I. INTRODUCTION
In order to preserve the polarization of signals, waveguide circuits are required to have a common center axis, and they must conform to the following conditions: Each cross section must be two-plane symmetric, such as in quadruple-ridged waveguides [Reference Sun and Balanis1, Reference Vanin, Wollack, Zaki and Schmitt2] or in orthomode horn applications [Reference Zhang3–Reference Thiart, Rambabu and Bornemann5]. Alternatively, the entire component can possess rotational symmetry, e.g. [Reference Amari and Bornemann6–Reference Amari and Bornemann9]. Both features benefit from extended bandwidth, e.g. [Reference James, Clark and Greene4, Reference Olver, Clarricoats, Kishk and Shafai7], through the excitation of only those modes that satisfy structural symmetry along with that of the incoming electromagnetic field, e.g. [Reference Buckley, Stein and Vernon8, Reference Amari and Bornemann9].
Modern microwave filters are required to possess properties such as wide bandwidth, improved out-of-band performance, the ability to generate transmission zeros, and preservation of all polarizations. Although the mechanism to generate transmission zeros at finite frequencies in the filter response is well understood in standard dual-mode filters, e.g. [Reference Atia and Williams10, Reference Moretti, Alessandri and Sorrentino11], fulfilling the polarization-preserving specifications is not necessarily a straightforward design exercise, e.g. [Reference Rosenberg and Schneider12].
Recently, inline rectangular waveguide filters have been proposed which utilize the resonances of TM11 modes to create filters and the fundamental-mode TE10-mode bypass coupling to generate transmission zeros [Reference Amari, Rosenberg and Bornemann13]. Unfortunately, this design does not lend itself to two-plane cross-section symmetry due to the mandatory offsets between connected waveguides. The equivalent scenario in circular waveguide technology would employ TM01-mode resonators. However, the TM01 will not be excited by an axial symmetric connection to the fundamental TE11 mode of a circular waveguide. (The reader is referred to [Reference Lee, Lee and Chuang14] for plots of modal field distributions in rectangular and circular waveguides.)
Therefore, we utilize the circular waveguide TM11 mode to create an equivalent to the so-called singlet design in rectangular waveguide [Reference Amari, Rosenberg and Bornemann13]. It will be demonstrated in this paper that this approach leads to the design of rotationally symmetric, thus polarization-preserving, circular waveguide filters and allows the creation of transmission zeros. Moreover, a bandpass filter using TM11-mode resonators will be much shorter than one realized with conventional TE11-mode cavities.
This work is an extension of [Reference Bornemann and Yu15] but it incorporates a number of additional aspects. First, we will explain the fundamental connections between TM11-mode resonators and their adjacent circular waveguide sections. Different scenarios are discussed which will later be exploited for the design of lowpass, highpass, bandpass, and bandstop filters. We also demonstrate the limiting aspect of TM01-mode resonators as opposed to TM11-mode resonators. Finally, examples of bandstop and lowpass filters are added which demonstrate that the entire range of filter categories can be designed to preserve polarization of the filtered signals.
II. DESIGN CONSIDERATIONS
For the purpose of initiating the discussion with respect to an individual filter category, we start with a regular circular waveguide iris filter that is straightforwardly designed using cascaded half-wave cavities separated by irises that act as impedance inverters [Reference Matthaei, Young and Jones16, Reference Uher, Bornemann and Rosenberg17]. The inset of Fig. 1 shows the filter which is designed for a center frequency of 11.5 GHz, a bandwidth of 500 MHz, and a return loss of 22 dB. Stopband requirements were selected such that seven cascaded TE11-mode cavities are obtained. This filter is used as a first prototype to satisfy the demand for the preservation of polarization. WC 80 (20.24 mm in diameter) input and output waveguides are selected and are maintained throughout the paper for better comparison between filters or filter components. The fundamental TE11-mode cutoff frequency of the ports is 8.68 GHz. Due to rotational symmetry, the next excited higher order mode is TM11 with a cutoff frequency of 18.1 GHz. Thus, the operational bandwidth under consideration of rotational symmetry is about 9–18 GHz. Similar principles are applied in other on-axis connected circular waveguide components such as, e.g., corrugated horns [Reference Olver, Clarricoats, Kishk and Shafai7]. The cutoff frequencies of the two modes involved in this work are
where c is the speed of light and d is the diameter of the waveguide.
Fig. 1. Performance of a seven-resonator regular circular waveguide iris filter designed for 500 MHz bandwidth at 11.5 GHz (cf. Table 1).
