Inline Waveguide Pseudo-Elliptic Filters Using Non-Resonating Modes with Folded-Waveguide Resonators

In this paper, a new structure to implement inline waveguide pseudo-elliptic filters using folded-waveguide (FWG) resonators along with non-resonating modes has been proposed. The basic structure is an FWG resonator fed by a rectangular waveguide through inductive irises. The TE10 mode of the feeding rectangular waveguide excites both dominant and first higher order modes in the FWG structure. The dominant mode of the FWG acting as a non-resonating mode creates an alternate energy path from source to load thus producing a transmission zero (TZ), while the first higher order mode acts as the resonant mode resulting in a pole. The structure therefore acts as a singlet capable of realizing a pole and a TZ either below or above the passband depending on the offset of the feeding irises from the center of the FWG resonator and relative to each other. The structure also provides flexibility in shifting the FWG center axis relative to the feeding rectangular waveguides, thus giving more coupling control and realizing wider range of singlet responses. The proposed singlet structure does not require any increase in the cross-sectional size relative to the standard rectangular waveguide. Two prototype filters using the proposed singlet structures have been designed, manufactured and tested. Measurements show good agreement with the simulations, verifying the feasibility of the proposed structure.


I. INTRODUCTION
P SEUDO-ELLIPTIC bandpass filter responses can implement a sharper transition from passband to stopband because of their ability to realize finite frequency transmission zeros (TZs) and are therefore preferred by modern microwave transceivers [1], [2]. For applications requiring low losses and/or high power handling capability, these filters are implemented in waveguide technology [3]. Several methods for realization of TZs in waveguide technology have been proposed in the literature, including cross-coupling between non-adjacent resonators and the use of multi-mode cavities [3]- [9], extracted pole synthesis method [10] and the use of inverters based on frequency-dependent coupling sections [11]. A relatively new approach of realizing pseudo-elliptic response that has attracted attention in the last decade is the use of non-resonating modes to create additional energy paths [1], [12], [13]. The non-resonating modes can either be propagating or evanescent at the passband of the filter and they can provide additional energy paths and thus can be used to implement arbitrarily located finite frequency transmission zeros (TZ) while still keeping the inline topology. In this regard, the earliest work makes use of asymmetric irises in a rectangular waveguide [14], [15] to realize TZs but the non-resonating (T E 20 ) mode vanishes very quickly along the cavity, resulting in week source to load coupling and thus the flexibility in terms of positioning of TZs is limited. To overcome this limitation, resonating mode of T E 201 has been utilized in an enlarged width rectangular waveguide cavity [16]- [18] to realize the finite frequency TZs. A relatively compact singlet (A singlet [19] is a structure that produces a pole along with a transmission zero.) implementation is possible by using T M 110 mode as the resonant mode [20]. To achieve further compactness, the concept of T M dual-mode cavity has been combined with the idea of non-resonating modes [21]- [23].
The above mentioned techniques lead to an increase in the cross-sectional size relative to the standard all-pole waveguide iris filters. For applications where the increase in cross-section is undesirable, inline waveguide pseudoelliptic filters capable of realizing finite frequency TZs can be implemented using the methods proposed in [24]- [28]. Pseudoelliptic waveguide filters can be realized by using rectangular ridge resonators as singlets [24], [25] to produce TZs below and/or above the passband. These filters can easily be manufactured using computer numerical control (CNC) milling of solid aluminum and do not require any additional components, making these structures economical and repeatable for production purposes. However, these advantages are achieved at the expanse of slightly lower unloaded quality factors in these filters. Pseudo-elliptic filters can also be implemented using dielectric disks appropriately located in a rectangular waveguide [26], [27]. However, for these filters to work, each dielectric resonator has to be precisely positioned at a certain angle in the waveguide leading to a tedious fabrication process. Pseudo-elliptic characteristic can also be realized by using dual post resonators [28] inside a rectangular waveguide. A TZ below the passband is achieved by having different post heights, while TZ in the upper stopband is implemented by using rotation of dual-post resonators. The dual-post resonators however have to be manufactured separately and then push fit (or screw fit) into the waveguide, thus making the manufacturing more involved.
