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3D-Printed Electromagnetic Band-Gap Band-Pass Filter Based on Empty Single-Ridge Waveguide

In this work, a new method for the design of band-pass filters in an empty waveguide by using periodic structures is presented in a theoretical and experimental way. The proposed filter topology uses a periodic profile of two heights in the central part of an Empty Single-Ridge Waveguide (ESRW) for the generation of an Electromagnetic Band-Gap (EBG), and it provides simple unit cell design parameters to produce specific dispersion characteristics. The implementation of two band-pass filters with different fractional bandwidths (36% and 55%) and different characteristics in the rejection band is proposed, where to minimize the mismatch in the ESRW filter’s passband, a tapering technique is used that follows a Kaiser distribution for the EBG. To experimentally validate the design concept, a band-pass filter with a fractional bandwidth of 55% centered on 5.45 GHz is manufactured, using a low-cost 3D printer, which allows the rapid manufacture of complex geometries at a very reduced price. The experimental results validate the simulated response of the implemented ESRW band-pass filter.


I. INTRODUCTION
Waveguide band-pass filters play an important role in current communication systems, because they have a series of advantages in terms of insertion loss and high-power signal handling capability [1]. Normally this type of device is implemented using simple rectangular or cylindrical resonators [2]- [5] and they are usually manufactured using numerical control milling techniques that have a high cost [6]. Currently, additive manufacturing using 3D-printing techniques has aroused great interest in different industrial sectors, moving from academic research to commercial exploitation [7], due to the ability to print complex mechanical prototypes quickly and at a reduced price as in aerospace structures [8], [9], electronic circuits [10], [11] and high frequency devices [12]- [17] for filtering applications [18]- [22]. Taking advantage of these features can lead to faster waveguide band-pass filter development and much more complex geometries that are not feasible to implement when using traditional machining technologies. For example, in [18] band-pass filters are implemented using conical posts as a design element and in [20] they use 3D printed grooved spherical resonators to increase the rejection band of the filter.
On the other hand, periodic structures have been used in recent years for filtering applications [22], where the periodicity in the direction of propagation of electromagnetic waves results in the presence of allowed and prohibited frequency bands [23], [24], as predicted by analyzing the dispersion diagram. These characteristics make it possible to produce microwave filters with reduced dimensions, obtaining a more selective response in frequency with respect to other filters that are not periodic, providing a greater tolerance to manufacturing errors.
Recently, some articles have been published on periodic structures in Substrate Integrated Waveguide (SIW) [25], [26] and in rectangular waveguide devices [27] or Single-Ridge Waveguide (SRW) [28]. The main drawbacks in these works are the high return losses that occur when using this type of structure and the attenuation caused by the presence of the dielectric.
This work presents the design and manufacturing process of different band-pass filters in Empty Single-Ridge Waveguide (ESRW), by analyzing the dispersion diagram of a periodic unit cell. However, in this case, 3D-printing techniques are adopted to implement an empty structure that is subsequently internally metalized to reduce the attenuation caused by the dielectric. In addition, a matching technique is used to improve return losses in the entire pass band of the filters designed in a smaller size.
The paper is organized as follows. In section II, the design and analysis of the ESRW unit cell is carried out using the dispersion diagram of the Floquet modes of the infinite periodic structure, to determine the allowed and prohibited bands controlled by different design parameters. Section III presents the design of two band-pass filters with different fractional bandwidths, where tapering techniques in the different sections of the unit cells are used to yield appropriate return loss responses in the filters' pass bands. Section IV describes the manufacturing and measurement process of one of the filters implemented by means of a lowcost 3D printer, and finally the conclusions of this work are drawn in section V.

II. STUDY OF THE UNIT CELL IN A PERIODIC STRUCTURE
In order to characterize a periodic structure in the frequency domain, the dispersion diagram associated with its unit cell must be calculated. This has been done for the structure presented in this work using the eigenmode solver tool of the commercial software Ansys HFSS (see for example [29] and [30]). The unit cell is a hollow rectangular waveguide with periodic ridges on one side, as shown in Fig. 1: the (copper) metallic faces have a conductivity of 5.80·10 7 S/m and the inner material is air (with relative dielectric permittivity ε r = 1.0 and loss tangent tan δ = 0). This structure is different from the one presented in [28], where PLA (polylactic acid) is used as the inner dielectric material, incurring in the presence of considerable losses in the pass band of the final prototype.

