Corrugated parallel-coupled line bandpass filters
with multispurious suppression
J.-T. Kuo and M.-H. Wu
Abstract: Corrugated coupled stages are devised to design bandpass filters with multispurious suppression. Through two-port analysis, the suppression is found critically dependent on equality of the even and odd mode phase velocities at even harmonics of design frequency fo. Quarter-wave (l/4) corrugated stages are tuned to allocate inherent transmission zeros at both 2foand 4fo, and 6fo for some structures, so that the circuit is free of spurious at these frequencies. Coupled stages with proper coupling lengths are arranged to cancel unwanted peaks at 3fo, 6fo and 7fo, and tapped input/output scheme is employed to tackle those at 3fo and 5fo. Measured data of designed filters show rejection levels better than 30 dB in upper stopband. Three circuits are fabricated and measured to demonstrate the idea.
1 Introduction
Parallel-coupled line filters have been widely employed in the radio frequency (RF) front ends of microwave/wireless communication systems. The circuit is popular since it is reliable, suitable for mass repetition and easy to design. Synthesis formulas have been well documented for the even and odd mode characteristic impedances, and hence linewidth and gap size, of each stage [1]. For microstrip realisation of a quarter-wave (l/4) stage, since the coupled lines have unequal modal phase velocities, spurious responses arise at even multiples of the design frequency, or 2mfowith m being a positive integer. The first spurious at 2fo is of the most concern since it degrades the passband sym-metry and is responsible for blocking extension of upper rejection band. This problem can be resolved with many effective approaches, including the capacitive compen-sation [2], the wiggly-line [3], the corrugated structures [4, 5], the stacked configuration [6], the over-coupled stages [7, 8], the suspended substrate [9], the uniform dielectric overlay [10], the periodically non-uniform coupled-line[11], the perturbed ground planes[12, 13].
An ideal bandpass filter has no spurious response in the upper stopband. Even though the spurious at 2fo can be effectively suppressed by the above-mentioned methods, unwanted passbands at 3fo, 5fo, 7fo and so on also arise because of the distributed nature of the circuit. The stepped-impedance resonators[14, 15] can be used to syn-thesise bandpass filters with a wide upper stopband. For the multi-layered resonators in [14], a wide aperture in ground plane is etched underneath the high impedance line to lift up the impedance ratio. The first high unwanted jS21jpeak occurs at 8.5fo. The circuits in[15], on the other hand, have a fully planar structure. In addition to designing resonator geometry to push the fourth resonance to as high frequency as possible, the tapped input/output structures
are incorporated to suppress the two leading resonances, so that the upper stopband is extended up to beyond 8fo.
Recently, multispurious suppression has been a hot research topic. The wiggly-line in[16]is a significant exten-sion of[3]. For a seventh-order filter, measured data of the leading four spurious passbands show rejection levels exceeding 30 dB. The optimisation of the strip widths, however, may take long simulation time since the circuit has many non-uniform lines which require fine discretisa-tion for accurate characterisadiscretisa-tion. The undesired responses at 2foand 3focan be eliminated either by imposing capaci-tive terminations [17] or by incorporating concept of the effective even and odd mode characteristic impedances [18] to the coupled stages. In [19], stages of l/4, l/6, and l/8 in electrical lengths are tuned to cancel the spurious at 2fo, 3fo and 4fo, respectively. In [20], a new class of parallel-coupled line filters with broad stopband response is designed based on synthesis of bandpass prototypes with pre-defined upper stopband characteristics. In [21, 22], spurious peaks are suppressed by choosing the con-stitutive resonators having identical fundamental frequency but staggered higher order resonances. The dual behaviour resonator filter in [23] achieves the spurious suppression by integrating a low-pass filter in the bandpass filter. In [24], a modified stopband-extended filter is implemented utilising the multiple transmission zeros placed at specified frequencies. In[25], double split-end quarter-wave stepped-impedance resonators are devised to construct a sixth-order filter with a bandwidth of 8.5fo with at least 37.8 dB of attenuation. In [26], the multispurious elimination is achieved with periodic stepped-impedance resonators.
This paper combines the ideas of the corrugated stages [4], overcoupled stages [19] and the tapped input/output scheme [15] to design a parallel-coupled bandpass filter with good rejection in a wide upper stopband. Parallel-coupled filters with properly designed corrugated stages are known to be free of spurious at 2fo[4]. It is a new devel-opment in this work that properly tuning the geometric par-ameters of such a stage can allocate the inherent zeros precisely at both 2foand 4foso that the spurious at these fre-quencies can be eliminated. Some of the stages are validated to have a further transmission zero at 6fo. Furthermore, coupling lengths of some stages can be properly adjusted
#The Institution of Engineering and Technology 2007 doi:10.1049/iet-map:20060130
Paper first received 9th June and in revised form 7th December 2006 The authors are with the Department of Communication Engineering, National Chiao Tung University, 1001 Tahsueh Rd., Hsinchu 300, Taiwan
to allocate zeros, and hence to cancel the undesired trans-mission, at 3foand 6foor at 7foso that the upper stopband can be further extended.
