диафрагмированные волноводные фильтры / 0b8b7b1f-cbd8-469f-acbf-bdbb88652c09

1 Introduction

Periodically loaded E-plane waveguides have in the past been employed for the realisation of bandpass filters with miniaturised resonators [1, 2], integrated lowpass–bandpass filters with improved stopband performance [2, 3] as well as dispersion free waveguides [4, 5]. Miniaturisation of resonators and dispersion linearisation is achieved by virtue of the slow-wave effect, according to which the phase progress per physical unit length is increased compared to that in a homogeneous transmission line. The miniaturisation factor of a periodically loaded transmission line typically depends on the periodicity as well as the internal geometry of each unit cell, as was shown for the case of periodically loaded E-plane waveguides in [3]. Slow-wave regions in periodic transmission lines are typically separated by electromagnetic band gaps, which are exploited to produce the lowpass response in the case of E- plane filters [2, 3]. In [3] it was demonstrated that transmission zeros emerge in the bandgap due to the resonant nature of the periodic loading.

Despite such favourable characteristics, propagation in periodic structures is associated with increased thermal losses. This deteriorates the quality factor of miniaturised periodically loaded resonators, which reflects in higher insertion loss and poorer selectivity in filters of higher order. The trade-off between miniaturisation and quality factor is general in microwave resonators beyond periodically loaded ones (e.g. dielectric filling, step impedance resonators etc.). This in turn poses the requirement for a quantitative assessment between the miniaturisation and the quality factors. For the case of periodically loaded E-plane resonators, such an assessment has yet to appear.

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In addition, the design of traditional E-plane filters is based on K-inverter prototypes [6], which typically yield sufficiently good estimation for the initial dimensions of the filter for an optimisation procedure to rapidly converge [7]. The extraction of the equivalent K-inverter can be based on the transmission coefficient across the evanescent coupling septa [8], which also yields an electrical length that has to be subtracted from the otherwise half wavelength resonator. Although this synthesis procedure is accurate for filters with homogeneous resonators [9], it fails in the case of filters with periodically loaded resonators. Although a first estimation of the half wavelength length for a periodically loaded waveguide can be obtained by the dispersion relation [3], the coupling between the resonant modes is not readily obtained as the transmission coefficient of the fundamental TE mode across the coupling septum. In addition, the length in the case of periodically loaded resonators is discretised and therefore subtraction of the electrical length predicted by [8], which in general is not commensurate with the periodicity, is not straightforward. As a result, periodic E-plane filters of higher order have yet to appear in the literature despite the practical interest that they have attracted [10–12].

In this paper we therefore initially present a thorough investigation of the quality factor for periodically loaded resonators with different miniaturisation factors. Using numerical and experimental results, we draw design curves that quantify the trade-off between miniaturisation and unloaded quality factor. Subsequently, we outline a design procedure for periodically loaded E-plane filters of higher order and demonstrate both numerically and experimentally an example of a fifth-order prototype. The insertion loss of the filter prototype is commensurate with that predicted by

IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 7, pp. 818–822 doi: 10.1049/iet-map.2010.0396

the unloaded quality factor of its resonators, further confirming the validity of this study.

2 Unloaded quality factor

The quality factor of a resonant system is in general defined as the average energy stored per period of oscillation over the energy loss per second [13]. For a resonator in isolation, energy loss comes solely through thermal dissipation and this case corresponds to the unloaded quality factor definition, QU [13]. A resonator that is excited externally is invariably coupled to other circuitry. As a result some power exchange with the external circuitry occurs and the overall quality factor, or loaded Q, QL, is in practise reduced compared to QU. For reasonably narrowband resonators, it can be shown that the loaded and unloaded quality factors, QL and QU, are given by

where v0 and Dv3dB are the resonant frequency and the 3 dB bandwidth of the resonator, respectively. S21(v0) is the

insertion loss at v0 (in natural units).

