On the Development of Metal 3D Printed Bandpass Filter With Wide Stopband Based on Deformed Elliptical Cavity Resonator With an Additional Plate

This paper presents a newly developed metal 3D printed waveguide bandpass filter based on a modified deformed elliptical cavity resonator with an additional plate. The modified resonator has excellent design freedom allowing semi-independent control of centre frequency, bandwidth, and transmission zeros. Furthermore, transmission zeros can be placed close and far from the passband to improve frequency selectivity and stopband’s width and rejection, respectively. Theory of the resonator, filter design with an example, and simulated and experimental results with extensive discussion are given. As proof of concept, a 5-pole Ku-band wideband filter was designed and manufactured using Selective Laser Melting, a metal 3D printing technique. Even though the prototypes were tested as-built, i.e. without any surface postprocessing, the simulated and measured results have good agreement and consistency between all four printed prototypes. Moreover, the prototypes were manufactured in one piece, significantly improving measured results compared to the previous two-piece prototypes. INDEX TERMS Deformed cavity resonator, metal 3D printing, selective laser melting, waveguide filter, wideband bandpass filter, wide stopband.


I. INTRODUCTION
Microwave resonators are a fundamental part of any wireless communication systems. Due to the development of new communication applications, which require higher data transfer rates and thus use higher frequency bands, and the ever-increasing number of users of those applications, the spectrum must be fully utilised [1]. This puts stringent requirements on microwave filters. Waveguide filters are commonly used at higher frequencies as they have desired properties such as low insertion loss, high selectivity, and high power handling. However, as a trade-off, waveguide filters generally have a large footprint. Therefore, developing new filter designs with improved performance and reduced footprint is essential to support the development of future communication applications.
The associate editor coordinating the review of this manuscript and approving it for publication was Xi Zhu .
Additive manufacturing (AM) or 3D printing offers various advantages: no tooling costs, reduced lead time, rapid prototyping, lightweight structures, and reduced material waste. However, perhaps the most significant advantage of AM is design and geometrical freedom, which allows improving performance and reducing the footprint of microwave components, and integrating different functions [2]- [5]. Metal 3D printing is a relatively new technique as only Laser Beam Powder Bed Fusion (LB-PBF) technologies have achieved widespread use in the industry [6]. This manufacturing technique is not without disadvantages as it has a rough surface finish and limited dimensional accuracy. However, if utilised to the fullest, design freedom can offer excellent performance despite these disadvantages [3], [7], [8]. Moreover, there are mitigating techniques to reduce the impact of rough surface finish and dimensional accuracy [9]. This paper presents a newly designed and monolithically 3D printed metal waveguide filter based on a deformed VOLUME 10, 2022 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/  elliptical cavity resonator with an additional plate, a modified version of the resonator from [10]. The additional plate serves three functions: splits one of the propagating modes into two, thus introducing extra pole and transmission zero (TZ); suppresses spurious resonances by forming narrow iris; and acts as a support for the overhanging top surface, which aids in manufacturing the model in one piece. Previously, a 5-pole filter prototype of this type was 3D printed in two pieces using Selective Laser Melting (SLM), and the preliminary results were reported at a conference [11]. Compared to the conference paper, this article extends the theory, adds filter design, adds one piece filter prototype experimental results, and expands the discussion of simulated and measured results. The paper outline is as follows. Section II explains the construction of the resonator and working principle. Section III demonstrates how to design a filter based on the resonator. Section IV gives prototype filter's simulated and experimental results and further discussion, including sensitivity analysis and comparison with other works.

A. RESONATOR STRUCTURE AND DIMENSIONS
The inside volume of the resonator with key dimensions is shown in Fig. 1. The resonator is constructed by subtracting two identical ellipsoids from the sides of the main ellipsoid, adding a rectangular aperture, subtracting the plate to form a narrow inner iris, and then adding standard waveguide ports. Note that all ellipsoids have the same dimensions in x and y directions. The first interesting thing about this resonator is that the waveguide ports are part of the resonator, contrary to the typical resonator. Without them, two of the three propagating modes would not be formed.