A. DESIGN OF THE ESRW UNIT CELL
To carry out the design of the ESRW with a cutoff frequency of the fundamental mode f 1 = 4.0 GHz, the equations extracted from [31], [32] have been used. The inclusion of the periodic structure in the central part of the waveguide (Fig. 1) modifies the propagation constant of the fundamental mode of the ESRW. More specifically, the cutoff frequency of the fundamental mode depends not only on the design parameters of the ESRW, but also on the dimensions of the central periodic section.
To keep the cutoff frequency of the fundamental mode at f1 = 4.0 GHz, the dimensions transverse to the direction of propagation are modified and optimized using the HFSS electromagnetic software. The final dimensions of the unit cell are a = 21.28 mm, b = 3.78 mm, t = 0.63 mm, and l = 7.10 mm. The period D of the unit cell is determined using the equation = /4 obtained from [22], for a central frequency of the EBG f EBG = 8.35 GHz (H = 3.15 mm and D = 15.0 mm) so that it acts directly on the fundamental mode. Fig. 2 shows the dispersion diagram of the first five Floquet modes of the periodic structure, with the cutoff frequencies of the first two modes being f 1 = 4.0 GHz and f 2 = 11.50 GHz, respectively (and f B = 5.22 GHz).

B. PARAMETRIC ANALYSIS OF THE ESRW UNIT CELL
The modification of the period D of the unit cell has a direct impact on the behavior of the ESRW dispersion diagram, in terms of the cutoff frequency of the different Floquet modes in the periodic structure as well as in the single-mode bandwidth and fractional bandwidth [28], [29] (defined as = -1 /�( • 1 ) and = ( 2 -1 )/ 1 respectively).      3 shows the dispersion diagram of the Floquet modes for different lengths of the period D of the unit cell. As can be seen in Fig. 3, for a constant height H = 3.15 mm, the fractional bandwidth increases significantly as the period D decreases. This is due to the fact that by reducing the period of the unit cell the forbidden band shifts to higher frequencies. It is also shown in Fig. 3 that the single-mode bandwidth increases significantly by reducing the parameter D, obtaining the maximum bandwidth with D = 13.50 mm. This is because the second Floquet mode begins to propagate at higher frequencies when that period is used in the unit cell. Table I resumes the performance of the structure when parameter D is modified. In [28] the minimum achievable bandwidth was limited by transmission losses occasioned by the excessive tan δ value of the PLA material employed for implementing the filters. In this work the empty conception of the designs permits the achievement of narrow band band-pass filters. As seen in Fig. 3 narrow band designs are attained for longer lengths D of the periodic cell. For a Δ = 2.5% design (f1 = 4.0 GHz and f B = 4.10 GHz) the required unit cell length is D = 44.50 mm, which increases the total size of the filter.
On the other hand, the modification of the height H of the central section also has a direct effect on the distribution of the Floquet modes of the periodic ESRW with respect to the fractional and single-mode bandwidth. Fig. 4 shows the dispersion diagram for different values of H. It is worth highlighting that for a period D = 13.50 mm the fractional bandwidth of the fundamental mode decreases as H increases and the single-mode bandwidth increases until it reaches a constant value B = 2.91. This result has been summarized in Table II.

C. STUDY OF THE ESRW FINITE PERIODIC STRUCTURE
In order to obtain a high rejection level in the prohibited band-gap shown in Fig. 2, a large number of unit cells (periods) must be considered [22]. However, the larger the  Step height   number of periods is, the longer the structure implementation becomes. Fig. 5 shows two different finite implementation (with D = 13.50 mm and H = 3.15 mm) of the ESRW periodic structure.
The simulated (with Ansys HFSS) scattering parameters of both finite implementations are compared in Fig. 6. As expected, the rejection level in the band-gap bandwidth is higher for the longer finite implementation, while there is a good agreement between simulations and the cutoff frequency (f1 = 4.0 GHz) and the band-gap frequency limits (f B = 6.57 GHz and f 2 = 15.65 GHz).