In this paper, we introduce a new singlet structure for realization of inline waveguide pseudo-elliptic filters, using non-resonating modes with folded-waveguide (FWG) resonators. The singlet is comprised of an FWG resonator which is fed by input and output waveguide sections using inductive irises, as shown in Fig. 1. By adjusting the location of irises with respect to the FWG center axis, a TZ either below or above the pole can be implemented. Additionally the proposed singlet offers flexibility in adjusting the center axis of FWG relative to the center axis of feeding rectangular waveguides. The filters realized using these singlets, do not require any increase in the cross-sectional size with respect to the standard rectangular waveguide. Two prototype pseudoelliptic filters are designed, manufactured and tested. The experimental results of these prototype filters are close to the simulations and thus validate the proposed singlet structure and the overall filter topologies.
Pseudo-elliptic filters presented in [16], [17] make use of T E 201 mode as a resonant mode while the bypass coupling is provided by T E 10 mode which acts as a non-resonating mode. To achieve a pole at the desired frequency in the passband the width of the rectangular cavity implementing T E 201 has to be significantly increased beyond that of the feeding waveguide. The center of the enlarged width cavity can be easily adjusted relative to the center axis of the feeding waveguides to achieve flexibility in realization of the finitefrequency TZ at the desired location. A singlet structure presented in [29] and also explained in [13] again uses an enlarged width rectangular waveguide cavity but with T E 301 mode as the resonating mode. The width dimension of the realized singlet is significantly larger than that of a standard rectangular waveguide. On the other hand, the singlet structure proposed in this paper combines an FWG resonator with the concept of non-resonating modes. The resonating mode used to excite the FWG is T E 201 like mode while T E 10 acts as the non-resonating mode. The main advantage of the proposed singlet arrangement over those presented in [16], [17], [29] is that the proposed structure does not require any increase in cross-sectional dimensions beyond those of a standard waveguide. However, the enlarged width cavity based singlets of [13], [16], [17], [29] do present great flexibily in setting the location of TZ at the required frequency by just adjusting the center of singlet cavity relative to the centers of feeding rectangular waveguides. This is easily achieved since these singlets are not constrained by the crosssectional dimensions of the feeding waveguides. In contrast the FWG based singlet structure presented in this article, has to fit inside the normal cross-sectional size of a standard waveguide and therefore we cannot simply offset the whole FWG structure relative to the feeding waveguides. To address this constraint, we have introduced a dimension r which can be adjusted to effectively shift the FWG center axis relative to that of the feeding waveguide, as shown in Fig. 1 without any increase in cross-sectional size. To the best of our knowledge, this way of using an FWG structure with feeding irises to realize a singlet structure has not been reported before. The filters based on the oversized rectangular cavity singlets of [13], [16], [17], [29] may provide a good solution for implementations at THz and millimeter-wave frequencies because of the smaller wavelengths (and hence smaller cavity dimensions) involved at those high frequencies. However in the lower microwave spectrum (<30GHz), this increase in cross-sectional area makes the filter bulky and presents a limitation particularly in "retrofit" [13] applications where all-pole filters of a pre-existing microwave system need to be upgraded by better performance (pseudo-elliptic) but similar size filters. The filters designed using the singlet structures proposed in this paper may be good candidates for such applications because their cross-sectional dimensions would be the same as the pre-existing all-pole waveguide filters.

