диафрагмированные волноводные фильтры / 79608e75-0a83-438a-8f4c-4fdf5d07d2a1

Abstract—Folded E-plane cross-coupled waveguide filter topologies provide high selectivity by virtue of their transmission zeros. A design method for these topologies is presented below. The method is explained in physical terms and is validated by both simulations and experiments.

Index Terms—Bandpass filters, coupling matrix, crosscoupled filter, transmission zeros.

I. INTRODUCTION

With the continuing increase in the complexity of wireless communications systems, the requirements from bandpass filters (BPFs) become ever more stringent. Classical design methods of BPFs apply to direct coupled filters, where coupling occurs between adjacent resonators only [1]. These designs suffer from the absence of transmission zeros, which results in relatively inferior slope selectivity. An improvement to this end has been accomplished by the topology that includes cross-coupling between non-adjacent (non-sequential) resonators that has the benefit of adding the necessary transmission zeros.

Cross-coupling in the waveguide filter may be implemented by folding the direct coupled filter in either the E- or H-plane and carving an aperture in the common wall thereby formed between the corresponding resonators, see [2-7] for the H-plane case. In general, the coupling coefficients may be found from the two dominant resonant frequencies of the coupled resonators [8], [9]. The dependence of coupling coefficients on resonator dimensions was found in the paper [7] by mode matching method and used in the design of cross-coupled filter folded in the H-plane. A similar method was suggested in [10] for design of a cross-coupled filter folded in the E-plane. However, detailed design procedure is still needed.

The primary objective of this work is to develop a method for designing cross-coupled E-plane folded filters in rectangular waveguide structures. This method would reduce the need for relatively costly numerical optimization of all physical dimensions (e.g., [11]). The method as presented below combines coupling and network theories with limited numerical optimization into a numerically efficient procedure, as described in Section

II. It is then demonstrated in Section III with a design example that includes simulations, fabrication and testing, showing good agreement between simulations and experiments. Conclusions are drawn in Section IV.

II.THEORY

The concept of coupling between two adjacent resonators is reviewed first, with the coupling coefficient defined as the ratio between coupled to stored electromagnetic energies. This coefficient can be related to the resonant frequencies in lumped-elements equivalent circuit analysis [8] by

correspond to the cases where electric and magnetic walls, respectively, are set at the symmetry plane between the resonators.

The coupling coefficients definitions, although derived from lumped elements theory, can be used for distributed elements as well, provided that the frequency band is

sufficiently narrow about resonance. The sign of k is determined by nature of the coupling, i.e., electric or magnetic. The resonant frequencies for the distributed elements are related to the physical structure and can be found using EM simulation or by experiment. Given the small size of the two resonator arrangement, such simulations would be quite efficient. We examine two forms of equivalent circuits that apply to electric and magnetic coupling, respectively, over this band.

Fig. 1 shows an equivalent lumped lowpass prototype for a cross-coupled filter with elements gm defined in the standard manner, following Levy [12]. The synthesis procedure suggested in [12] for the J -inverter that represents the cross-coupling is iterative, as follows:

Fig. 1. Cross-coupled lowpass filter prototype network [12].

where a are the transmission zeros of the

lowpass prototype. The first iteration is defined in [8], [12]. Having obtained the lowpass filter prototype elements, the coupling coefficients can be determined via [8]

where FBW is the fractional bandwidth of the desired

matrix of the filter. Elements not in these sub-diagonals are set as zeros.

The most intuitive configuration for realizing crosscoupling in a rectangular waveguide is attained by folding the standard direct-coupled filter into two levels, as shown

in Fig. 2 for the case N 4 . Direct iris couplers are seen between resonators 1 and 2, and between resonators 3 and 4. Resonators 2 and 3 are also coupled directly via an aperture in the common wall between the two levels. An aperture in the common wall between 1 and 4 would realize, however, cross coupling between these two

resonators, expressed by the coefficient M14 .

Fig. 2. Folded direct coupled filter.

The positions and dimensions of the apertures now need to be determined. To this end, we examine each pair of coupled resonators separately. Recalling (1), we use eigenmode solution type in the EM simulator (HFSS®) to find the resonant frequencies of the structure in Fig. 3. According to [8], the two lowest resonant frequencies of

two identical coupled resonators would be fe and fm in

(1).

representing directand cross-coupling, respectively, are elements in the appropriate sub-diagonals in the coupling

Fig. 3. Two coupled resonators with an aperture in the common wall.

Coupling coefficients can have, in general, either positive or negative sign. Cross-coupling coefficients will have an opposite sign relative to the direct coupling coefficients [10]. Since both types of coupling need to be realized, the designer should have control over the nature of the coupling (electric or magnetic). It is now shown that this nature of coupling is determined by the lateral ( z - directed) position of the aperture in the common wall. Two options for this position are considered: (1) near the side wall and (2) at the center of the common wall. Analysis is simplified by symmetry that allows for the reduction the problem into two smaller sub-problems with magnetic and electric wall boundaries, respectively, at the symmetry plane.