2 Zeros of a cascade of two coupled-line stages One purpose of using a relatively high 1r substrate is to make circuit size smaller. For a traditional parallel-coupled line filter on such a substrate, however, the spurious peaks will be much higher than those on substrates of low 1r. For example, the jS21j peak at 2fo for a substrate of 1r’ 10 can be close to 0 dB [8, 11, 16]. Since the purpose of this work is to demonstrate the effectiveness of coupled-line corrugation on elimination of unwanted harmonics, the RT/Duroid 6010 microwave laminates with 1r¼ 10.2 and thickness ¼ 1.27 mm are adopted.
Fig. 1shows the layout of a uniform coupled-line stage. Given circuit bandwidth D and maximally flat or Chebyshev passband function, the linewidth Wu and gap size Suof each stage can be determined after the even and odd mode characteristic impedances, Zoeand Zoo, are calcu-lated by the conventional synthesis formulas[1].
The zeros of a coupled stage must be accurately allocated at 2mfo, m ¼ positive integer, so that entire circuit can be free of spurious at these frequencies. The reason is investi-gated as follows. The Z-parameters of the coupled stage in Fig. 1can be derived as[1]
Z11¼Z22 ¼ j1
2(Zoe cot ueþZoo cot uo) (1) Z12¼Z21 ¼ j1
2(Zoe csc ueZoo csc uo) (2) where ueand uodenote the electrical lengths of the stage for the even and odd modes, respectively. At fo, ue’ uo’ p/2. For simplicity, consider a first-order filter with a cascade of two such stages. The forward transmission can be derived by directly multiplying two identical ABCD matrices converted from the Z-matrix given in (1) and (2)
1 jS21j2¼1 þ j Z11 Z2 21Zo (Z11Z22Z12Z21Z2o) 2 (3) where Zois the reference port impedance. At frequencies near 2mfo, ue’ uo’ mp, the cot and csc functions approach infinity so that the magnitude of each Z-parameter is much larger than Zo. Thus, (3) can be reduced to 1 S21 2’ 1 þ j ZoeZoo Zo Z11 Z2 21
cotuecotuoþcscuecscuo
2
(4) Substituting (1) and (2) into (4) yields
1 jS21j2’ 1 þ ZoeZoo Zo x (1 þ cosuecosuo) (ZoesinuoZoosinue)2 2 (5a) x ¼ ZoesinuocosueþZoosinuecosuo (5b)
The inherent zero of jS21jcan be obtained by enforcing ZoesinuoZoosinue¼0 (6) Obviously, the conditions
ue¼uo¼mp (7)
are a possible solution to (6). In this case, not only will there be no spurious peaks near 2mfobut also the response will have a notch at 2mfo, because the zero of the denominator in (5a) has one order higher than that of the numerator in (5b). When (6) holds but (7) does not, however, jS21j response will present a large peak at the frequency where x ¼ 0 which is close to the zero frequency given in (6). This point will be further demonstrated in next section. 3 Corrugated coupled-line stage
Evolved fromFig. 1, the corrugated coupled stage inFig. 2 has S ¼ Su, W ¼ Wu– T/2 and period P. The variables T and d stand for the length and width of the corrugation ‘teeth’, respectively. Along the coupled lines inFig. 1, the odd mode propagates faster than the even mode. Because of the corrugation inFig. 2, that is, T = 0, the even and odd modes migrate to the c and p modes, respectively. The c mode travels along the outer conductor edges, whereas the p mode propagates along the rectangular zigzag trace between the strips. Thus, it is possible to make these two modes have identical phase velocities when number of periods or the teeth length T are properly chosen.