II. SINGLET BASED ON A FOLDED-WAVEGUIDE (FWG) RESONATOR
Folded-waveguide (FWG) structure can be thought of as a reduced height rectangular waveguide, folded at one or more suitable axis locations. The conceptual development of two commonly used FWG topologies in literature [30]- [33], is shown in Fig. 2. The equivalent reduced height rectangular waveguide along with electric field distribution of dominant (T E 10 ) and first higher order (T E 20 ) modes are shown in Fig. 2(a). Fig. 2(b) shows the FWG achieved by two foldings of the reduced height rectangular waveguide. Fig. 2(c) is achieved by one folding at the center of the reduced height rectangular waveguide. Note that for both types of FWGs, the location of the FWG center axis is fixed and is marked by a cross in the Fig. 2.
In contrast to the conventional FWG shown in Fig. 2(b), we propose a slightly different FWG structure with a provision of adjusting the center axis of the FWG with respect to the feeding rectangular waveguide, thus offering more flexibility by allowing shifting of the dominant FWG mode electric field maximum relative to the feeding rectangular waveguide electric field maximum. The cross-sectional view of this proposed FWG structure is shown in Fig. 3, along with the electric field distribution of the first two modes. The parameter r can be used to adjust the location of the FWG center axis.
We propose a new singlet structure by using the FWG structure of Fig. 3 which is capable of realizing both a pole and a TZ either in the lower or in the upper stopband. The FWG is fed by input and output inductive irises from two rectangular waveguide sections which results in a singlet structure. Different views of this singlet are shown in Fig. 1. The cross-sectional views of different parts of the proposed singlet are shown in Fig. 4. The dominant T E 10 mode of the feeding rectangular waveguide sections can excite both the dominant (T E 10 like mode) and the first higher order (T E 201 like mode) modes in an FWG section, thus resulting in a singlet capable of realizing a pole and a TZ. Whether the TZ is implemented below or above the passband depends on the location of irises with respect to the FWG center axis and with respect to each other. Fig. 5 shows the top view of the singlet structure along with magnetic field lines for the resonating and the nonresonating modes. When both irises are at FWG center axis, resonating mode is not excited leading to only source to load coupling. When these irises are shifted away from the FWG center axis, the result is a pole and a TZ. A TZ is realized below the passband when the irises are on same side with   respect to the FWG center while a TZ above the passband is implemented when the irises are located on opposite sides of the FWG center axis. It is worth mentioning that the proposed singlet structure can also be fed by FWG sections instead of rectangular waveguide sections.
The proposed singlet response can be represented by a 3×3 VOLUME 4, 2016 x y x y x y   normalized coupling matrix (CM) given below [19].
The routing and coupling scheme for this singlet structure is given in Fig. 6. A TZ can be implemented at different locations from the proposed singlet by changing certain physical parameters of the structure, as explained in the cases below.

A. IRISES CENTERED AT THE CENTER AXIS OF FWG
In this case the feeding input and output irises are positioned at the center axis of FWG [see Fig. 5(a)]. As shown by the magnetic field lines of Fig. 5(a), the resonating mode of FWG (T E 201 like mode) because of its odd symmetry can not be excited by the even symmetry dominant T E 10 mode of the rectangular waveguide and therefore does not create any pole or any TZ i.e. M S1 = M 1L = 0. Bypass coupling between source and load, however is non-zero (M SL = 0), because the feeding rectangular waveguide does excite the dominant (T E 10 like) mode of the FWG.

B. IRISES ON SAME SIDE OF FWG CENTER AXIS
When the centers of the two irises are shifted away from the FWG center axis and when both irises are located on the same side relative to the FWG center axis [see Fig. 5(b)], the feeding T E 10 mode of the rectangular waveguide can excite both dominant and first higher order modes of the FWG, resulting in realization of a pole and a TZ. As shown in Fig.  5(b), the magnetic field lines for dominant mode of FWG are in phase with those of resonating (first higher order) mode of FWG at the cavity output and hence the TZ is realized below the passband. Note that this result is also consistent with the singlet implementation of [16]- [18], using an oversized rectangular waveguide cavity with T E 201 as resonant mode. As shown in Fig. 7, location of the resulting TZ can easily be varied by changing dimension d of input and output irises (here d 1 = d 2 = d). Additionally the parameter r can be used to shift the location of FWG center axis. This shift in location of FWG center axis is particularly useful for this case of same side irises, since it offers more flexibility in realization of required singlet response and since both irises are located on same side. All couplings (M S1 = 0, M 1L = 0 and M SL = 0) are non-zero for this case. Fig. 7(a) shows singlet response against variations in d, for r = 4 mm. The inset shows the cross-sectional views of the different portions of singlet structure, for r = 4 mm along with the location of FWG center axis. TZ below the passband is realized for d = 1, 2, 3, ...10 mm. For d=11 mm, the two irises are very close to the FWG center axis, and therefore the resonating mode is not excited. Thus neither pole nor TZ is realized for d=11 mm.
Similarly, Fig. 7(b) and (c) shows the singlet responses for r=7 mm and r=10 mm, respectively. Note that Fig. 7 shows the responses for a singlet with dimensions d 1 of input iris and d 2 of output iris as equal and thus M S1 =M 1L , which is not essentially required by the singlet structure and thus these dimensions can be set independently to achieve the desired response.

