Inline Waveguide Filter With Transmission Zeros Using a Modified-T-Shaped-Post Coupling Inverter

This letter reports the design techniques for a class of inline waveguide bandpass filters with sharp-rejection capabilities at the lower stopband based on a novel nonlinear-frequency-variant-coupling (NFVC) structure. The proposed NFVC consists of a modified-T-shaped metallic post (MTP) that is placed at the center of the waveguide broad wall with its open arms lying along the waveguide width. The engineered NFVC structure produces a first-order bandpass filtering transfer function with a pair of transmission zeros (TZs) located below the passband range. To demonstrate the usefulness of the proposed NFVC inverter, a 9.9-GHz third-order inline waveguide bandpass filter prototype with two TZs is developed and tested. It consists of two half-wavelength cavity resonators coupled together via the conceived MTP coupling structure. The measured results are in close agreement with the electromagnetic (EM) simulated ones, thus validating the devised waveguide filter design principle.

As a further contribution, the design of a class of inline generalized Chebyshev waveguide bandpass filters based on an alternative NFVC structure is reported in this letter. The proposed NFVC inverter consists of a modified-T-shaped metallic post (MTP) that is placed inside the waveguide, where the open arms of the post are lying along the width of the waveguide. This type of NFVC inverter is capable of producing two TZs along with a pole that is located above the pair of TZs. Note that although the referred case where the pole is located above the TZs is considered here, the pole could also be positioned either below or in between the TZs. Using this NFVC inverter to couple two half-wavelength cavity resonators, a third-order waveguide bandpass filter prototype with two TZs is designed, fabricated, and tested to verify the practical usefulness of the proposed NFVC structure.

II. MODIFIED T-SHAPED NFVC INVERTER
An MTP placed inside a waveguide, as shown in Fig. 1(a) and (b), behaves as an NFVC inverter that can produce two TZs along with a transmission pole in its frequency 2771-957X © 2022 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See https://www.ieee.org/publications/rights/index.html for more information. response. The plot shown in Fig. 2, which corresponds to the transimpedance parameter (Z 21 ) for an example of the devised MTP NFVC inverter and its equivalent circuit, confirms the creation of a TZ pair and a pole located at a frequency above the two TZs. This structure can be modeled as two mutually coupled series-type resonators separated by a section of transmission line (TL), as can be seen in the equivalent circuit depicted in Fig  thick, and they are vertically aligned with the top surface of the main post. The center-to-center separation between the side arms of the post along the length of the waveguide is 2.2 mm. It is evident from the results shown in Fig. 3(a) that by introducing a small TL section (θ TL ) between the side arms, the pole frequency is reallocated above the TZs. Besides, by further increasing its length, the pole frequency moves further away from the TZ pair. The length of side arms could be mainly used to allocate the TZs and the pole in a widely adjustable spectral range as revealed by Fig. 3(b). Furthermore, the width and the height of the short-ended vertical post serving as mutual coupling are useful parameters to control the frequency separation between the two TZs. Note that the two TZs could be brought closer to each other by either decreasing the height of the vertical post or by increasing the width of the vertical post along the waveguide width as demonstrated in Fig. 3(c) and (d), respectively. Note that the TZ pair could be relocated to above the pole location by simultaneously increasing the width (W w ) of the vertical post and decreasing the length of side arms (L SA ). It must be remarked upon that a comparable response could be obtained by replacing the MTP with a pair of closely spaced partial-height posts, as reported by Sandhu et al. [12]. However, considering practical reasons, the current approach may be more suitable since very closely spaced posts are difficult to fabricate. Besides, it is tedious to place tuning screws in a tight space to allow postfabrication tuning for such an arrangement. A performance comparison of the proposed MTP inverter with related prior-art inverter configurations-in terms of the number of TZs and created additional poles, the minimum number of resonators required to implement the coupling inverter, need of cross couplings, and flexibility of locating the TZ either below or above the passband-is presented in Table I. III. FILTER DESIGN EXAMPLE To validate the concept, a third-order bandpass filter with 150-MHz bandwidth, 9.9-GHz center frequency, and two TZs located in the lower stopband at 9.675 and 9.78 GHz has been designed. This filter prototype consists of two WR90 half-wavelength cavity resonators that are coupled through the conceived MTP NFVC network. To design the filter as per the given specifications and topology, a coupling matrix is synthesized using the approach in [24]. The synthesis is carried out directly in the passband domain and by using the coupling-network model shown in Fig. 1(c). The synthesis process involves solving a structured inverse nonlinear eigenvalue problem. Based upon the coupling-matrix entries and prespecified TZ locations, the MTP NFVC inverter is extracted to realize a pole and two TZs. Specifically, such process provides the electrical length of waveguide cavities θ res = 177. Adding such a TL section allows us to flip the pole frequency above the TZ pair. Once the MTP NFVC inverter is extracted, then the remaining waveguide cavity resonators and external coupling inverters are derived using standard waveguide filter design theory. The external couplings are realized as H-plane inductive window couplings. Fine numerical tuning is finally performed using the zero-pole optimization technique of the commercial finite element method (FEM)-based microwave electronic design automation (EDA) software InvenSim. The E-field pattern of the final optimized filter is shown in Fig. 4.

IV. EXPERIMENTAL VALIDATION
The photograph of the final fabricated prototype is given in Fig. 5. The filter is manufactured in two pieces-body and top lid-using computer numerical control (CNC) machining. The post is fabricated separately and then fit in the base wall of the filter and heat-treated to solidify the connection to the base unit. The post dimensions are as follows: W w = 3.76 mm, W L = 4.2 mm, H v = 5.24 mm, L SA = 4.45 mm, θ TL = 2.2 mm, and H SA = 2.44 mm. Tuning screws with a diameter of 3 mm are placed at the center of waveguide cavities, at the external coupling windows, and at the edges of the side arms of the MTP to allow postfabrication tuning so that to compensate for manufacturing inaccuracies. A comparison between the EM-simulated and measured power transmission and reflection responses of the built third-order bandpass filter prototype is shown in Fig. 6. The measured prototype has a minimum in-band return and insertion loss of 19 and 0.56 dB, respectively, 135-MHz bandwidth, and a center frequency of   9.847 GHz. The spurious resonance at about 13 GHz is caused by the MTP inverter. As can be seen, a reasonable agreement is obtained between EM-simulated and experimental results in terms of passband return-loss level, filter bandwidth, presence of two TZs below the passband, and spurious performance, so that the engineered waveguide filter principle is fairly verified. It is observed that the whole filter response is slightly shifted to a lower frequency due to manufacturing errors in the dimensions of the MTP inverter and the outer cavities. Some minor discrepancy in terms of passband insertion loss is attributed to the added lossy tuning screws. Note also that the filter pole associated with the MTP has a lower quality factor Q (2963) when compared to a regular cavity resonator. This results in a reduced average unloaded Q for the filter of 5100. Nevertheless, the measured prototype verifies this design principle of the waveguide filter.
V. CONCLUSION This letter presents a design technique to implement inline waveguide bandpass filters with two TZs at the lower passband side based on a new class of MTP NFVC inverter. The proposed inverter produces a transmission pole located above a pair of TZs, unlike its simple T-shaped postinverter counterpart where the pole is located between the two TZs. This is achieved by moving apart the arms of the T-shaped inverter, which allows reallocating the pole above the TZ pair. A 9.9-GHz proof-of-concept prototype of a third-order waveguide bandpass filter based on the proposed MTP NFVC inverter is designed, fabricated, and measured to validate the engineered filter principle and design theory.