B. THEORY
The typical, base case resonator response is shown in Fig. 2, with all dimensions displayed in Table 1. This base case will be used throughout the paper for parametric studies. The three poles in the passband are the TE 101 mode of the rectangular cavity, and quasi-TE 102 and quasi-TE 103 modes of the split elliptical cavity. The latter two modes are not formed without the waveguide ports. The H-fields of all three modes are shown in Fig. 3.
The second interesting thing about this resonator is that the first two spurious resonances typically are not excited at around 20.5 GHz from the TM 110 modes of the split elliptical cavity. Hence, these resonances are not visible in Fig. 2. The rectangular apertures have to be moved up or down to break the symmetry of excitation of the spilt elliptical cavity in order to excite the TM 110 modes. Therefore, the stopband   is increased significantly without any effort. In addition, TM 110 modes might prove useful in designing dual-band filters, as discussed later.

C. TRANSMISSION ZEROS
The pair of TZs is mainly controlled via the r u parameter, as shown in Fig. 4. Notice how TZs are squeezing the modes when r u is equal to 1 mm or 5 mm. That indicates how TZs can suppress mode resonances. However, as passband modes can also be suppressed, TZs cannot be moved too close to the passband. Hence, transmission zeros cannot significantly improve selectivity and are instead used to obtain a wide stopband with high rejection.   Furthermore, each TZ can be controlled individually by changing subtracted ellipsoids, c.f. Fig. 1b, to be nonidentical. The base case for this example is shown in Fig. 5 with the side view of the corresponding 3D model. Figs. 6 and 7 illustrate how to control the first TZ and second TZ using r u1 and r u2 , respectively. In both cases, only one TZ moves while the frequency of the another TZ and all the modes remains essentially unaffected.
At this point, it might seem like both TZs are formed and controlled by the second passband's modes, especially looking at Fig. 6. However, this is not the case, as it is possible to move TZs below the passband by shifting TE 101 mode above both quasi modes, i.e., by changing the sign of the coupling coefficient, as shown in Fig. 8. Several parameters had to be changed to swap the modes: a aperture = 9.5 mm, a res = 6.75 mm, r u = 5.2 mm, and w inner iris = 9.5 mm. Enlarging the elliptical cavity and decreasing the width of the rectangular aperture puts quasi-TE 102 at 14.42 GHz, quasi-TE 103 at 14.965 GHz, and TE 101 at 16.45 GHz. It should be noted that the TE 101 mode's H-field remains unchanged, but both quasi modes have quite different H-fields. Fig. 9 shows the frequency response when TM 110 modes of the split elliptical cavity are excited, resulting in a potential of dual-band response. The dual-band response is achieved by either moving rectangular apertures and the inner iris or the elliptical cavity lower or higher to disturb the symmetry of excitation of the TM 110 modes. The inset of Fig. 9 shows a side view of the resonator where the elliptical cavity was moved lower by 1 mm. Note that a small aperture of 3.75 mm × 1.75 mm is added between the split elliptical cavities to improve TM 110 modes' coupling. All other dimensions are the same as in Table 1.

D. DUAL-BAND RESONATOR
The response shown in Fig. 9 is dual-band, with the first passband being wider and having one extra pole due to the TE 101 mode than the second passband. There are two TZs between the bands to increase the rejection and two more TZs above the second passband. However, quasi modes and TM modes are mainly controlled by the split elliptical cavity parameters. Therefore, it might be hard or impossible to control centre frequencies, bandwidths, and TZs positions entirely independently. More degrees of freedom might help to improve independent control. For example, different parameters of the main elliptical cavity in x and y directions (c.f. Fig. 1), non-identical subtracted ellipsoids from the sides, or even non-identical split elliptical cavity halves. In addition, coupling between these various modes might be hard to realise and control, thus additional coupling structures might be needed. All of this remains to be explored in the future.