In the course of the EM simulations we seek the dependence of the resonant frequency on aperture length for a rectangular aperture with a fixed aperture width w =1.5 mm in Option (1) and a square aperture in Option (2), for which the reduced configurations are shown in Fig. 4(a) and 4(b) respectively.

Fig. 4. Aperture configurations in the finite thickness wall between the two levels. Cavity dimensions are 10.7 mm along the x-axis, 4.3 mm along the y-axis and 8.64 mm along the z-axis. The thickness of the common wall is t=1.5 mm. The resonant frequency of a resonator with

no aperture for the dominant TE101 mode is 22.3 GHz.

(a)Option (1): near-side-wall coupling aperture.

(b)Option (2): square coupling aperture at the center of the common wall.

The result for Option (1) is seen in Fig. 5(a). One can see that fe fm therefore the coupling coefficient is

positive, and the coupling mechanism for this Option (1) is magnetic, as has been shown in [8]. Results for Option

(2) are seen in the graphs in Fig. 5(b). In this case, fe fm . We thus have a negative coupling coefficient,

therefore an aperture at the center of the common wall produces electric coupling.

Fig. 5. Resonant frequency dependence on the aperture length of

resonator in (a) Option (1) (Fig. 4(a)) and (b) Option (2) (Fig.4(b)).

The aforementioned coupling mechanism can be also understood by noting that the electric (magnetic) field of

the dominant TE101 mode has a maximum at the center of the resonator (near the side wall). Therefore, the respective coupling mechanisms are of electric and magnetic nature, respectively. This effect also enables a rapid determination the dimensions of the coupling apertures, since the optimization needs now to be performed only for a single parameter at a time.

III. CASE STUDY

A cross-coupled filters has been designed in a WR-187 waveguide (whose dimensions are a=47.55 mm, b=22.15 mm, its passband is 4.95 - 5.05 GHz and the center frequency is 5 GHz) according to the method described above, and then fabricated and tested. The required

characteristics are set forth as follows: passband return loss=30 dB, filter order is N=4, two transmission zeros are located at 4.828 GHz and 5.178 GHz. In the first step of the design, the lowpass prototype filter elements with the following coupling matrix are obtained:

It becomes evident from the structure of the matrix that the filter contains both direct and cross coupling

mechanisms, where M12 , M23 and M34 represent direct couplings while M14 represents cross-coupling.

The next step is the determination of the physical parameters in the waveguide embodiment. For the direct coupled resonators, iris widths and resonators lengths are obtained with the mode matching method [13] as follows: irises #1 and #4 have a width of 21.63 mm and the width of irises #2 and #3 is 12.96 mm. Resonators #1 and #4 length is 33.20 mm while the length of resonators #2 and #3 is 36.66 mm.

Resonators coupled through an aperture in the common wall require a different approach. Since the coupling coefficient between resonators 2 and 3 has a positive sign, the aperture in the common wall should be placed near the

sidewall. In contrast, the coupling coefficient M14 has a

negative sign, therefore the aperture should be positioned at the center of the common wall. Using structures such as the ones in Fig. 3 (with the appropriate aperture position) and Eq. (1) the aperture dimensions are optimized for the

desired values of M14 and M 23 . The length and width of

the aperture between resonators 2 and 3 are found to be 16.54 mm and 14.89 mm, respectively, and the size of the square aperture facilitating the cross coupling is 10.10 mm.

As a final design step, all different elements (irises, resonators) are combined into the full filter configuration as shown in Fig. 6 (including coax to waveguide transitions). Simulation of the entire filter gives the result shown in Fig. 7. This simulation coincides very well with the design as predicted.

Using all the above parameters, the physical filter was fabricated. It is composed of three parts (Fig. 8): two identical parts with iris-divided cavities and coaxial probes and a plate with holes realizing coupling between the first two parts.

Measurements were performed using Agilent Technology Vector Network Analyzer E8364A. In general, the

measured, simulated and predicted results are in good agreement as can be observed from Fig. 7. Various tolerance analyses showed that it is safe to assume that the main cause for the deviation of the manufactured filter result from the simulated result is the inaccuracies in the realization of the probe rather than the body of the filter.

Fig. 6. Full filter structure.

Fig. 7. Simulation and test results of final filter structure.

Fig. 8. Three parts of the fabricated filter.

IV. CONCLUSIONS

In this work, a procedure has been proposed and demonstrated for designing a folded E-plane filter with transmission zeros while using a direct method with very few optimization steps. A coupling matrix representing the relations between the resonators comprising the filter was constructed. In order to add transmission zeros to the frequency response of the filter, a cross-coupling mechanism between non-adjacent resonators had to be realized. For establishing the capability to design apertures used in the cross-coupling mechanism, an investigation of coupled resonators was carried out. In particular, the effect of the coupling aperture position within the waveguide wall on the nature of the coupling (electric or magnetic), has led to conclusions which enabled the constructing of the entire cross-coupled filter with a minimalistic use of iterative optimization. A filter was designed and fabricated and the measured results proved to be in a good consistence with the predicted performance.

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