Jen-Tsai Kuo, Member, IEEE, and Eric Shih
Abstract—Bandpass filters with an optimal rejection band-width are designed using parallel-coupled stepped impedance resonators (SIRs). The fundamental ( ) and higher order resonant harmonics of an SIR are analyzed against the length ratio of the high- and low- segments. It is found that an optimal length ratio can be obtained for each high- to low-impedance ratio to maximize the upper rejection bandwidth. A tapped-line input/output structure is exploited to create two extra transmission zeros in the stopband. The singly loaded ( ) of a tapped SIR is derived. With the aid of , the two zeros can be independently tuned over a wide frequency range. When the positions of the two zeros are purposely located at the two leading higher order harmonics, the upper rejection band can be greatly extended. Chebyshev bandpass filters with spurious resonances up to 4 4 , 6 5 , and 8 2 are fabricated and measured to demonstrate the idea.
Index Terms—Microstrip, microwave filter, spurious response, stepped impedance resonator (SIR), transmission zero.
I. INTRODUCTION
I
N THE RF front-end of a modern communication system, bandpass filters with wide stopband and high selectivity are usually required to enhance the overall system performance. Over the past 30 years, the parallel-coupled microstrip filter has been one of the most commonly used filters due to its planar structure, ease of synthesis method, and low cost [1]. It is known that the traditional parallel-coupled microstrip filters suffer from the spurious responses at , twice the passband frequency, which may seriously degrade the attenuation level in the stop-band and passstop-band response symmetry [2]. It results from the deviation between the even- and odd-mode phase velocities of each coupled section in the filter. As a result, the width of the upper stopband is less than and this could limit the applica-bility of the filter. Many methods [3], [4] have been proposed to overcome this problem.The stepped impedance resonators (SIRs) have been found advantageous in designing microstrip bandpass filters [5]–[9] with good stopband performance. One of the key features of an SIR is that its resonant frequencies can be tuned by adjusting its structural parameters, such as the impedance ratio of the
Manuscript received July 25, 2002; revised November 19, 2002. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 90-2213-E-009-062, and by the Ministry of Education and the National Science Council Joint Program under Contract 89-E-F-A06-2-4.
The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 300, Taiwan, R.O.C. (e-mail: [email protected]).
Digital Object Identifier 10.1109/TMTT.2003.810138
high- and low- segments. As a result, the first spurious har-monic can be much higher than . For example, the design in [7] completely suppresses the resonance with an induc-tive effect, and the first parasitic response is observed at fre-quencies close to . A combination of different SIR struc-tures can also be adopted for a bandpass filter with wide stop-band [8], [9]. Nonconventional SIRs [8] can be used to con-struct high-performance bandpass filters with the control of spu-rious responses outside of a selected bandwidth over a very large frequency range. In [10], low-pass structures are integrated within bandpass topologies. By adjusting low-pass filter cutoff frequencies, harmonic resonances of distributed bandpass fil-ters are attenuated. A multilayer solution is proposed to over-come some design limitations, such as realizable characteristic impedance values.
Filters with a tapped-line input can save space, as well as cost, since the first and last sections of the filter are eliminated [11]. A further benefit is that two independent extra transmission zeros in the stopband can be easily created without requiring complex coupling between resonators [12]–[14]. It means that, without altering the passband response, we can apply tapped couplings to both the first and last resonators to fully control the positions of the two extra zeros. This is a very useful feature for practical receivers in rejecting image frequencies and enhancing the re-jection level in the stopband of a bandpass filter.
In this paper, we aim at designing a bandpass filter with a very wide stopband possessing a satisfactory rejection level. To this end, first, SIRs are used as building blocks to push the second, third, and fourth resonances to as high frequencies as possible. The filter is synthesized based on a parallel-coupled structure. With proper input and output tappings, two transmission zeros are then created and devoted to cancel the first higher order reso-nance so that the filter has a very wide stopband up to the second spurious resonance. Finally, the two zeros are used to suppress the first and second higher order harmonics in a one-on-one manner, so that the stopband of the filter can be extended up to the third higher order harmonic or the fourth resonant frequency of the SIR.