To compare the quality factors for periodically loaded resonators, we have designed a series of four periodically loaded E-plane resonators and one uniform E-plane resonator, all resonant at the same frequency. The realisation is based on all metal insert split-block housing E-plane topology [1–7], which is compatible with traditional printing techniques. A schematic layout of the resonator geometry is shown in Fig. 1. The ridge waveguide loads behave as series LC loads [3]. The operation of the periodic E-plane resonator is typically below the resonant frequency of the LC loads. By reducing the gap, s, the guided wavelength shrinks so that if all other dimensions remain constant, the resonant frequency would drop. To maintain the resonant frequency at a fixed value, for every lower value of the gap, s, we also reduce the

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separation Lr. Commercial finite-element method (HFSS) is used for the numerical optimisation of the pairs (s, Lr) in order to produce five resonators all at the same central frequency. A homogeneous resonator (where s ¼ b) was designed also using HFSS. In all cases the input and output coupling septa were fixed at r ¼ 5 mm. The thickness of the metal insert is assumed 100 mm for all resonators. The transverse dimensions of all six resonators are given in Table 1.

The simulated transmission coefficient for all resonators of Table 1 is shown in Fig. 2. The resonant frequency is approximately 8.82 GHz in all cases. In the simulations, the metal insert is assumed to be made of copper and the waveguide housing of aluminium. As shown, the 3 dB fractional bandwidth and the minimum insertion loss increases for more miniaturised designs. To quantify the increase of the fractional bandwidth, Fig. 3a shows the loaded quality factor, QL, as calculated from the responses of Fig. 2 and using (1). Fig. 3b shows the estimated unloaded quality factor as calculated from (2). In both graphs the quality factors are plotted against the length of the resonator, L, normalised to the length of the homogeneous resonator with s ¼ b. As shown in Fig. 3b, the unloaded quality factor reduces approximately linearly with the normalised resonator length. The loaded quality factor, QL, depicted in Fig. 3a is about one order of magnitude reduced compared to the unloaded quality factor, QU. This is attributed to the power exchange between the resonator and the input and output port, which is determined by the reflectivity of the coupling discontinuity. Longer coupling septa, r, produces more weakly coupled resonators that correspond to higher loaded quality factors.

To validate the above, the designed resonators have been fabricated and experimentally tested. A routing procedure was employed for the etching of a copper foil of thickness 100 mm. A split-block waveguide housing milled in aluminium was employed. A photograph of the prototypes is shown in Fig. 4. The resonators were experimentally

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Fig. 3 Simulated and measured

a Loaded

b Unloaded quality factors for the resonators of Table 1

Fig. 4 Tested prototypes

measured and the experimentally determined loaded and unloaded quality factors are superimposed in Fig. 3. Good agreement is observed between the simulated and measured results. This suggests that despite the rough edges on the copper insert as a result of the routing procedure, the quality factor is not significantly deteriorated. Fig. 3b provides a useful guide for the designer, in order to predict the insertion loss of a higher order filter based on the degree of miniaturisation.

3 Filter application

In this section, we demonstrate the synthesis of higher-order periodic E-plane bandpass filters by means of a fifth-order example with Chebyshev response. Based on the study above, the design is based on resonators with gap, s, of 3 mm. For an application requiring resonators with an unloaded Q of above 1600, the prototype provides about 70% reduction in the resonator length when compared with the conventional resonator operating at the same frequency. The lower and upper cutoff frequencies were chosen as 8.5 and 9 GHz, respectively, and the passband ripple 0.4 dB, so

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that the maximum return loss in the passband is approximately 10 dB. Rather than employing a K-inverter prototype, we employ a coupling matrix prototype for the filter. From the coupling matrix prototype, we can extract the split of the even and odd resonance for each pair of resonators employing a weakly excited pair of resonators coupled according to the prototype, which is known to be related to the coupling coefficient [14]. For the coupling coefficient between the first and second resonators, k12 ¼ 0.04 according to the prototype, the weakly excited transmission coefficient that yields the even and odd resonance is shown in Fig. 5.

We subsequently optimise the periodic resonator and coupling septum dimensions in order to match the response of Fig. 5. To reduce the free variables, we set Lw ¼ Lr. The procedure is repeated for the coupling septum between resonators 2 and 3 in order to realise the coupling coefficient k23 ¼ 0.033 required by the prototype. Following this procedure and considering the symmetry of the physical structure as seen from either of the two ports, initial values for all filter dimensions are obtained except the first and last septum. Matching the external quality factor is employed to obtain the remaining parameter. Once the initial dimensions for the filter are obtained, an optimisation procedure is employed to produce an equiripple response.

