Miniaturized Monolithic 3-D Printed Bandpass Filters Using Nested Spherical Resonators

This article reports on a new class of highly miniaturized nested spherical bandpass filters (BPFs) alongside a monolithic integration concept. Miniaturization is achieved by a nested spherical resonator approach, i.e., by incorporating a postsupported sphere within the volume of a hollow spherical resonator. In this manner, the resonant frequencies of the first four modes are lowered when compared to conventional unloaded spherical resonators, allowing for smaller BPFs to be materialized. Alternative BPF topologies using the fundamental mode as well as a combination of higher order modes are considered to further miniaturize the filter volume. Scalability to higher order transfer functions through cascaded configurations of multimode resonators is explored. Monolithic integration is enabled by digital additive manufacturing (AM) techniques using stereolithography apparatus (SLA). For proof-of-concept validation purposes three prototypes were designed, manufactured, and tested. They include: 1) a two-pole BPF with a passband centered at 2.72 GHz, fractional bandwidth (FBW) of 33.1%, and minimum in-band insertion loss (IL) of 0.15 dB; 2) a three-pole BPF with a passband centered at 6.94 GHz, FBW of 13.65% and minimum in-band IL of 0.07 dB; and 3) a six-pole BPF with a passband centered at 6.88 GHz, FBW of 12.77% and minimum in-band IL of 0.3 dB.


Miniaturized Monolithic 3-D Printed Bandpass
Filters Using Nested Spherical Resonators challenging requirements that these systems need to fulfill.
To facilitate operation in interference-dominated environments, RF filters with high unloaded quality-factor (Q) need to be incorporated in their RF transceivers of these systems to acquire the desired bands of interest with the minimum possible loss penalty while suppressing the unwanted interference.Low-weight and small form factor are other important considerations that need to be considered for their RF design.Three-dimensional resonators using rectangular [1], [2], spherical [3], cylindrical cavities [4], or dielectric-loaded cavity resonators [5] have been extensively investigated for the realization of low loss and highly performing RF filters.Compared to their planar transmission counterparts, such as microstrip, coplanar waveguide, and substrate integrated waveguide (SIW) technologies, they exhibit significantly higher Q (up to 10×) and can handle high levels of RF power (>50 dBm) [6], [7], [8].However, they are large in size, particularly, when designed for low frequencies of operation.Capacitively-loaded coaxial cavity resonator bandpass filters (BPFs) [9], [10] or ridge-waveguide BPFs [11], [12] have recently been proposed as miniaturization alternatives for 3-D RF filtering at the expense of Q and in-band insertion loss (IL).In yet another approach, multimode resonators are used within the filter volume to miniaturize their size by adding poles or transmission zeros (TZs) in the filtering transfer function without needing to cascade additional resonators.Notable demonstrations of this approach include a waveguide dual mode filter [13] using transverse magnetic and nonresonating modes, a multimode waveguide filter [14] using the epsilon-near-zero (ENZ) channel, a waveguide filter using the parallel multimode spherical cavity [15] and a BPF [16] using high-Q quadruple-mode rectangular waveguide resonators.However, all of the aforementioned configurations are expensive and bulky.Furthermore, they need to be manufactured as split-blocks and assembled through screws requiring postfabrication tuning.
Plastic-/resin-based monolithic integration approaches using SLA AM have been recently proposed for the realization of open-ended RF components such as the ones using rectangular waveguides [41] and spherical resonators [42], [43], [44].Monolithic SLA-based integration concepts were also recently demonstrated for fully enclosed coaxial-resonator based configurations such as single-and multiband BPFs and diplexers [9], [25], [45] with high levels of complexity.However, most of the SLA-based fully enclosed passive RF components still need to be divided into several parts to allow for internal metallization, and as such they are manufactured as split blocks and are assembled with screws or other fixtures.For example, in [46], a split-block SLA-based coaxial filter using SIR resonators is demonstrated with an IL of 1.1 dB whereas in [47], a SLA-based split-block four-pole filter using conical resonators is presented.In the plastic-/resin-based SLA AM filter in [45] slots were added to facilitate metallization of its internal walls [45].
In this article, new types of RF filters using novel single-mode and multimode metal nested spherical resonator configurations are investigated for the first time for the realization of high-Q and highly miniaturized 3-D filters.The center frequencies of filters were chosen close to the ones of applications within the FR1 band, i.e., sub-6 GHz.The proposed concept is based on the incorporation of a postsupported fully metallic sphere within a hollow sphere.In this manner, the resonant frequencies of the first four resonant modes are significantly lowered when compared to a conventional hollow spherical resonator having the same outer dimensions.Further miniaturization is achieved by exploiting multiple resonant modes within a single resonator to create high-order filtering transfer functions.
This article is organized as follows.Section II introduces the theoretical foundations of the proposed nested spherical resonator concept alongside its application to the realization of compact and high-order RF filters.In Section III, practical design and integration aspects are discussed through the design, manufacturing, and testing of three BPF prototypes.Finally, the major contributions of this work are outlined in Section IV.