Accurately allocating inherent zeros of a l/4-stage at both 2foand 4fois important for extending the upper stop-band.Fig. 3shows the circuit tuning at 2foand 4fo. The hori-zontal axis is frequency normalised with respect to fo¼ 2.45 GHz. Suppose that the uniform coupled stage is defined by the conventional synthesis [1] with Su¼ 0.4 mm and Wu¼ 0.6 mm. Based on [4], with S ¼ Suand W ¼ 0.4 mm, stage A is tuned at d ¼ 0.6 mm and T ¼ 0.53 mm to allocate the inherent zero at 2fo pre-cisely, as shown in Fig. 3a. If the stage is ideal, it is expected to have further null transmission at 4fo, 6foand so on. Its second zero, however, is at 4.08fo, that is, 2% away from 4fo. Based on (6), it can be deduced that ue.uo. 2p, and hence be.bo, provided Zoe. Zoo. Here, the subscripts e and o, respectively, stand for c and p modes, and be and bo are their corresponding phase constants. There are degrees of freedom to further tune the stage at 4fo. As the first step, the teeth width d is reduced to 0.5 mm. Both zeros shift to frequencies higher than 4fo. The teeth length T is then extended to slow down the phase velocity of the p mode, that is, to increase bo. The solution, that is, stage B, is obtained with the two inherent zeros at 2foand 4fowith T ¼ 0.58 mm, as shown inFig. 3a.
The responses for cascade A (two A-stages) and cascade B (two B-stages) in the vicinity of 4foare shown inFig. 3b. Both cascades possess the same zeros as inFig. 3a. Cascade
B presents no peak near 4fo, as expected. For cascade A, however, there is a – 2 dB spurious peak at 4.04fo, the midway position between the zero and 4fo. For further dem-onstration, response of a cascade X is also plotted. It con-sists of two identical l/4-stages with zeros at 4.04fo. The spurious peak locates at around 4.02fo and its level is reduced to 215 dB. This suggests that for the particular circuit configuration the zero should be no more than 1% deviation from 4foif a peak level of better than 15 dB is pursued.
The forward transmission of an ideal l/4-stage has a zero at 2mfo when the coupling length is a multiple of 1808. Similarly, a coupling length of l/6 at fowill be a multiple of 1808 and create a zero at 3mfo[19]. Therefore for generat-ing zeros at both 2mfoand 3mfousing the same corrugated coupled line, six periods are adopted in a stage. Note that the corrugation pattern has been trimmed for 2mfo, only trivial tuning is required for recovering from possible offsets, because of microstrip dispersion, when shifting 2fo and 4foto 3foand 6foby using four of the six periods for inter-stage coupling. There is also a need to suppress the spurious at 7fo. The stage is then designed to have seven periods and four of them (4/7 l/4) are used in the coupling length. The transmission zeros are thus allocated at 3.5mfo.
Fig. 4 presents the geometric parameters of corrugated stages in Fig. 2 for simultaneously tuning zeros at 2foand 4fo. The results for S ¼ 0.2, 0.6 and 1.0 are inFig. 4aand
those for S ¼ 0.4, 0.8 and 1.2 in Fig. 4b. All dimensions are in millimetre. When S is increased or d is decreased, the corresponding T is increased. This is because that for the odd mode propagation with increasing S or decreasing d, the coupling between the two strips decreases and hence the wave travels faster. The ‘teeth’ length is then increased to compensate or slow down its phase velocity. For the sol-utions in solid dots, the stage has an extra zero at 6fo within 0.5% accuracy. This property is of course useful for extra elimination of the undesired spurious at 6fo, but it exists only for structures with relatively small W and S.
Based on the data inFig. 4with simple interpolation, one can have good initial guess for geometric parameters which are not provided inFig. 4. Note that the simulation circuit consists of only a coupled stage which can be one-fifth or one-sixth of a whole filter. So the simulation time can be only 1/30 of that required by a full circuit, since the CPU time is proportional to square of number of discretisation cells.
4 Experiments
Three circuits are designed and fabricated for dem-onstration. The first circuit is a third-order Chebyshev filter with fo¼ 2.45 GHz, 0.1 dB ripple and D ¼ 8%. The circuit layout is inFig. 5aand the simulation and measured
Fig. 4 Geometric parameters for a corrugated stage with zeros simultaneously tuned at 2foand 4fo
a S ¼ 0.2, 0.6, 1.0
b S ¼ 0.4, 0.8, 1.2, all in millimetre. fo¼ 2.45 GHz, substrate:
1r¼ 10.2, thickness ¼ 1.27 mm
Fig. 3 Geometry tuning of a corrugated stage a jS21jresponses for the coupled stages
b jS21j responses of cascades of two identical stages near 4fo.
W ¼ 0.4, S ¼ 0.4. Stage A: d ¼ 0.6, T ¼ 0.53. Stage B: d ¼ 0.5, T ¼ 0.58. Stage X: d ¼ 0.5, T ¼ 0.57, all in millimetre
responses are inFig. 5b. The two end stages are tuned to suppress the spurious responses at 2fo and 4fo, and the two stages in between are tuned at 3fo, so that the upper stopband is extended up to 5fo. The two middle stages use
only l/6 for interstage coupling. The circuit is symmetric about its centre, so that only the dimensions of the first and second stages are given in the figure caption. It is inter-esting to note that near 2fo, 3foand 4fo the glitches show cancellation of the zeros and the spurious by each other. Note also that two end stages are employed to tackle the same spurious. This is because that bandwidth of the unwanted spurious is increasingly wider as its harmonic order is higher.