During the final optimisation, it is helpful to consider the fact that the lower coupling coefficient associated with resonators 2 and 3 compared to that associated with resonators 1 and 2 is also associated with a greater phase negative electrical length that should be absorbed within the half-wavelength resonators [8]. This is typically the case in traditional E-plane filters and as a result resonators more distant from the ports tend to be electrically shorter. For filters with homogeneous resonators, electrically shorter resonators correspond to physically shorter lengths. However, rather than reducing the resonators’ lengths in order to absorb the negative electrical lengths, which is relatively inconvenient for periodically loaded resonators with discrete periodicity, reduction of the electrical length can be achieved by increasing the gap s of the unit cell [3]. This is known to reduce the effective electrical length of the resonator [3]. This procedure has been found to converge rapidly to good initial values for the filter.

Using the technique outlined above, the final filter dimensions have been produced and are given in Table 2. The measured and simulated responses are shown in Fig. 6. A photograph of the prototype is shown in Fig. 7. Good

Fig. 5 Even and odd resonance obtained from weakly excited pair of resonators as obtained from the filter prototype for the coupling between the first and second resonator (k12 ¼ 0.04) and the optimised dimensions of the periodic E-plane filter

IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 7, pp. 818–822 doi: 10.1049/iet-map.2010.0396

Table 2 Dimensions of designed filter with uniform and periodically loaded resonators

For definition of the variables, refer to Fig. 7a

agreement between the measured and simulated response is observed. A frequency discrepancy of about 60 MHz (Fig. 6b), corresponding to less than 1%, observed between the simulation and measurement is mainly attributed to fabrication tolerances due to the routing procedure. For benchmarking purposes, a traditional E-plane filter with hollow E-plane resonators has also been designed. The relevant dimensions are also shown in Table 2, while the simulated response is superimposed in Fig. 6 with dashed grey line.

The overall size of the filter with the periodic resonators is about 75% that of the traditional filter, while the reduction in the size of the resonator is about 72% compared to the uniform one. This difference is attributed to the requirement for increased coupling septa. The measured passband loss at the second reflection null of the experimental prototype is about 0.5 dB, which can be further reduced with a selection

Fig. 6 Simulated and measured response of a 5th order filter prototype with periodically loaded resonators. For comparison, the simulated response for a 5th order filter with uniform resonators is also shown

a S12 (dB)

b S11 (dB)

IET Microw. Antennas Propag., 2011, Vol. 5, Iss. 7, pp. 818–822 doi: 10.1049/iet-map.2010.0396

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Fig. 7 5th order E-plane filters

a Schematic representation of E-plane filters with uniform and periodic resonators. Dimensions as in Table 1

b Photograph of the fabricated prototype

of higher conductivity materials. The study in the previous section predicts that for a resonator reduced by about 70%, the unloaded quality factor would be approximately 1600. Accordingly, the insertion loss of the fifth-order filter is estimated to be around 0.4 dB. The consistent agreement between the two values proves the validity of this study. Significantly, an improvement in the out-of-band rejection is observed. This improvement is partly attributed to the longer evanescent mode septa required for the realisation of the impedance inverters. The presented prototype is compatible with the low-cost fabrication techniques of E-plane filters.

4 Conclusion

The quality factor of periodically loaded resonators has been theoretically and experimentally assessed for increasing miniaturisation. Periodically loaded E-plane filters were used as an example. It was demonstrated that the unloaded quality factor reduces linearly with the miniaturisation factor. A fifth-order Chebyshev filter was designed, fabricated and tested. The prototype filter had about 0.5 dB of insertion loss, 75% reduced size and improved stopband performance. The predicted insertion loss from the unloaded Q factor was in good agreement with the measured loss of the filter validating the presented study.

5References

1 Goussetis, G., Budimir, D.: ‘Compact ridge waveguide filter with improved stopband performance’. IEEE MTT-S Int. Microwave Symp., Philadelphia, USA, June 2003, pp. 953–956

2Goussetis, G., Budmir, D.: ‘Novell periodical loaded e plane filter’,

IEEE Microw. Wirel. Comp. Lett., 2003, 13, (6), pp. 193–195

3Goussetis, G., Feresidis, A.P., Kosmas, P.: ‘Efficient analysis, design and filter applications of EBG waveguide with periodic resonant

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