II. DESIGN AND ANALYSIS OF THE FILTERS A. Nested Spherical Resonator Concept
The geometrical details of the proposed nested spherical resonator concept are shown in Fig. 1(a).It is composed of Resonant frequencies of the first four resonant modes of a conventional hollow spherical resonator and the proposed nested spherical resonator.In both examples, the radius of the outer cavity r 2 is 14 mm.a spherical cavity that is loaded by a postsupported metallic sphere.The resonator loading makes the resonant frequencies of the four first resonant modes to be lower than a conventional unloaded spherical resonator allowing for smaller RF filters to be realized.To get a better insight on the resonant capabilities of the proposed resonator concept, eigenmode simulation analysis is performed in ANSYS HFSS for the first four resonant modes whose electric field distribution is provided in Fig. 1(b).As noticed, the second and the third modes are degenerate and can be used for compact BPF design.To demonstrate the size-compactness merits of the proposed resonator concept, Fig. 2 provides a comparison of the resonant frequencies of the first four modes of a conventional hollow spherical resonator and the proposed nested spherical resonator.Evidently, the nested spherical resonator resonates at lower frequencies than those of a conventional hollow spherical resonator with the same dimensions.To further demonstrate the resonant capabilities of the proposed resonator concept, Fig. 3 depicts how the resonant frequency and Q of the first four modes can be controlled by altering the geometrical parameters of the resonator.As shown, in Fig. 3(a), the first four modes and their Q decrease when the radius r 1 of the inner sphere increases.Furthermore, the resonant frequency and Q of the first and the fourth modes increase, while those of the two degenerated modes (second and third modes) remain almost constant when r p increases as shown in Fig. 3(b).Moreover, the resonant frequency of the four modes decreases while their Q increases with the increase of the radius of the outer spherical cavity r 2 , as shown in Fig. 3(c).
To verify the usefulness of the four different modes in the realization of compact and high-performing BPFs, three different filter topologies have been designed and analyzed and are presented in Sections II-B-II-D.

