synthesized response is shown in Fig. 4 for various example cases that show how the TZ can be placed at frequencies lower or higher than the passband and how alternative equi-ripple and maximally flat passband profiles can be obtained.As shown, the location of the TZ is controlled by the sign of the cross coupling between resonators 1 and 3, m 13 , and the resonant frequency of resonator 2, m 22 .
For practical realization purposes, a transfer function with a TZ above the passband and a connectorized coaxial excitation scheme is used as external coupling (Q EXT ) to facilitate the interconnection of these filters with other SMA-based RF components.The probes are coupled to the two capacitively loaded resonators (1, 3) by connecting their apex to the upper wall of a rectangular waveguide section that is short-circuited at a distance λg/4 from the probe.Q EXT strength depends on the relative location of the SMA connector to the  resonator (e.g., L E ), the size of the apex of the RF probe (D E , H E ), and the iris width (l W ). Its dependence on some of these parameters is provided in Fig. 5(a), e.g., by increasing l w and D E , Q EXT decreases.The inter-resonator coupling (K x y ) between resonators 1 and 3 is performed through an iris (W I , H I , L I ), whereas the inter-resonator coupling between resonators 1 and 2 and 2 and 3 is performed through a slot (W x y , L x y ) that needs to be placed as far as possible from the RF probe to avoid cross coupling from the RF input-output to resonator 2. The K x y dependence on selected geometrical details is shown in Fig. 5(b).As shown, K x y increases by increasing L x y and increasing W I .

III. EXPERIMENTAL VALIDATION
To validate the folded hemispherical three-pole/one-TZ BPF concept, a filter prototype was designed for a center frequency of 8 GHz and a 3-dB fractional bandwidth (FBW) of 6% (Q EXT = 18.47,K 12 = 0.0381, and K 23 = 0.0175) using the design guidelines in Section II.To realize the filter as a single block, SLA manufacturing was used alongside a commercially available Cu platting process that facilitates a 50-µm Cu thickness.As such, release holes with a diameter of 1 mm were added on all of the filter walls as shown in Fig. 6 to facilitate metallization.The size of the holes and their  location has been determined so that they do not generate additional loss at the passband.Furthermore, the filter orientation and the number of support structures need to be appropriately selected so that the filter is printed as a single block.The CAD model details for SLA manufacturing are provided in Fig. 6.Specifically, in this case, the part is first rotated by 25 • around the x-axis and then by 25 • around the z-axis and all of the support structures are placed outside of the part with most of them being beneath it.Although in theory any shape can be manufactured with AM, in this case, manufacturing is constrained by the fact that all of the utilized support structures need to be placed outside of the filter body so that they can be removed after platting.
The prototype is shown in Fig. 7(a) before (left) and after (right) Cu platting.RF characterization was performed with a Keysight N5224A PNA and is shown in Fig. 6.The three-pole/one-TZ BPF exhibits following characteristics, the center frequency of 8 GHz, the FBW of 4.62%, and the minimum in-band IL of 0.28 dB (Q eff of 1400).A comparison with its corresponding EM simulated response is also shown and appears to be in good agreement.Table I provides a detailed comparison of the proposed folded hemispherical BPF concept with state-of-the-art 3-D printed coaxial, spherical (half mode [7], full size [5], and dual mode [6]), elliptical [11] 3-D BPFs, and folded CNC machined waveguide BPFs [3].As shown, this is the only monolithic folded hemispherical BPF and facilitates smaller physical size than the rest of its spherical, hemispherical, and elliptical single-mode counterparts.In relation to the work in [8], it allows for TZs to be implemented in a compact manner using a folded geometry.When compared to [13], this work is based on a different resonator concept.In [13], coaxial resonators are used that typically have lower Qs and lower RF power handling.

IV. CONCLUSION
A class of hemispherical BPFs have been presented for the realization of three-pole/one-TZ transfer functions.A geometrical configuration using folded capacitively loaded and unloaded hemispherical resonators has been used for size compactness.A monolithic integration concept using SLA 3-D printing has been demonstrated for further miniaturization and enhanced performance as opposed to conventional CNC-machined split-block concepts.The operating principles of the folded hemispherical BPF concept were validated experimentally through the manufacturing and testing of a threepole/one-TZ BPF at the X -band.

Fig. 5 .
Fig. 5. (a) Q EXT as a function of the iris width l W and the RF probes diameter D E .(b) K X Y as a function of the slot length L X Y and as a function of the iris width W I .