диафрагмированные волноводные фильтры / 33ef695a-d0bd-430e-ac0d-2458a80ef005

Abstract—This paper presents the procedure for extracting the generalized coupling coefficients between resonating and nonresonating nodes of extracted pole sections. The extraction allows obtaining key design parameters for inline extracted pole filters directly from results of full-wave simulations. A third-order waveguide -plane bandpass filters with -wavelength resonators is designed to prove feasibility of the proposed concept. The bandpass filter with a generalized Chebyshev response has 300-MHz bandwidth at 9.45 GHz and reveals wide stopband due to three transmission zeros arranged at predicted frequencies. Performance of the fabricated filter is evaluated through measurements, the experimental results obtained show very good agreement with simulations.

Index Terms—-plane filters, extracted pole filters, nonresonating nodes, pseudoelliptic filters, waveguide filters.

I. INTRODUCTION

WAVEGUIDE bandpass filters are generally preferred and are used extensively for transmitter/receiver front-end applications when low-loss and high power handling is required. Filter structures with sophisticated transfer functions, such as elliptic and pseudoelliptic filters, are used that enable realization of very sharp rejection skirts and are also relatively compact. They are based on well-established techniques, such as the use of cross couplings between nonadjacent resonators [1], use of the extracted pole cavities [2], and use of

the nonresonating node (NRN) concept [3].

Inline extracted pole filters with NRNs have become very popular due to several of their benefits: they are able to produce maximum number of transmission zeros, equal to order of the filter, without direct coupling between source and load; the filters exhibit the property of modularity, since positions of transmission zeros can be controlled independently in this type of filters by adjusting resonant frequencies of the resonators. An inline extracted pole filter consists of several direct coupled extracted pole sections, and for applications where maximum of transmission zeros is not required, resonators (see Fig. 1).

A model of these inline filters is known to reveal scaling properties, which can be conveniently described by generalized cou-

Manuscript received February 19, 2011; revised August 22, 2011; accepted August 29, 2011. Date of publication October 10, 2011; date of current version December 14, 2011.

The authors are with the Wireless Communication Research Group, School of Electronics and Computer Science, University of Westminster, London W1W 6UW, U.K. (e-mail: d.budimir@wmin.ac.uk).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2011.2168967

Fig. 1. Coupling scheme of an inline extracted pole filter with NRNs.

pling coefficients introduced by Macchiarella [4]. Generalized coefficients must not be confused with classical coupling coefficients that cannot be used for characterization of couplings where NRNs are involved. An approach to use the generalized coupling coefficients for implementation of inline extracted pole filters in waveguide technology have been demonstrated in [5].

Generally, conventional waveguide technology is the most popular to accommodate the inline extracted pole filters [6]–[8]; microstrip [9], combline [10], hybrid microstrip, and substrate integrated waveguide [11] realizations have also been reported. However, despite the numerous advantages offered by waveguide cavities, fabrication of the structures is complex, time consuming, and quite expensive. An alternative approach is available when -plane inserts are used within the existing waveguide housings. -plane technology offers a very convenient way of realizing waveguide filters with uniform rectangular cross section [12].

This paper therefore presents an -plane waveguide solution for a third-order inline extracted pole filter of the generalized Chebyshev type with three transmission zeros forming a wide stopband. The filter is implemented by extraction of generalized coupling coefficients directly from results of full-wave simulations using expressions derived from analysis of single and interacting extracted pole sections. The new formula are particularly useful for analysis of extracted pole sections with nonstandard discontinuities involved where traditional techniques lead to bulky calculations. Feasibility of the approach is demonstrated through experiment.

II. EXTRACTION OF GENERALIZED COUPLING COEFFICIENTS

In Fig. 2, an equivalent schematic circuit of single doubly loaded extracted pole section is presented, where and

Here, is a low-pass prototype frequency variable that can be obtained from real frequency by standard bandpass to lowpass transformation

(3)

where is the center frequency of the filter to be designed and is its fractional bandwidth.

A. Internal Coupling and Susceptance of Resonator

Typical frequency responses of the doubly loaded extracted pole section prototypes are shown in Fig. 3. The prototype circuit exhibits a transmission zero at and a reflection zero at

(4)

(5)

It can be noticed from (1) and (2) that the circuit in Fig. 2 can be completely described by three parameters: two ratios , and frequency invariant susceptance . In terms of the generalized coupling coefficients proposed in [4], the first ratio has a meaning of the squared coupling coefficient between resonating and nonresonating nodes divided by the normalized bandwidth, while the second is the reverse external quality factor

(6)

Fig. 3. Frequency responses of doubly loaded extracted pole sections: (a) for and (b) for .