Figure 1 shows the performance of the regular circular waveguide iris filter. Its length from first to last iris is 131.5 mm with 1 mm thick irises (cf. Table 1). It is observed that the main passband is well represented but a second wider passband appears already at 15.5 GHz, thus limiting the applicable bandwidth of such a design.
Table 1. Dimensions of circular waveguide filters (diameters d n and lengths l n in mm).
In order to improve the stopband performance, the use of frequency-dependent radial inverters was suggested [Reference Zhang3] as they are capable of adding transmission zeros in the lower and/or upper stopband of a bandpass filter. As is demonstrated in [Reference Zhang3, Reference Bornemann and Yu15], however, the introduction of such transmission zeros fails to prevent the second passband due to the fact that the electrical lengths of the resonators remain largely unchanged.
In order to eliminate the second passband in Fig. 1 or move any higher-order resonances beyond 18 GHz, shorter cavities must be used whose resonances are independent of length. This is possible by using resonating TMmn0 modes. As pointed out earlier, the TM010 mode will not be excited by a rotationally symmetric connection to a fundamental-mode circular waveguide. Therefore, we are investigating in Fig. 2 the use TM110-mode cavities. In all three cases, the cavity has a diameter of 36 mm and a length of 2 mm. These dimensions have purposely been selected to provide a resonance that is lower than the passband in Fig. 1. It will aid in demonstrating in Fig. 3 the limitations of an offset-connected TM010-mode resonator.
Fig. 2. Performance of a TM110-mode cavity coupled to standard circular WC 80 waveguides: (a) direct coupling, (b) through large irises, and (c) through small irises.
Fig. 3. µWave Wizard computation of an offset-connected TM010-mode cavity with the transmission zero to the right of the resonance.
Figure 2(a) shows a direct connection between the TM110-mode cavity and WC 80 waveguides whose TE11-mode cutoff frequency is at 8.68 GHz. The resonator produces as notch at 10.15 GHz similar to an E-plane stub in a rectangular waveguide. The resonance close to the cutoff frequency occurs due to the fact that at and close to cutoff, all sections support fundamental-mode propagation. This is also known from rectangular waveguide lowpass [Reference Matthaei, Young and Jones16] and bandpass filters [Reference Rosenberg, Amari and Bornemann18].
The basic shape of the curve in Fig. 2(a) implies that this direct connection can be exploited for either lowpass or bandstop filter applications. In the progress of connecting the TM110-mode cavity to the input/output guides through irises, Fig. 2(b) shows coupling through a large iris. Compared to Fig. 2(a), we observe that the resonance close to cutoff vanishes as the iris prevents propagation at or close to cutoff. The notch frequency in Fig. 2(a) moves downwards to 9.45 GHz due to the fact that the irises contribute to the notch effect. A new resonance appears at 9.85 GHz. Moreover, another resonance (towards 12.5 GHz and beyond) might possibly be exploited for highpass applications.
As coupling through the irises decreases (Fig. 2(c)), the TM110-mode resonance becomes more pronounced (9.62 GHz), and the notch frequency moves down to 9.2 GHz. This is the basic shape required for bandpass filters.
For a bandpass filter designer, the question as to whether the transmission zero in Fig. 2(c) can be moved from being below to above the resonance is of fundamental importance. The answer is negative if rotational symmetry is to be maintained. Since the structure in Fig. 2(c) represents a singlet [Reference Amari, Rosenberg and Bornemann13], the transmission zero is generated by a bypass coupling of the TE11 mode through the TM110-mode resonator. Note that the cavity is too short for the TE11 mode to resonate. In order to move the transmission zero to the other side, the polarity of the main resonator must be reversed [Reference Amari, Rosenberg and Bornemann13]. However, this is only possible if the feeding waveguide is moved off the center of the cavity. Figure 3 shows such a structure and its performance. Indeed, the transmission zero appears on the right side of the resonance. However, the resonating mode is no longer a pure TM110 but a combination of modes including the TM01 mode, which is excited due to the off-center connection. The cutoff frequency of the TM01 mode is
and it will propagate in the input/output ports above 11.34 GHz, thus severely limiting the operational bandwidth in addition to the loss of the polarization-preserving property of the filter.