C. IRISES ON OPPOSITE SIDES OF FWG CENTER AXIS
For the case where input and output irises are located on opposite sides of FWG center axis [see Fig. 5(c)], a TZ above the pole can be realized. Note that the magnetic field lines for the non-resonating (dominant) and the resonant (first higher order) modes are out of phase at the output of the singlet of Fig. 5(c), hence creating a TZ in the upper stopband. We can adjust the location of TZ by changing dimension d 1 for input and d 2 for output irises, as shown in Fig. 8.
Here all coupling coefficients namely M S1 , M SL and M 1L are non-zero because the feeding T E 10 mode of rectangular waveguide can excite both dominant and first higher order modes of FWG.
Note that Fig. 8 shows the singlet for which parameter d 1 of input and d 2 of output are set in terms of a new parameter d, as d 1 = d and d 2 = a − d − w 2 . The insets show the crosssectional views at different locations of the singlet structure. Fig. 8(a) shows the singlet response for r = 4 mm. Here only d = 1 mm and d = 2 mm are shown, because when d becomes further greater, the irises are no longer on opposite sides of FWG center axis and will therefore realize TZ below the passband instead (see subsection II-B). It is worth mentioning, that for ease of explanation, dimensions d 1 of input iris and d 2 of output iris have been expressed in terms of a variable d (d 1 = d and d 2 = a − d − w 2 ). However, this constraint is not required by the structure and the two parameters can be set independently of each other, thus giving more flexibility in implementation of the desired singlet response.

III. THREE-POLE/ONE-TZ WAVEGUIDE FILTER
Higher order filters can be designed by cascading multiple singlets or by cascading a combination of singlets and resonators [19], [24], [34]. In this section, a three-pole filter with one TZ below the passband is designed. The TZ below the passband is realized by using the singlet with irises on same side of FWG center axis [see Fig. 5(b)]. For this filter, the remaining two poles are realized by using FWG cavity resonators operating in dominant T E 101 like mode. Fig. 9 shows the routing and coupling scheme for this three-pole filter. The CM is of the form Optimization based technique [35] is utilized to synthesize the CM for the required filter response. Fig. 10 shows the CM VOLUME 4, 2016 and FWG3 section, respectively using the equation below [24], [34].
where f 0 is the center frequency and F BW is the fractional bandwidth. For this design f 0 = 10.2 GHz and F BW = 0.3/10.2 = 0.0294. The resulting resonant frequencies are as given in Table 1. Note that the positive values of M ii leads to resonant frequencies smaller than f 0 while negative values of M ii results in resonant frequencies greater than f 0 .
To design this filter from the CM (3), an equivalent circuit shown in Fig. 11 has been synthesized using a procedure similar to [25]. The equivalent circuit consists of input and output couplings (K S1 and K 3L ), singlet (S2) represented by its CM and folded waveguide sections operating in dominant resonant modes (F W G 1 and F W G 3 of lengths 1 and 3 , respectively). Basic microwave network theory [36] has been utilized to represent each block of Fig. 11 as an ABCD matrix. The individual ABCD matrices are then multiplied to obtain the overall ABCD matrix of the filter.
where ABCD matrices for input coupling, output coupling and FWG sections operating in dominant mode can be given as [37], [36] ABCD KS1 = 0 jK S1 j/K S1 0  11. Equivalent circuit for the three-pole/one-TZ waveguide filter.
where β is the propagation constant of the dominant (T E 10 like) mode in each FWG section. Singlet S-parameters are determined from its CM of (1), using equations from [35] and using unity fractional bandwidth (F BW = 1). These S-parameters are converted in to the required ABCD matrix by using the standard conversion tables [36]. The complete equivalent circuit of Fig. 11 is then synthesized using optimization based method [25], [35]. The required design parameters to be synthesized are given as the vector below The objective function used is given in (7), where ABCD parameters are determined for each iteration and each frequency point, using (5) and then ABCD to S-parameter conversion is carried out to obtain the overall S-parameters [25].
w=1 |S 11 (f w )| − 10 −RL/20 2 (7) where f pu and f zv represent the desired poles and TZs frequencies, respectively. W pu and W zv are the weights for  The synthesized equivalent circuit parameters are given in Table 2. Fig. 10 shows the frequency response for the synthesized equivalent circuit, and it agrees well with the CM response. Using the synthesized equivalent circuit of Table 2, the physical dimensions of the filter can be obtained. Iris widths of the input and output irises can be determined from the values of K S1 and K 3L , respectively, using the well known waveguide iris filter design method [38]. The physical lengths for the two FWG sections ( 1 and 3 ) are already determined and are given in Table 2. To determine the physical dimensions of the singlet S2 (shown in Fig. 1), full wave electromagnetic (EM) simulations are carried out to achieve a close match of the simulated S-parameter response to that of the singlet CM. Fig. 12 compares the coupling matrix response with the simulation response, which indicates the two responses are almost similar particularly in the vicinity of the pole and the TZ. The phase response of the simulated singlet is then matched to that of the singlet CM to determine the locations of the input/output reference planes for the simulated singlet structure.
The resulting dimensions are then used to create the filter structure in HFSS. Some optimizations are required in this full wave EM simulator, since the equivalent circuit model does not account for all the higher-order mode effects. Optimizations are performed in HFSS and the resulting response is plotted in Fig. 10, which indicates that the HFSS response is in good agreement with the CM and circuit responses.
Manufacturing of the designed three-pole filter is carried out by CNC milling of aluminum. The manufactured prototype filter is shown in Fig. 13.
A vector network analyzer is used to measure the S-VOLUME 4, 2016   Fig. 15.