This paper is organized as follows. Section II analyzes the resonance characteristics of an SIR. For an SIR with given impedance ratio, Section III investigates the optimal length ratio to obtain a maximally wide stopband. Section IV addresses the tuning of the extra transmission zeros, and formulates the singly loaded of a tapped SIR. Section V gives the design procedure for an SIR filter, and Section VI presents some simulation and experimental results.
Fig. 1. Structure of an SIR with tapped input.
Fig. 2. Normalized resonant frequencies of an SIR.
II. RESONANTPROPERTIES OF ANSIR
Shown in Fig. 1 is the structure for a typical SIR, of which an infinite number of resonant frequencies exist. Each resonance has either a symmetric (even-mode) or an antisymmetric (odd-mode) voltage distribution on the resonator. The fundamental resonance occurs in the odd mode, and the first higher order resonance in an even mode, and so forth. The conditions for determining the resonance frequencies of an SIR are given as [5]
(odd-mode) (1) (even-mode) (2) where is the impedance ratio of the SIR defined as
(3) When , at . It can be seen from (1) and (2) that the resonant frequencies of an SIR can be tuned by changing the value of and the lengths of the high- and low- segments. A simple root-searching program can be em-ployed to calculate the resonant frequencies of the structure. For a microstrip SIR on a dielectric substrate with
and high- , Fig. 2 plots the resonant frequencies against for the fundamental, first, second, and third higher order modes for , 0.8, and 2.5. These reso-nant frequencies have been normalized with respect to the fun-damental frequency of a uniform impedance resonator (UIR), i.e., . It can be observed that, if , the plot has extreme values as is varied from 0 to 1. All these extreme frequencies occur at different values of . For example,
Fig. 3. Ratios of the three leading higher order resonant frequencies to the fundamental frequency of an SIR forR = 0:2; 0:3; and 0:4.
when , the fundamental resonant frequency has a min-imum at , and the second higher order resonance has two minimal frequencies at and and a max-imal frequency at .
III. OPTIMALDESIGN FORWIDESTOPBAND
Since the object of our design is to look for a bandpass with a stopband as wide as possible, the space between each of the higher order resonant frequencies and the fundamental one is of the paramount concern. Thus, Fig. 3 plots the higher order resonant frequencies normalized with respect to their associated for and . Here, the optimal value for obtaining a maximal for each value is clearly indicated. For example, when , should be 0.68, 0.5, and 0.4 to locate the higher order resonant frequencies at , , and , as indicated by points , , and , respectively. It is to be noted that the smaller the values of are, the larger the maximal ratio is. In [5], several maximal to ratios are also plotted versus values, but they are limited to the case of or . If can be changed freely, as indicated in Fig. 3, the maximal values of and can be increased significantly.
IV. CREATE TRANSMISSIONZEROS BY TAPPING THEI/O RESONATORS
The couplings between the feed lines and end resonators of a bandpass filter can be performed with gap or tapped coupling. When the latter is used, the singly loaded of a resonator should be calculated. The value of should be determined by the filter specification, which specifies the passband response. For a tapped resonator, the value is given as [5]
(4) where is the load impedance seen by the resonator looking into the load at the tap point, is the operation frequency, and is the total susceptance of the resonator seen by the feed line at the tap point. Thus, for the tapped SIR in Fig. 1 can be derived as follows.
where (5b) and (5c) When (5d) where (5e) (5f) (5g) and (5h) One can easily validate that these results reduce to those in [5] when .
Assume that the right-hand-side portion of the tapped SIR in Fig. 1 is coupled with the next SIR in a filter. Based on [12] and [13], an extra transmission zero can be created via this tapped coupling. The frequency of the zero is determined by treating the cascaded nonuniform line sections to the left-hand side of the tap point as a quarter-wave open stub so that the input impedance at the tap point is virtually short circuited.