(7)

Taking into account (4) and (5), and can be extracted from measured or simulated data using the following expressions:

(8)

(9)

It can be concluded from (6) and (9) that susceptance of the nonresonating node should be negative to arrange a transmission zero above the filter’s passband, and positive to place it below the passband (see Fig. 3). At the same time, (9) shows that space between two eigenmodes is determined by the absolute value of the generalized coupling coefficient . Finding solutions that allow obtaining higher , stopband performance of the designed filters can be improved by placing transmission zeros farther from the passband.

III. EXTERNAL QUALITY FACTOR

Similarly to the conventional resonators case [13], the external quality factor can be extracted from evaluation of the 3-dB

Fig. 4. Extraction of generalized external quality factor.

which yields

(10)

(11)

It is seen from (6) and (7) that and always have the same sign determined by the value of . Consequently, plus should be chosen in (11) if and minus otherwise.

Extraction of the generalized quality factor from simulated or measured frequency response of an extracted pole section with a transmission zero in the upper stopand using its 3-dB bandwidth is illustrated in Fig. 4.

Equations (8), (9), and (11) form a set of expressions for complete characterization of a single extracted pole section with arbitrary implementation. This is particularly important for structures where values of inverters or susceptances cannot be calculated directly or these calculations are bulky. Example of such a structure will be presented below.

A. Coupling Between Adjacent Asynchronously Tuned Sections

Assume that two arbitrary extracted pole sections are connected through an admittance inverter of value , as shown in Fig. 5. For analysis, the inverter is replaced with its equivalent II-network, and the condition for natural resonance of the circuit can be written as

(12)

where and are the input admittances looking at the right and the left of reference plane – of Fig. 5. The input admittances can be expressed as follows:

(13)

(14)

Fig. 5. Schematic representation of two adjacent extracted pole sections connected through admittance inverter.

where

(15)

Let us denote the positions of poles and zeros revealed by both single sections examined individually as , , , and , respectively. Assuming that external couplings of the structure are very weak, i.e., , and applying substitutions from (4) and (5) in (15), after several manipulations, we obtain the following:

(16)

where

(17)

is defined as squared generalized coupling coefficient between two asynchronously tuned extracted pole sections. Roots and of (16) correspond to two self-resonances of the system in terms of normalized frequency. Solving (16) and combining roots, we obtain the expression for experimental evaluation of the squared generalized coupling coefficient

(18)

Locations of the poles and transmission zeros used for extracting the generalized coupling coefficient between two asynchronously tuned extracted pole sections are shown in Fig. 6. Note that the procedure requires very low external couplings at input and output of the structure and preliminary evaluation of poles and of each individual extracted pole section.

B. Coupling Between Extracted Pole Section and Resonator

A schematic used for extraction of a generalized coupling coefficient between an arbitrary resonator with resonance frequency at normalized frequency and an extracted pole section connected through admittance inverter is shown in Fig. 7.

The circuit can be solved using the technique applied in Section III-A, and (12)–(14) are correct for this case, as

Fig. 6. Extraction of generalized coupling coefficient between adjacent asynchronously tuned extracted pole sections.

Fig. 7. Schematic representation of extracted pole section connected with resonator through admittance inverter.

well as (15) being correct for . However, here we use

in (13) to represent the resonating node. Making the assumption that external couplings are weak, the

initial eigenmode equation can be converted into the following:

(19)

where

(20)

is the squared generalized coupling coefficient between the resonating node and nonresonating node of the extracted pole section divided by the normalized bandwidth.

Expression for extraction of the generalized coupling coefficient can be obtained using two roots and of (19)

(21)

The extraction procedure of the generalized coupling coefficient between the extracted pole section and resonator using positions of poles and transmission zeros is illustrated in Fig. 8.

Fig. 8. Extraction of generalized coupling coefficient between extracted pole section and resonator.

Fig. 9. Low-pass prototype circuit for the third-order filter.

Fig. 10. View of the -plane insert for implementation of the entire filter.