III. BAND-PASS FILTERS DESIGN
After studying in section II the periodic ESRW and determining the influence of the design parameters of the empty unit cell (D and H) with respect to f 1 , f 2 , f B , B and Δ, the implementation of the finite structure is performed in section III, which describes the design process for two bandpass filters with different bandwidths. Two main drawbacks that occur when using periodic structures for filtering applications must be taken into account in the implementation of these filters as shown in Fig. 6. On the one hand, to obtain a high level of attenuation in the rejection band, a large number of unit cells must be considered, increasing the total length of the filter. On the other hand, when truncating of the periodic structure, an significant ripple appears in the transmission and reflection coefficients of the EBG, due to the difference between the characteristic impedance of the input and output ports, Z0, and the Bloch impedance in the periodic structure [22], leading to unacceptable levels of return losses. To solve this problem an impedance matching using a Chebyshev transformer was carried out in [28]. In this present work, to figure out this issue, tapering techniques are used [35], managing to improve the levels of return losses in the entire filter pass band and to reduce the total size of the structure compared to that presented in [28].

A. DESIGN OF A BAND-PASS FILTER WITH A FRACTIONAL BANDWIDTH OF Δ = 36%
The first designed filter, shown in Fig. 7, consists of a periodic ESRW with 8 unit cells and Δ = 36% at a design center frequency of 4.85 GHz. The design of the filter parameters, period of the unit cell D = 17.0 mm and height of the central section H = 3.15 mm, has been carried out directly from the information provided in Fig. 3 and Tab. I. To reduce the impedance mismatch introduced by the EBG structure, a tapering function is used, which is normally employed in Fiber Bragg Gratings (FBG) [36], [37]. The tapering technique is applied directly by modifying the length of the central section in each period Li of the finite structure, as shown in Fig. 7. The distribution of these lengths for each unit cell follows the next equation [38]:   where L i is the length of the central section of the unit cell in the i-th period, L max is the maximum central length (D/2), T(z) is the tapering function that is applied in the various unit cells and z i is the distance between the i-th period and the center of the structure. The improvement in the frequency response achieved using this technique can be understood as a progressive matching of the Bloch impedance to the characteristic input and output impedances produced by the tapering function. The tapering function used follows a Kaiser window distribution given by the equation [32]:

FIGURE 6. Simulated scattering parameters of two finite implementations of the EBG ESRW with 4 unit cells (grey lines) and 8 unit cells (black lines). Also shown are the lowest (fB) and highest (f2) frequency of the band-gap and the cut-off frequency of the fundamental mode (f1) in the periodic structure (dotted grey lines).
where I 0 is the modified Bessel function of the first class. The tapering function is applied directly on the 8 lengths (L i ) of the central section of the periods of the structure, applying equations (1) and (2). Table III shows the normalized coefficients of the Kaiser tapering function and the different lengths of the unit cells once applied. To make the connection of the ESRW filter in a simple way and keep losses low, a taper of length l t has been used to match the ESRW structure to the input and output filter 50 Ω inverted microstrip lines, as shown in Fig. 5. To calculate the taper width (W t ) at the ESRW ends, the mode impedance of the fundamental mode at the center frequency of the band-pass filter (f = 4.85 GHz) is calculated using the HFSS simulation software. The simulated filter response is shown in Fig. 8, together with the cutoff frequency f 1 of the first Floquet mode and the lower frequency f B of the band-gap, which define the filter passband. The passband ranges from 4.0 GHz to 5.70 GHz and has a single-mode rejection band that extends up to 12.50 GHz, which coincides with the cutoff frequency f 2 of the second Floquet mode (Fig. 3). The out of band rejection is better than 30.0 dB, despite the fact that only 8 periods have been used in the finite structure. The return losses are greater than 20.0 dB in the passband and the insertion losses are less than 0.60 dB from 4.05 GHz to 5.65 GHz. The electrical characteristics considered for the PLA substrate in the simulations are εr = 2.8 and tan δ = 0.02 [39], while copper is used to implement the inverted microstrip input and output lines (σ = 5.80 ·10 7 S/m).

B. DESIGN OF A BAND-PASS FILTER WITH A FRACTIONAL BANDWIDTH OF Δ = 55%
The second ESRW filter implemented is shown in Fig. 9, with the same number of periods as the previous filter. In this case, the upper frequency of the passband is increased to   Finally, a taper of length l t is included at the input and output ports of the ESRW as shown in Fig. 7, to match the structure to the 50 Ω inverted microstrip transmission lines.
The simulated filter response is shown in Fig. 10, as well as the cutoff frequency f 1 of the first Floquet mode and the lower frequency f B of the band-gap. The filter's passband ranges from 4.0 GHz to 6.90 GHz, which is larger than the previous filter and has a single-mode rejection band that extends up to 16.0 GHz with out of band rejection better than 30.0 dB. Return loss is larger than 20 dB throughout the passband while insertion loss is less than 0.6 dB from 4.05 GHz to 6.85 GHz.