B. Two-Pole BPF Using Two Coupled Resonators
The first BPF topology is based on two coupled nestedspherical resonators as shown in Fig. 4. Its two-pole passband is formed by the fundamental mode (mode 1) of each resonator.Furthermore, three-TZ are present in the out-of-band power transmission response resulting in a two-pole/three-TZ transfer function as shown in Figs.5-7.The TZs are due to the multiple RF signal paths that are created within the filter due to the presence of the multiple modes.These RF signals destructively add at frequencies where the TZs are present.Figs.5-7 also show how the poles and the TZs of the transfer function can be controlled by altering different geometrical parameters.Specifically, Fig. 5, demonstrates how the radius r p of metal post and the rotation angle θ 1 of the input port affect the BPF performance.As shown in Fig. 5(a), larger r p , makes the two poles and the TZ 1 move to higher frequencies while TZ 2 and TZ 3 remain constant.From Fig. 5(b), it can be seen that the two poles move closer to each other with the increase of θ 1 whereas the TZ 1 and the TZ 3 remain almost constant and the TZ 2 moves closer to the passband.Furthermore, in Fig. 6(a) the effect of r 1 is studied.Specifically, the poles and TZs move to lower frequencies with the increase of r 1 .The frequency details of the transmission poles, as r 1 is altered, are depicted in Fig. 6(b).To further facilitate control of the passband bandwidth by altering the coupling between the fundamental modes of the two resonators, a part from each of the inner metal spheres is removed as shown in Fig. 4(b).
The filters's performance as a function of the gap g and radius r c are provided in Fig. 7.As shown, when r c (i.e., the radius of coupling area between two spheres) increases the in-band return loss (RL) level becomes higher.Moreover, TZ 1 moves to a higher frequency, and TZ 2 and TZ 3 move opposite to each other.When g increases, the passband poles come closer to each other allowing for narrower BW states to be obtained.Furthermore, the TZ 1 moves to higher frequencies, and the TZ 2 and TZ 3 will move opposite to each other when the g increases.According to the above analysis, the bandwidth and positions of the TZs and poles can be adjusted by controlling r c and g.
Taking into consideration the aforementioned parametric studies, the two-pole BPF transfer function can be designed as follows.
1) Perform eigenmode simulations and determine the dimensions of the nested spherical resonator in Fig. 1 so that the fundamental mode (mode 1) of the resonator resonates at the desired center frequency f c .
2) Connect two resonators as shown in Fig. 4 and perform a set of parametric studies as shown in Figs.5-7 to determine the passband BW, TZs, and filter matching levels.
3) Select g and r c for the desired filter BW to be obtained.4) Specify the location of the TZs by altering r p (TZ 1 ), θ 1 (TZ 2 ), r 1 , r c , and g (TZ 1 , TZ 2 , and TZ 3 ).
5) Through r c and θ 1 specify the desired RL level.
A design example for a center frequency of 2.77 GHz and fractional BW of 36.1% is shown in Fig. 8.To facilitate metallization of the internal surfaces of the filter after being monolithically manufactured through SLA 3-D printing, nonradiating holes need to be added on its surfaces to allow for the copper platting solution to flow through the filter volume.As such, the location and the size of the holes need to be carefully analyzed by electromagnetic (EM) simulations to ensure that they do not affect the BPF's performance.Fig. 8 shows a comparison of the BPFs S-parameters in the presence or absence of holes as well as for different hole sizes with radius r h between 0.5 and 0.9 mm.As shown, the presence of the holes has a minimal impact on the RF Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.performance of the filter.For these states, the effective Q remains almost constant and around 1086 in the case of the surface holes with radius r h = 0.75 mm.As such, holes with a radius of 0.75 mm are used for the practical realization of the two-pole BPF as well as the rest of the BPF examples in this article.

C. Three-Pole BPF Using a Single Nested Resonator Configuration
The geometrical details of the three-pole BPF using a single nested spherical resonator are depicted in Fig. 9.As shown in Fig. 3(b), the frequencies of the second, third, and forth modes (i.e., mode 2-4) will come close to each other with the decrease of r p which makes it possible to form passband based on these three modes.Furthermore, three TZs appear in the out-of-band response with the overall transfer function having three-poles and three TZs.To demonstrate the design principles of this filtering configuration, the power transmission and reflection response of the three-pole/three-TZ for alternative geometrical parameters are shown in Fig. 10.Specifically, Fig. 10(a) shows that when r p increases, the third pole (mode 4) moves to higher frequencies whereas the frequencies of the rest of the poles remain constant.Besides, the frequency of TZ 1 and TZ 3 remains unaltered, and the stopband isolation at high frequencies worsens with the increase of r p .In Fig. 10(b), it is shown that the in-band RL can be adjusted by altering θ 2 , and the high frequency stopband isolation at high frequencies worsens with the increase of θ 2 .On the basis of these results, the bandwidth, RL, and stopband isolation of this filter can be adjusted through r p and θ 2 .Moreover, since this filter and the two-pole BPF use the same resonator, the effect of the inner radius of the metallic sphere r 1 is similar to that of the two-pole BPF.Based on the aforementioned performance trade-offs the BPF transfer function can be designed as follows.
1) Through eigenmode simulations determine the dimensions of the nested spherical resonator in Fig. 1 so that modes 2-4 resonate around the desired center frequency f c .
2) Perform a set of parametric studies as shown in Fig. 10 to determine the passband BW, TZs, stopband isolation, and the filter matching levels.
3) Select r p for the desired filter BW to be obtained.4) Specify the location of the TZs by altering r p (TZ 2 ) and r 1 (TZ 1 , TZ 2 , and TZ 3 ).
5) Adjust the stopband isolation by altering r p and θ 2 .6) Through θ 2 specify the desired RL level.