IV. DESIGN EXAMPLE

In order to demonstrate how inline extracted pole filters can be designed by extraction of generalized coupling coefficients, we synthesize a third-order bandpass filter with three transmission zeros implemented as an -plane insert within conventional rectangular waveguide. The filter is designed to satisfy the following specifications:

•center frequency: 9.45 GHz;

•ripple passband: 9.3–9.6 GHz;

•return loss: 20 dB;

•transmission zeros: 10.6, 11.6, 12.7 GHz.

A. Synthesis

To synthesize the filter prototype meeting the above requirements, the direct synthesis technique for inline filters with nonresonating nodes presented in [10] has been applied, which

yields the following expressions for the -parameters of the prototype:

(22)

(23)

(24)

(25)

After extraction of values of admittance inverters and frequency invariant susceptances, we get the following cir-

. Thereafter, the generalized coupling coefficients to be designed can be calculated from (6), (7), and (17). Note that we use squared generalized coupling coefficients to characterize the circuit for convenience and in order to simplify and minimize mathematical operations. Their values obtained from

B. Implementation

Generally, the calculated filter prototype can be designed in a variety of technologies. In this case, we decided to realize the filter in a conventional rectangular waveguide using an -plane

all-metal insert placed between two halves of a housing, as this approach has proven its feasibility in the design of direct coupled filters in past years. Let us consider the entire insert with three direct coupled extracted pole sections, shown in Fig. 10. Each section is comprised of a rectangular fin between two septa, and it can be represented by means of a low-pass prototype with the schematic circuit given in Fig. 2, where the fin is considered as a quarter-wavelength-type resonator grounded to the upper wall of the waveguide while having the opposite end open. The NRN between the two septa acts as inductive susceptance because it can be considered as an -plane resonator operating below resonance frequency. Interaction between the resonator and the NRN is complex for analysis in this structure since the NRN section is inhomogeneous, and thus it is more convenient to use (4), (9), (11), and (18) for extraction of generalized coupling coefficients.

Extraction is carried out according to the following procedure. First, dependence of the position of the transmission zero over the resonator’s height is examined for the single symmetrical section with arbitrary septa and length of the NRN.

are then fixed for all three sections, and are adjusted until the projected generalized coupling coefficient is reached. After this, external factors are adjusted for input/output sections by changing lengths of the input/output septa. Finally, coupling coefficients between adjacent sections are examined for the corresponding couples by changing lengths of septa between them. Results of extraction for the designed filter are shown in Fig. 11.

Fig. 12. Comparison of initial simulation results and the ideal frequency response of the filter.

Fig. 13. Simulated and experimental responses of the fabricated filter.

C. Simulation and Experimental Results

From the dependencies plotted as a result of extraction, the initial dimensions of the filter have been obtained. Their values are summarized in Table I.

Fig. 12 compares the initial frequency response of the filter, obtained as a result of the full-wave simulation of the structure

with dimensions extracted from plots in Fig. 11, with the ideal frequency response defined by (22)–(25). It is evident that the curves show significant discrepancy. More than 20-dB worse rejection of the simulated filter in the upper stopband can be explained by imperfect nonresonating nodes, which, in fact, contain frequency-dependent susceptances. Transmission zeros of the filter are shifted toward the passband due to capacitive interaction of the resonators with septa, which results in decrease of their resonant frequency. It is also evident that return loss in the passband should be improved by tuning.

The filter with initial dimensions was then optimized with respect to the manufacturing tolerances in the full-wave simulator CST Microwave Studio. From comparison of the initial and optimized dimensions of the designed filter that are presented in Table I, it is evident that the design procedure based upon the proposed extraction of generalized coupling coefficients provides good initial approximation of dimensions of the -plane insert. However, 19% and 41% deviations between the initial and final values of and do not follow the pattern shown by the other dimensions. This happened due to composition of the unbalanced cost function that considered positions of transmission zeros as the most important goal. Consequently, the optimizer shifted the transmission zero at 12.5 GHz to the specified 12.7 GHz by decreasing parasitic coupling between the septa and resonator, which resulted in reduction of the septa’ lengths and yielded slightly wider passband of the filter than specified.

In order to validate the analysis and confirm the introduced design approach, the structure has been realized using standard -plane waveguide technology. The insert of the shape shown in Fig. 10 has been cut out of the copper foil with a thickness of mm and placed between two halves of housing of rectangular waveguide (WG-90, cross section: 22.86 10.16 mm) made of brass.