The frequency of the transmission zero can be tunable if the tap point can be freely sliding on the I/O SIRs. However, the value of the SIRs cannot be changed since it has been de-termined by the filter specification. When the tap point is chosen for a prescribed zero frequency, based on (4), the value can be altered correspondingly to keep the value unchanged. If the required does not equal 50 , a quarter-wave trans-former can be employed to perform the impedance transforma-tion. Fig. 4 plots the simulation responses with tunable trans-mission zeros for a third-order SIR filter with GHz and fractional bandwidth %. The high- segment has , and . Herein, the fabricated and simu-lated circuits use the RT/Duroid 5880 substrate with
and thickness mm, and the circuit simulator is IE3D.1 1Zeland Software Inc., Fremont, CA, Jan. 1997.
(a)
(b)
Fig. 4. Creating tunable transmission zeros by tapping the input and output resonators. (a) Circuit scheme. (b) Simulated responses. Related parameters are:
f = 2:45 GHz, 1 = 10%, R = 0:7, and = . The =( + ) values
for generating the decreasingf are 1.213, 1.273, and 1.322, and those for the increasingf are 0.787, 0.727, and 0.678.
Fig. 4(a) shows the slide of feed lines, together with an impedance transformer, on the input and output resonators. The tap position is defined in Fig. 1, and the corresponding changes of the zeros in the lower and upper stopbands are shown in Fig. 4(b). It is to be noted that the passband responses are unchanged when the locations of the zeros are changed. It is obvious that the two zeros created by the input and output tappings can be freely chosen to locate in either only one or both of the stopbands. This has been validated with measurements [15].
V. FILTERDESIGNPROCEDURE
In designing a parallel-coupled bandpass filter with SIRs, the interstage couplings required for prescribed filter function are given as [16]
(6) where ’s are the element values of the low-pass filter proto-type, is the fractional bandwidth, and is the order of the filter. This coupling coefficient is used to determine the spacing between two adjacent SIRs. In obtaining the coupling coeffi-cient of an isolated pair of SIRs by a circuit simulator, each SIR is spaced with a gap to the feeding line. The coupling coefficient is calculated as
TABLE I
SPECIFICATIONS ANDDIMENSIONS OF THEFOUREXPERIMENTALFILTERS
where and are the resonant frequencies in the transmission response. Both conductors and dielectric substrate are assumed loss free to have and with good accuracy. For the first and last SIRs, the external ’s are given as [16]
(8a) (8b) If the SIRs are considered lossless, in (5) should be in ac-cordance with these external ’s. In other words, given the load impedance, the tapped positions of feed lines at the end res-onators should be determined by the external ’s in (8).
VI. SIMULATION ANDMEASUREMENT
From Fig. 3, the value of should be chosen as low as pos-sible for constructing an SIR filter with a wide stopband. A mi-crostrip line, however, has a realizable impedance value with upper as well as lower limits. These limits depend on dielec-tric constant and thickness of the substrate, resolution of layout in fabrication process, and size of the circuit. In this study, the high- segment is chosen to have a linewidth of 0.4 mm, which has a characteristic impedance and, more impor-tantly, a tolerable metallic loss.
In the following experimental examples, the values of are chosen to be 0.2 and 0.4. The corresponding linewidths for -segments are given in Table I. The unloaded- can be de-graded in accordance with small . By invoking the full-wave simulator, the unloaded of the SIRs with and are found to be 170 and 172, respectively. The substrate parameters with regard to circuit loss, i.e., S/m and are included in the simulation. For all the particular cases shown below, the calculated external
values for the input and output resonators are no more than 17.2, which is much less than the values of the resonators. Thus, in addition to the conductor and dielectric losses, the radiation effect, which is possibly further enhanced by the impedance junctions of the SIR, can be neglected. These loss factors are not taken into account for the ease of design.