Fig. 6ashows the layout andFig. 6bthe performance of a fifth-order corrugated filter with a passband function iden-tical to that ofFig. 5. Tapped input/output is used to create zeros for cancelling the spurious at 5fo. The zero frequency can be determined by the open stub whose electrical length is one quarter wavelength at the design frequency. Impedance transformers are inevitable since the tapped points are chosen by the zeros[15]. As shown in Fig. 6a, since tapped input/output couplings are employed to provide necessary couplings between the feed lines and the input/output resonators, there are only four parallel-coupled stages in the circuit. The two end parallel-coupled stages are designed to tackle the spurious 2fo and 4fo and the other two create transmission zeros at 3fo and 6fo. The two end stages also possess zero transmission at 6fo so that it also provides spurious suppression to the circuit at this frequency. The measured and simulation results show that the upper stopband covers a band up to 6.5fowith rejec-tion levels better than 30 dB. Generally, the posirejec-tions of the two end stages can be interchanged with the other two, pro-vided that the coupling coefficients follow the synthesis for-mulas. The outer stages of a bandpass filter, however, Fig. 5 Performance of a third-order corrugated bandpass filter
a Layout
b Simulation and measured results. W1¼ 0.2, S1¼ 0.37, T1¼ 0.52,
d1¼ 0.4, W2¼ 0.57, S2¼ 1.09, T2¼ 0.52, d2¼ 0.4, all in millimetre
Fig. 6 Performance of a fifth-order corrugated bandpass filter a Layout
b Simulation and measured results. W1¼ 0.28, S1¼ 0.39, T1¼ 0.48,
d1¼ 0.5, W2¼ 0.61, S2¼ 1.15, T2¼ 0.54, d2¼ 0.4
Fig. 7 Performance of the second fifth-order corrugated bandpass filter
a Layout
b Simulation and measured results. W1¼ 0.282, S1¼ 0.37, T1¼ 0.46,
usually need more couplings than the middle ones. Thus l/ 4-stage will be preferred to l/6-stages as outer stages.
Fig. 7ashows the layout andFig. 7bthe performance of the second fifth-order filter. The two end parallel-coupled stages tackle the spurious 2fo, 4foand 6fo. The two middle stages have seven periods, and four of them are used to establish interstage couplings. Note that the second transmission zero of the coupled seven-period stages at 7fois utilised for the spurious suppression, instead of its first zero at 3.5fo. This stage is essentially important for further extending the upper stopband. The tapped input/output structures are then designed to cancel the spurious at 3fo and 5fo. The measured results show that the upper stopband is extended up to 7.8fowith rejection levels better than 30 dB. The poss-ible passband degradation can be an important issue when corrugated structures are used in circuit design. The zoomed inset inFig. 7shows the detailed passband response. The inband insertion loss is only 1 dB. This is very close to that of a design with only uniform coupled stages. The effect of corrugation on filter performance will become sig-nificant at higher frequencies, say, beyond 10 GHz[27].
5 Conclusion
Microstrip bandpass filters are designed to have multispurious suppression up to more than 5fo, 6foor 7fowith corrugated coupled stages by allocating transmission zeros at harmonics of the passband. The design is purposely demonstrated on a substrate of relatively high dielectric constant. Based on the analysis of a first-order bandpass filter, the tuning of each zero relies on a sufficiently small deviation (1% or less) of the even and odd mode phase velocities at the harmonics for successful spurious suppression. For precisely allocating the inherent transmission zeros at both 2foand 4fo, geometric parameters of corrugated coupled stages are presented. The structure is very flexible since 2/3 and 4/7 of a l/4-stage used for interstage coupling can effectively eliminate the unwanted peaks at 3fo and 7fo, respectively. The tapped input/output configuration is also incorporated to eliminate the undesired responses at 3foor 5fo.
Based on the proposed approach, bandpass filters of higher orders can be designed to have an even wider rejec-tion bandwidth since more coupled stages can be config-ured. Bandpass filters of no more than fifth-order with an upper rejection band covering over 10fo, however, can be a tough task if a high rejection level is required.
6 Acknowledgment
This work was supported in part by the National Science Council, Taiwan, under Grants NSC 94-2213-E-009-073 and 94-2752-E-009-003-PAE.
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