III. FILTER DESIGN A. PARAMETRIC STUDY
The third interesting thing about the resonator is that when combining two or more resonators, TE 101 modes of the rectangular apertures merge, hence contributing one pole per filter. Therefore, it is quite challenging or even impossible to work out sensible and physically accurate filter topology and corresponding coupling matrix representing physical filter. Even though the filter topology may be unobtainable, few things are certain. The filter order, N , is always an odd number because it is equal to 2×n+1, where n is the number of split elliptical cavities. The maximum number of TZs the filter can have is N − 1, corresponding to the number of split elliptical cavities. Additionally, higher order filters should be possible to realise as was shown with an 8-pole filter based on the previous resonator model in [10]. Although the optimisation part of the design is bound to be more complex and time consuming compared to the example given in this section.
Another way to demonstrate filter design is by performing a parametric study to understand how each parameter affects the frequency of the modes and pair of transmission zeros. The obtained parametric curves are then used to get initial guess for optimisation which yields the final filter parameters. Normalised lowpass frequency, f n , (1) is used for easier and more accurate comparison: where f is the frequency, f 0 is the centre frequency, and FBW is the fractional bandwidth. The base case response in Fig. 2 is used to normalise the frequency, with f 0 = 15.25 GHz and FBW = 22.61%. Refer to Fig. 1 for parameter definitions. Fig. 10 displays normalised lowpass frequency of all propagating modes and pair of TZs for a aperture , b aperture , w inner iris , and h inner iris . a aperture affects all three modes. w inner iris has a similar effect on TE 101 and quasi-TE 103 while not affecting quasi-TE 102 . b aperture and h inner iris have more control over all three modes. Notice how quasi modes converge to the same frequency as h inner iris approaches zero. This confirms that both quasi modes are split from one mode, as noted before. b aperture and h inner iris are the two parameters that can bring all three modes near each other, thus controlling the bandwidth (BW). While the centre frequency decreases as  these parameters decrease, it can be increased with a aperture while maintaining similar separation between the modes. However, if printing in one piece, the dimensional accuracy and gap limit of SLM, puts restrictions on realisable BW. The smallest gap size to print without any reduction in quality is around 0.3-0.5 mm. Thus lowest FBW possible should be about 5-10%. None of these four parameters heavily affect the pair of TZs.
Figs. 11 and 12 show normalised lowpass frequency of all propagating modes and pair of TZs for a res and b res , and r v and r u , respectively. a res is the main parameter to control quasi modes, as it affects them more than TE 101 . However, it also heavily affects TZs. b res or r u can be used to correct that. Both can control TZs in almost the same manner with little effect on the frequency of the modes. Notice how whenever the thickness of the split elliptical cavity decreases, c.f. Fig. 1b, either r u approaches b res or vice versa, the pair of TZs starts squeezing the quasi modes. As mentioned earlier, this limits how close TZs can be placed to the passband. r v only affects modes and TZs frequencies when smaller; as it increases, the effect decreases. However, r v cannot be set too small because it will create an overhanging surface, c.f. Fig. 1b, which cannot be 3D printed or would reduce the manufacturing quality. Fig. 13 shows normalised lowpass frequency of all propagating modes and pair of TZs for t plate , t, and l port . t plate and t have some effect on quasi modes while almost none on TE 101 ; hence can be used as additional control parameters. l port increases the frequency of all modes at shorter lengths; therefore, it cannot be set too small. There are also some other restrictions imposed by the manufacturing method. For example, t and t plate cannot be set below 0.5-0.6 mm because thin walls are prone to warping due to thermal stresses [12].
To summarise: a aperture , b aperture , w inner iris , h inner iris , and a res control centre frequency; b aperture and h inner iris control bandwidth; and a res , b res , and r u control transmission zeros. The remaining parameters, namely, r v , t plate , t, and l port , are not that influential thus are used for fine-tuning.