IV. FOUR-POLE/TWO-TZ WAVEGUIDE FILTER
A four-pole pseudo-elliptic waveguide filter with two TZs, one above and another below the passband is designed by using two singlets and two half-wave rectangular waveguide sections. The singlet implementing lower stopband TZ makes use of irises on same side of FWG center axis [see The routing and coupling scheme for this four-pole filter is shown in Fig. 16 To synthesize this CM, same methodology as explained in Section III is utilized. The resulting CM is shown below as (9). The S-parameter response of the CM of (9) is given in Fig. 17. The equivalent circuit of Fig. 18 is utilized to design this filter. The design procedure is optimization based and is similar to the one explained in Section III. ABCD matrices are utilized to represent each block of the equivalent circuit. Table 4 shows the synthesized parameters for the equivalent circuit of Fig. 18.   For each singlet (S1 and S3), the S-parameter response of the CM [see Table 4] is matched to that of the physical structure using HFSS. The comparison of CM response with full wave simulation response for each singlet is shown in Fig. 19. The simulated response compares well with the CM response for each singlet in the vicinity of pole and TZ locations. Using the same steps as in Section III, the resulting dimensions are implemented in HFSS for this fourpole filter. The simulated S-parameter response is shown in Fig. 17, which also compares it with CM and circuit responses. The simulated response is in good agreement with the CM and circuit responses particularly in close proximity of the passband and TZs locations.
Manufacturing of the filter is carried out using CNC milling and the manufactured prototype is shown in Fig. 20.
The filter's measured response is shown in Fig. 21, which also shows the HFSS simulated response. The measured S-parameters match well with the simulated results. The measured return loss is better than 19.16 dB in the passband. Tolerance analysis is performed in HFSS using a Gaussian distribution with standard deviation of 10 µm. For each iteration, all physical dimensions of the filter except for a   Table 5 compares the proposed structure with other previously reported non-resonting modes based waveguide pseudo-elliptic filters. The proposed filters are easily manufactured through CNC milling, are less sensitive to manufacturing errors, do not exceed the cross-sectional area dimensions of the feeding rectangular waveguides and achieve good unloaded quality factors and insertion loss.

V. CONCLUSION
A new class of inline waveguide pseudo-elliptic filters, using a novel singlet structure based on folded-waveguide (FWG) cavity resonator has been presented in this work. The FWG is fed by rectangular waveguides using irises that excite both resonant and non-resonating modes. The dimensions and locations of the irises are adjusted to achieve the required location of TZs. Proper placement of input and output irises relative to each other and relative to the FWG center axis, leads to an implementation of a TZ either below or above the passband. Additionally, FWG center axis may also be adjusted thus providing more flexibility in realizing the required singlet response.
The proposed singlet structures have been used to design a three-pole filter with one TZ below the passband and a four-pole filter with two TZs, one below and other above the passband. Manufacturing tolerance analysis of the four-pole filter indicates that the filter performance is not very sensitive to manufacturing errors, with worst-case return loss of 15.68 dB and TZs locations almost the same as their designed locations. The two prototype filters are manufactured through CNC milling process and measured using a vector network