IV. 3D-PRINTED FILTER PROTOTYPING
To demonstrate the practical feasibility of the theoretical analysis of ESRW band-pass filters described in the previous sections, the ESRW filter with a fractional bandwidth of Δ = 55% shown in Fig. 9 was manufactured and measured. The filter is divided into two different parts that were implemented individually (Fig. 11): the upper part (top view) formed by the inverted microstrip transmission line and the flat face of the ESRW, and the base (bottom view) which is formed by the ESRW itself. As shown in Fig. 11 the two pieces were manufactured using PLA filament and a low-cost 3D printer BQ's Prusa i3 Hephestos with a resolution of 0.015 mm in the XY axes and a 0.4 mm diameter nozzle.
The metallization of the two pieces of PLA that make up the ESRW filter is carried out following a multi-step process. First, a conductive copper coating is applied over the entire surface of the PLA parts. Second, an LPKF Protomat S42 numerical control milling machine is used to selectively remove the metal coating and make the transitions from the ESRW to the inverted microstrip lines of the filter's input and output ports, with the help of the positioning holes shown in Fig. 11 (top view). Finally, a conventional electroplating process is carried out to increase the thickness of the copper to ~ 35 µm (with an accuracy of ±5 µm) and the conductivity of the 3D-printed parts. The final result after metallization of the two pieces of the ESRW filter for a bandwidth Δ = 55% is shown in Fig. 12.
On the other hand, in order to incorporate the SMA connectors to the input and output ports of the ESRW filter, two grooves had to be left in the PLA in the central part of the inverted microstrip line to be able to correctly accommodate the central conductor of the coaxial connector, as shown in Fig  10. In Fig. 13(a) the process of connecting the SMA connector to the groove in the inverted microstrip line is shown, where a silver epoxy (RS PRO) is included to electrically join the SMA inner conductor to the transmission line. Fig. 13(b) presents the final prototype, including the connection of the screws to adjust the filter structure and the input and output ports of the vector network analyzer.
The S-parameters of the manufactured filter were measured using the Agilent PNA E8363C Vector Network Analyzer. The simulations and measurements of the Sparameters of the fabricated device are compared in Fig. 14, showing good agreement over the entire frequency range. The measured frequencies of the filter passband and rejection band are f1 = 4.0 GHz, f B = 6.79 GHz and f 2 = 15.80 GHz, which are very similar to those simulated with HFSS f 1 = 4.0 GHz, f B = 6.80 GHz and f 2 = 15.80 GHz (this small deviation from the EBG cutoff frequency could be due to small variations in the filter manufacturing process in the 3D printer and to the tolerance in the PLA surface metallization process). The measured insertion loss in the filter passband is 0.89 dB, very close to the insertion loss of 0.50 dB obtained in the simulation (this variation could be due to the losses introduced by the SMA connectors that are not included in the HFSS simulation). The return loss measured in the filter passband is better than 16.50 dB, which confirms the matching approach followed in the filter design (the small disagreement could be due to minor variations in the relative permittivity of the PLA or manufacturing errors in the printing of the filter and in the milling process of the inverted microstrip).

V. CONCLUSIONS
In this work, a simple topology for the design of band-pass filters in an empty waveguide is presented through the use of periodic structures. The design process for band-pass filters is based on including an EBG in an ESRW, using two different heights in the central section of the waveguide in order to produce the desired frequency response. The analysis of the dispersion diagram of the unit cell of the periodic structure allows to study the different design parameters of the filter frequency response, providing total control over the required specifications of the filter. The cutoff frequencies of the passband and the forbidden band are mainly determined by the height of the center section and the period of the periodic structure, respectively. To validate the configuration and the proposed design method, two band-pass filters with different fractional and single-mode bandwidths are presented, improving the response of the reflection coefficient in the passband by using a Kaiser EBG distribution in the different lengths of the central sections of the unit cells, providing excellent transmission in the ESRW filter passband. Finally, a band-pass filter with a 55% fractional bandwidth is manufactured using a low-cost 3Dprinting technology and an electroplating process to metallize the structure. The measured filter response is consistent with electromagnetic simulation and provides good passband and rejection band performance through the use of a simple-todesign structure with low insertion losses and high return losses. The results presented in this work are very promising and open a wide range of possibilities for 3D-printing technology to develop and design microwave filters in an easy way in ESRW that present: very low transmission losses, wide working bandwidths and rejection, with a very reduced cost and manufacturing time.