For the design of rotationally symmetric circular waveguide filters, a coupling matrix approach according to [Reference Amari, Rosenberg and Bornemann13] can be used in principle. However, it was experienced that the repeated computations of direct and bypass couplings for each resonator required large amount of manually input data. Therefore, the respective scenarios of Fig. 2 are selected for the filter characteristic at hand, and optimized [Reference Madsen, Schaer-Jacobsen and Voldby19] to satisfy a predefined response. The computational speed of the coupled-integral equation technique (CIET) permits the filter design to be completed within normal time frames. For instance, the analysis of the filter in Fig. 4 with 500 frequency points, TE/TM1n modes up to 1000 GHz and TE/TM1n-mode basis functions up to 200 GHz requires 8 s on a dual-core 1.66 GHz PC with 2 GB of RAM.
III. RESULTS
In order to compare a bandpass filter based on TM11-mode cavities with the standard circular-iris filter displayed in Fig. 1, we use the properties displayed in Fig. 2(c) and cascade seven such resonators. Figure 4(a) shows the filter structure and its performance including comparison obtained with the µWave Wizard. Several differences are noted when comparing this response with that of Fig. 1. We obtain seven transmission zeros below the passband according to Fig. 2(c). The transition from stopband to passband between 11 and 11.25 GHz is significantly improved due to the appearance of the transmission zeros. This comes at the expense of a reduced stopband performance immediately above the passband. However, this apparent disadvantage is offset by the elimination of the second passband in Fig. 1 (15.5–18 GHz), where the filter in Fig. 4(a) maintains an attenuation of 35 dB. Note that the nearly constant attenuation level towards higher frequencies is in accordance with filter theory since all possible transmission zeros appear below the passband. Finally, the TM11-mode filter in Fig. 4(a) is considerably more compact as its length from first to last iris is only 31.4 mm (cf. Table 1) – a reduction by more than 75% compared to the filter in Fig. 1.
The practical filter designer is, of course, concerned about accuracy of fabrication and tolerances involved in the process. Therefore, Fig. 4(b) shows a tolerance analysis of this filter. One hundred trials with variations of up to ±25 µm are performed and plotted as thin gray lines in Fig. 4(b). It is demonstrated that the stopband performance displays a negligible influence on manufacturing tolerances of ±25 µm and that the passband return loss variations are well within the margins of tolerance analyses performed in [Reference Bornemann, Rosenberg, Amari and Vahldieck20]. It is thus apparent that this filter (and the ones presented later in this paper) is subject to the same dependencies for mass production as other filters with or without transmission zeros [Reference Bornemann, Rosenberg, Amari and Vahldieck20].
Another advantage of utilizing TM11-mode resonators becomes apparent when further investigating the capabilities of the singlet shown in Fig. 2(b). By increasing the length of the resonator as well as the diameters and lengths of the connected irises, two more resonances, which appear above that of the TM11 mode can be exploited. This is shown in Fig. 5, where the |S 11| minimum at 11.41 GHz is due to the TM110 resonance. The TE11-mode resonance in the iris causes the dip at 13.5 GHz, and that at 16.5 GHz is due to the TE111 resonance in the large cavity. Consequently, such a combination of axially connected circular waveguide sections is destined for quasi-highpass applications.
Fig. 5. Performance of an extended TM110-mode singlet incorporating resonances of irises and that of the TE111 mode.
Based on this investigation, a quasi-highpass configuration formed by seven TM11-mode resonators is assembled and optimized for a roll-off frequency of 11.21 GHz and return loss of 24 dB. The filter is shown in Fig. 6 (inset) together with its performance. The length including input and output irises is 63 mm and thus about twice as long as the bandpass filter depicted in Fig. 4. Note that rotational symmetry and thus polarization preservation has not been compromised.
For comparison with measurements the reader is directed to [Reference Rosenberg, Amari, Bornemann and Vahldieck21] where a comparable quasi-highpass filter in rectangular waveguide technology has been presented. The excellent agreement between measurements and CIET results in [Reference Rosenberg, Amari, Bornemann and Vahldieck21] as well as the verification of the filter response in Fig. 6 by the µWaveWizard validates the design procedure for polarization-preserving circular waveguide TM11-mode filters.