-parameters of the filter have been measured using the Agilent E8361A vector network analyzer. Fig. 13 shows the simulated and measured responses of the fabricated filter. The computed response reveals all three reflection and three transmission zeros, while the experimental curve does not depict one of poles,

clearly showing all three transmission zeros at frequencies very close to expected. The measured insertion loss at center frequency is 1.2 dB. Required return loss in the passband of better than 20 dB is successfully achieved. Overall agreement of computed and experimental responses is very good, taking into account fabrication and measurement errors. Photography of the fabricated structure is shown in Fig. 14.

V. CONCLUSION

In this paper, the extraction technique for generalized coupling coefficients between resonating and nonresonating nodes of extracted pole sections has been proposed. A third-order generalized Chebyshev bandpass filter with a maximum number of transmission zeros has been designed using the proposed design procedure and implemented in a rectangular waveguide. The structure is compatible with the split block housing and metal insert -plane technology, thus maintaining the low-cost and mass-producible characteristics. Experimental and simulation results have been presented to validate the argument.

ACKNOWLEDGMENT

[8]M. Fahmi, J. A. Ruiz-Cruz, R. R. Mansur, and K. A. Zaki, “Compact ridge waveguide filters using non-resonanting nodes,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 1137–1340.

[9]G. Macchiarella and S. Tamiazzo, “Synthesis of microwave diplexers using fully canonical microstrip filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2009, pp. 721–724.

[10]S. Amari and G. Macchiarella, “Synthesis of inline filters with arbitrary placed attenuation poles by using nonresonating nodes,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 10, pp. 3075–3081, Oct. 2005.

[11]B. Potelon, C. Quendo, J.-F. Favennec, E. Ruis, S. Verdeyme, and C. Person, “Design of bandpass filter based on hybrid planar waveguide resonator,” IEEE Trans. Microw. Theory Tech., vol. 58, no. 3, pp. 635–644, Mar. 2010.

[12]G. Goussetis and D. Budimir, “Compact ridged waveguide filters with improved stopband performance,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 953–956.

[13]J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001.

Oleksandr Glubokov (S’10–M’11) received the B.Eng. and M.Sc. degrees in telecommunications from the National Technical University of Ukraine, Kiev, Ukraine, in 2005 and 2007 and is currently working toward the Ph.D. degree at the University of Westminster, London, U.K.

Since October 2007, he has been with the Wireless Communications Research Group, School of Electronics and Computer Science, University of Westminster. His research interests include design and analysis of miniaturized microwave and millimeter-wave waveguide filters.

The authors would like to thank the anonymous reviewers for their helpful comments.

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D. Budimir (M’93–SM’02) received the Dipl. Ing. and M.Sc. degrees in electronic engineering from the University of Belgrade, Belgrade, Serbia, in 1981 and 1985, respectively, and the Ph.D. degree in electronic and electrical engineering from The University of Leeds, Leeds, U.K., in 1995.

In March 1994, he joined the Department of Electronic and Electrical Engineering, Kings College London, University of London. Since January 1997, he has been with the School of Electronics and Computer Science, University of Westminster,

London, U.K., where is currently a Reader of wireless communications and leads the Wireless Communications Research Group. He has authored or coauthored over 240 journal and conference papers in the field of RF, microwave, and millimeter-wave wireless systems and technologies. He authored

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pp.1341–1344.

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[7]J. R. Montejo-Garai, J. A. Ruiz-Cruz, J. M. Rebollar, M. J. PadillaCruz, A. Onoro-Navarro, and I. Hidalgo-Carpintero, “Synthesis and design of in-line -order filters with real transmission zeros by means of extracted poles implemented in low-cost rectangular -plane waveguide,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 5, pp. 1636–1642, May 2005.

(Artech House, 2000), and a chapter in Encyclopaedia of RF and Microwave Engineering (Wiley, 2005). He is a referee for IET Electronics Letters, IET Microwaves, Antennas, and Propagation, and the International Journal of RF and Microwave Computer-Aided Engineering. His research interests include analysis and design of hybrid and monolithic microwave integrated circuits, the design of amplifiers, filters and multiplexing networks for RF, microwave and millimeter-wave applications, and RF and microwave wireless system design.

Dr Budimir is a member of the Engineering and Physical Sciences Research Council (EPSRC) Peer Review College. He is a referee for the IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, the IEEE ANTENNAS AND WIRELESS PROPAGATION LETTERS, the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, and the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—PART II: EXPRESS BRIEFS.