B. OPTIMISATION
The filter design exploiting the novel deformed cavity and metal 3D printing capability is unique. It is an empirical method based on scaling the base case to the desired centre frequency and then using the parametric curves to manually vary several parameters to place TZs and simultaneously adjust bandwidth and centre frequency. After obtaining a reasonable initial guess, optimisation is used to achieve final filter specifications and parameter values. Full-wave simulation software CST Microwave Studio Suite [13] with Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) optimisation algorithm was used to optimise the filter. CMA-ES is a global optimiser well suited for complex problems with many variables such as this one.
A sample filter will be optimised to illustrate the filter design process. The simulations were run on a PC desktop with Intel(R) Core(TM) i7-4770 3.40 GHz CPU and 16 GB DDR3 RAM. The average time required to run a single simulation was approximately 44s. The filter has the same specifications as the manufactured prototype filter: N = 5, f 1 = 13.75 GHz, f 2 = 16.5 GHz, RL = 20 dB, and a wide stopband.
First, using the base case parameters scale to the desired centre frequency; in this case, no scaling was needed. Then position TZs and adjust bandwidth to obtain the initial guess for the optimisation. The first pair of TZs is placed close to the passband. From Fig. 12, it can be seen that the closest TZs can be set to the passband without affecting the modes is about f n = 2.5, which is around 19 GHz for this filter. The second VOLUME 10, 2022  pair of TZs is placed far from the passband such that the stopband would have constant rejection and be as wide as possible. In this case the second pair of TZs is placed around 29 GHz. The initial guess parameters and response for this example are shown in Table 2 and Fig. 14, respectively.
Next, optimisation goals and parameter intervals must be set. The filtering function of the sample filter, shown in Fig. 14, can be used to obtain optimisation goals around the passband. Optimisation goals in the stopband can be defined after obtaining the initial guess response. All parameters' intervals were set to ±20% of the initial value. The first round took 1705 runs, about 19 hours. It was stopped before convergence because one of the parameters, b aperture2 , reached the lower boundary. Even then, the response after the first round was close to optimal, as seen in Fig. 14. For the second round, all parameters were set to ±20% of the first round value. Except for b aperture2 , for which the lower boundary was set to −50%. The second round took 2209 runs, 26.4 hours, to converge.
As seen in Fig. 14, the final response is slightly better than after the first round in terms of selectivity and stopband. The upper passband edge is not exactly the same compared to the filtering function, but it is at the acceptable level for this example. If required, further optimisation could be performed with revised optimisation goals. The single TZ at 34.21 GHz is from the higher-order modes coupling. As shown in Table 2, all parameters, except b aperture1 , b aperture2 , and b aperture3 , remained well within ±20% of the initial value. Thus, the optimisation could have been done in a single round if a wider interval, e.g. ±50%, was applied to those three parameters.

IV. PROTOTYPE FILTER A. SIMULATED RESULTS
The prototype filter is a 5-pole, centred at 15.06 GHz with 18.26% FBW. The 3D model and S-parameters are shown in Figs. 15 and 16, respectively. The 5-pole filter has improved  performance compared to the previous 4-pole filter [10] whilst maintaining an almost exact footprint of 33.30 mm × 33.30 mm × 23.12 mm (only 0.17 mm longer). It should be noted that this filter and the filter in the optimisation example are not precisely the same but have similar dimensions.
The eigenmode-simulated unloaded quality factor (Q u ) of a single resonator for the TE 101 , quasi-TE 102 , and quasi-TE 103 modes is 2026, 2842, and 2558, respectively. The electrical conductivity σ = 3.56 × 10 7 S/m for aluminium was used in the simulation. The previous resonator model had Q u values of 2815 and 2842 for the TE 101 and quasi-TE 102 modes, respectively. A slight degradation of the Q u is observed due to the additional mode.
The weak I/O coupling S 21 response in Fig. 16 shows how spurious resonances are suppressed by TZ and narrow inner iris, at 28.8 GHz and 31.58 GHz, respectively. Together with TZs produced by higher-order modes coupling, c.f. Fig. 14, higher order filters could potentially suppress all second passband resonances between 28-36 GHz, thus increasing the stopband up to 40-45 GHz. This remains to be explored in the future.