D. Six-Pole BPF
To explore the scalability of the multimode resonator concept in Section II-C to higher order transfer functions, a filter geometry comprising two in-series cascaded multimode resonators (mode 2-4 contribute to the filter passband) and having a six-pole transfer function is considered next.The nested spherical cavity resonators are cascaded with an air-filled coaxial transmission line (TL) as shown in Fig. 11.Figs.12-14 demonstrate how the transfer function characteristics of the six-pole BPF are controlled by altering its geometrical parameters.As shown in Fig. 12(a) and (b) the in-band RL, bandwidth, and stopband isolation at high frequency can be adjusted by controlling r p and θ 3 .Furthermore, as shown in Fig. 13, the in-band RL and stopband isolation at high frequency can be adjusted by altering the radius r t1 of the inner conductor of air-filled coaxial TL and the length c l of air-filled coaxial TL affect the in-band RL and the outof-band spur performance as shown in Fig. 14.Taking into consideration the aforementioned design trade-offs the six-pole BPF is designed as follows.
1) Determine the dimensions of the nested spherical resonator in Fig. 1 according to eigenmode simulations so that modes 2-4 of the resonator can resonate around the desired center frequency f c .
2) Cascade two nested resonators with an air-filled coaxial TL as shown in Fig. 11 and perform a set of parametric studies as shown in Figs.12-14 to determine stopband isolation and filter matching.
3) Select r p for the desired filter BW to be obtained.4) Adjust the stopband isolation by altering r p , θ 3 , r t1 , and c l .
5) Through r t1 and θ 3 specify the desired RL level.

III. EXPERIMENTAL RESULTS
To verify the practical viability of the nested spherical resonator-based BPF concept, three filter prototypes Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.with a 50 µm layer resolution.Metallization was performed using a commercially available platting process that creates a 50 µm Cu-layer across all surfaces.

A. Two-Pole BPF
The manufactured prototype of two-pole BPF is shown in Fig. 15(a) and its corresponding CAD model for SLA monolithic 3-D printing is provided in Fig. 15(b).The RF measured response of the two-pole BPF is depicted in Fig. 15(c) and is summarized as follows: center frequency f c of 2.72 GHz and the 3-dB FBW of 33.1%.The measured minimum IL was measured around 0.15 and the passband RL around 23.2 dB.A comparison with the EM simulated response is also included and appears to be in good agreement successfully validating the proposed filter concept.

B. Three-Pole BPF
The manufactured prototype and CAD model of SLA manufacturing of three-pole BPF is shown in Fig. 16(a) and (b) respectively.The measured S-parameters are shown in Fig. 16(c) and its RF measured performance is summarized as follows: f c : 6.94 GHz, 3-dB FBW 13.65%, minimum in-band IL of 0.07 dB, and passband RL is better than 14.2 dB.A comparison with its corresponding EM simulated response is also provided in the same figure and appears to be in good agreement with the RF measured response.