Figures 4 and 6 demonstrate bandpass and highpass performances, respectively, of rotationally symmetric circular waveguide components. The up-to-now missing filter categories are bandstop and lowpass filters. According to Fig. 2(a), sandwiching a TM110-mode resonator between two fundamental-mode circular waveguides produces a frequency notch. In order to obtain a narrow stopband and, at the same time, maintain an acceptable return loss (e.g. 20 dB) in the rest of the frequency range, a number of TM110-mode cavities with very similar resonance frequency must be cascaded. The performance of such a seven-resonator filter is shown in Fig. 7. The 20 dB return loss bandwidth extends up to 11.14 GHz and beyond 11.86 GHz; the 20 dB rejection band lies between 11.3 and 11.7 GHz with more than 60 dB attenuation between 11.46 and 11.59 GHz.
Fig. 7. Performance of a bandstop filter with seven TM110-mode resonators; dimensions cf. Table 1.
A lowpass filter is obtained by following the same design principle as for the bandstop filter. However, the individual notch sections have to be initially designed for different frequencies to cover the entire stopband towards higher frequencies. This leads to a design that, contrary to the ones shown previously, is asymmetric in axial direction, but still maintains rotational symmetry. The inset in Fig. 8 depicts a lowpass filter formed by nine TM110-mode cavities, each of which contributes to one of the nine notch frequencies shown between 14.4 and 18 GHz in Fig. 8. The level of attenuation was set to 35 dB during optimization with a return loss of at least 20 dB in the passband from 9 to 14 GHz. As in previous examples, excellent agreement with results from the µWave Wizard is demonstrated.
Fig. 8. Performance of a lowpass filter with nine TM110-mode resonators; dimensions cf. Table 1.
IV. CONCLUSION
Novel designs of TM11-mode filters in circular waveguide technology are presented. It is demonstrated that such filters possess a number of significant advantages to standard circular waveguide iris filters. TM11-mode filters are smaller and do not suffer from a second passband within the frequency range of application. They allow for design flexibility with respect to lowpass, highpass, bandpass, and bandstop operation. All filter structures exhibit cross-sectional rotational symmetry and thus preserve the polarization of any input signal. Excellent agreement with the commercially available software package µWave Wizard validates the design approach. A tolerance analysis for the bandpass filter reveals dependencies comparable to other filters of comparable bandwidths. Structural dimensions of all filters are provided and can be scaled to different frequency bands.
Jens Bornemann received the Dipl.-Ing. and the Dr.-Ing. degrees, both in electrical engineering, from the University of Bremen, Germany, in 1980 and 1984, respectively. From 1984 to 1985, he was a consulting engineer. In 1985, he joined the University of Bremen, Germany, as an assistant professor. Since April 1988, he has been with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, where he became a professor in 1992. From 1992 to 1995, he was a fellow of the British Columbia Advanced Systems Institute. In 1996, he was a visiting scientist at Spar Aerospace Limited (now MDA Space), Ste-Anne-de-Bellevue, Québec, Canada, and a visiting professor at the Microwave Department, University of Ulm, Germany. From 1997 to 2002, he was a co-director of the Center for Advanced Materials and Related Technology (CAMTEC), University of Victoria. From 1999 to 2002, he served as an associate editor of the IEEE Transactions on Microwave Theory and Techniques in the area of Microwave Modeling and CAD. In 2003, he was a visiting professor at the Laboratory for Electromagnetic Fields and Microwave Electronics, ETH Zurich, Switzerland. From 1999 to 2008, he served on the Technical Program Committee of the IEEE MTT-S International Microwave Symposium. He has coauthored Waveguide Components for Antenna Feed Systems, Theory and Design (Artech House, 1993) and has authored/coauthored more than 220 technical papers. His research activities include RF/wireless/microwave/ millimeter-wave components and systems design, and field-theory-based modeling of integrated circuits, feed networks, and antennas. Dr. Bornemann is a registered professional engineer in the Province of British Columbia, Canada. He is a fellow IEEE and a fellow of the Canadian Academy of Engineering.
Seng Yong Yu received the B.Sc. degree from the Department of Physics, East China Normal University, Shanghai, China in 1985, the M.Sc. degree from the Department of Physics, Kumamoto University, Kyushu, Japan in 1996 and the Ph.D. degree from the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, in 2008. From 1985 to 1994, he worked as an electronic engineer in the Shanghai Shipbuilding Industry Co. From 1994 to 1997, he was an assistant research engineer in the Physics Laboratory of Kumamoto University, where he was mainly engaged in the design and development of cryogenic laser devices for spectroscopy. Since January 2009, he is a postdoctoral fellow at the University of Victoria. His research interests include numerical analysis and techniques in electromagnetics, modeling of guided-wave structures, computer-aided design, and efficient optimization using frequency domain simulation methods for various applications.