B. SENSITIVITY ANALYSIS
A sensitivity analysis was performed to examine the fabrication tolerance effect of the SLM on the prototype filter. The analysis is based on the Monte Carlo sampling (MCS) method [14]. For a more realistic sensitivity analysis, all 29 parameters of the filter were included. The dimensional tolerance of the SLM is ±0.1 mm. However, it should be  noted that this is the worst-case scenario, and the tolerance can be reduced by selecting the most appropriate build angle and other 3D printer settings.
Total 1000 simulation runs with uniformly random dimensions were performed to obtain the sensitivity analysis. The result is shown in Fig. 17. The filter is rather sensitive to the dimensional tolerance of ±0.1 mm. RL is mostly better than 10 dB, and there is a larger frequency variation at the upper edge of the passband. However, the wide stopband of the filter is maintained with similar rejection. Additionally, tuning screws could be added from both sides, or potentially top and bottom as well, of the filter for post-tuning which could improve inband response.

C. EXPERIMENTAL RESULTS AND DISCUSSION
The 5-pole prototype filter, shown in Fig. 18, was printed using SLM with RenAM 500Q AM system [23]. The material used was aluminium alloy powder, AlSi10Mg, which has appropriate electrical and mechanical properties [24], [25]. Four prototypes were printed monolithically at a 45 • angle as it offers the best surface finish for all inner surfaces. The inband and wideband results are shown in Figs. 19 and 20, respectively. As observed, the agreement between measured and simulated results is adequate. The measured IL at the centre frequency is about 0.65 dB, and the RL across the passband is lower than 12.7 dB. It should be noted that the test fixture included two waveguide to coaxial adapters, which were not de-embedded and that the inner surface was not  post-processed in any way. There is a larger frequency shift at the upper edge of the passband, as was predicted by the sensitivity analysis. The filter maintained a wide stopband with 40 dB rejection up to 33.3 GHz, about 2.21 times the centre frequency. The group delay is given in the inset of Fig. 19. The shift in group delay is caused mainly by the narrower bandpass of the measured filter and some extra delay from two waveguide to coaxial adapters.
The results are vastly better than the two-piece prototype [11]. Printing in one piece eliminated any assembly errors and potentially improved surface current flow as H-plane cut disturbed surface current of all propagating modes. Furthermore, the consistency of the results suggests that the dimensional inaccuracies or errors are similar, thus giving insight into reducing or eliminating them during the manufacturing process using optimisation methods [9]. Table 3 summarises a quantitative comparison of the prototype filter versus other reported bandpass filters. The prototype filter has the widest stopband compared to those filters manufactured using 3D printing. In addition, it has advantages of relatively easy design and manufacturing, compact footprint, and high power handling. Furthermore, IL and potentially RL could be improved with post-processing such as bead blasting, electro-polishing, or silver plating, and de-embedding the test fixture. This remains to be investigated in the future.

V. CONCLUSION
This paper presented a comprehensive study of a novel deformed elliptical cavity resonator with an additional plate to develop compact, high performance microwave filters exploiting advanced additive manufacturing technology. For demonstration, a 5-pole filter prototype was designed and monolithically manufactured using Selective Laser Melting. The measured results had a good agreement with the simulation and were an improvement over the previous prototype printed in two pieces. Furthermore, the filter is comparable with reported state-of-the-art bandpass filters. Several potential future R&D directions were identified in the paper, which include the design of a dual-band filter based on the same resonator.