C. Six-Pole BPF
The experimental validation of the six-pole BPF concept has been performed through the manufacturing and testing of the prototype in Fig. 17   performance is summarized as follows: f c : 6.88 GHz, 3-dB FBW 12.77%, minimum IL of 0.3 dB, and passband RL is better than 12.6 dB.

D. Comparison With the State of the Art
Table I shows a comparison of the proposed nested spherical BPF concept with state-of-the-art 3-D printed 3-D BPFs using traditional hollow spherical, waveguide, coaxial, and 3-D resonators with irregular shapes.As shown, the proposed filter concept has a smaller size than all the filters based on traditional hollow spherical resonators and exhibits lower IL.Furthermore, the proposed concept has been demonstrated for higher order transfer functions and is the only BPF concept based on nested spherical resonators that are reported in this work for the first time.New types of highly miniaturized spherical resonator-based BPFs were presented.They are based on nested spherical resonators whose resonant frequencies are lowered by incorporating a postsupported sphere.Miniaturized prototypes using the fundamental mode as well as a combination of higher order modes are demonstrated for the first time.A compact monolithic integration scheme using SLA 3-D printing is also proposed for size compactness and low weight when compared to conventional split-block integration schemes.Multiple BPF prototypes using the fundamental mode as well as a combination of three higher order modes were designed, manufactured, and tested at S-band and C-band.

Fig. 1 .
Fig. 1.(a) 3-D geometry of the nested-spherical resonator and (b) E-field distribution of the first four resonant modes.The geometrical parameters of the resonator are: r 1 = 9 mm, r 2 = 14 mm, and r p = 1.5 mm.

Fig. 3 .Fig. 4 .
Fig. 3. Resonant frequency and Q of the first four modes of the nested spherical resonator for alternative geometrical parameters: (a) r 1 , (b) r p , and (c) r 2 .

Fig. 5 .
Fig. 5. EM simulated S-parameters of the two-pole BPF in Fig. 4 as a function of different geometrical parameters.(a) r p .(b) θ 1 .The rest of the geometrical parameters are: r 1 = 7.1 mm, r 2 = 14 mm, g = 2 mm, and r c = 5.08 mm.

Fig. 6 .
Fig. 6.(a) EM-simulated S-parameters of the two-pole BPF in Fig. 4 as a function of r 1 .(b) EM-simulated resonant frequency of the two in-band poles as a function of r 1 .In these results, r 2 = 14 mm, r p = 1.5 mm, g = 2 mm, r c = 5.08 mm, and θ 1 = 45 • .

Fig. 8 .
Fig. 8. EM-simulated response of the two-pole BPF in Fig. 4 in the absence or presence of Cu-platting holes.(a) Filter performance in the absence or presence of holes with a radius r h = 0.75 mm.(b) Broadband filter performance for alternative hole sizes.(c) Passband performance for alternative hole sizes.The rest of the filter dimensions are r 1 = 7.1 mm, r 2 = 14 mm, r p = 1.5 mm, g = 2 mm, r c = 5.08 mm, and θ 1 = 45 • .

Fig. 10 .
Fig. 10.EM-simulated S-parameters of the three-pole BPF in Fig. 9 as a function of alternative geometrical parameters.(a) r p .(b) θ 2 .The rest of the geometrical parameters are: r 1 = 9.1 mm and r 2 = 14 mm.

Fig. 13 .
Fig. 13.(a) EM-simulated S-parameters of the six-pole BPF in Fig. 11 as a function of the radius r t1 of the inner conductor of the air-filled coaxial TL.(b) Characteristic impedance of the coaxial TL as a function of r t1 .The rest of the geometrical parameters are: r 1 = 9.2 mm, r 2 = 13.9 mm, r p = 1.91 mm, r t2 = 1.8 mm, c l = 31.6mm, and θ 3 = 60 • .
(a).Its corresponding CAD model for SLA manufacturing is shown in Fig.17(b) and the measured S-parameters are shown in Fig.17 (c).Its RF measured

TABLE I COMPARISON
WITH STATE OF THE ART 3-D PRINTED FILTERS IV.