Several SIR filters are designed and fabricated to validate the above-described findings. The circuit parameters and detailed dimensions of each filter are listed in Table I, where and represent the linewidth and length of the -segment of a single SIR, respectively, and is the spacing between the th and ( )th SIRs.
Fig. 5. Simulated and measured responses for an SIR filter.f = 1:5 GHz,
N = 3, R = 0:4, 1 = 10%, and passband ripple = 0.1 dB. Simulated result
for a UIR filter with identical specification is also plotted for comparison.
Fig. 5 shows simulated results for a third-order Chebyshev filter with a 0.1-dB ripple level. The filter has GHz, %, and . The simulation response of a UIR filter with identical passband specification is also plotted for comparison. The SIR filter has the first spurious passband at , as predicted in Fig. 3, which is much higher than the second spurious response of the UIR filter. Furthermore, this spurious response shows a bandwidth much smaller than the second spurious response of the UIR filter. In Fig. 5, a zoomed area is used to show in detail the fundamental passband perfor-mance of the designed bandpass filter. At the center of the pass-band, the insertion loss of the UIR filter is 0.1 dB better than that of the SIR filter. The experiment and simulation results have a good agreement.
The filter shown in Fig. 5 is redesigned with % and , and the results are shown in Fig. 6(a). Again, the response for a UIR filter is also plotted for comparison. The first spurious response for the SIR filter is pushed to , as well as predicted in Fig. 3. It is interesting to note that the shadow area in the plot demonstrates the improvement of stopband re-jection by the SIRs. As shown in the zoomed area, at the center of passband, the insertion loss of the UIR filter is 0.7 dB better than that of the SIR filter. Fig. 6(b) presents the photograph of the fabricated circuit.
It is worth mentioning that the attenuation level at frequen-cies around of the filter in Fig. 6(a) is better than that of Fig. 5 by 15–25 dB. In our numerical experiments, it is found
(a)
(b)
Fig. 6. (a) Simulated and measured responses for an SIR filter.f = 1:5 GHz,
N = 3, R = 0:2, 1 = 6%, and passband ripple = 0.1 dB. Simulated
result for a UIR filter with identical specification is also plotted for comparison. (b) Photograph of the fabricated circuit.
that the attenuation level in the upper stopband also depends on the bandwidth of the filter. The choice of the value, however, seems to dominate the attenuation level of the filter at . This finding leads us to use an SIR with a small for designing fil-ters of wide stopband with satisfactory rejection levels in the upper stopband.
In both Figs. 5 and 6(a), the two transmission zeros on both sides of the passband are created by tapping the input and output resonators without any tuning. It means that the tap point is de-termined by the value with , and there is no need using any quarter-wave transformer.
In Fig. 3, the peak value of for is 6.5. To reach this goal, we can use the two extra zeros, created by tap-ping the input and output resonators, to cancel the first spurious resonance at . The effectiveness of suppressing the spu-rious resonance using only one zero is, of course, different from that using two. This is investigated in Fig. 7 with a fifth-order bandpass filter, whose % and GHz. It is to be noted that should be used to have .
There are three plots shown in Fig. 7. For simulation , the tap points are determined by , and no impedance transformer is required. For simulation , one of the two zeros is located at the first spurious resonance ( or 5.6 GHz); and for simulation , both zeros are located at the first spurious resonance. Case has a peak spurious value of higher than 5 dB at , while cases and have approximately 30 and 45 dB, respectively. It is interesting to note that the
for plot at is below 40 dB. The reason is that is a higher order zero created by tapping at one of the end
res-Fig. 7. Simulation responses for an SIR filter with tap-controlled transmission zeros. For caseA, the tap points are determined by R = 50 . For case B, one of the two zeros is located at the first spurious resonance(3:76f ). For caseC, both zeros are at the first spurious resonance (3:76f ). f = 1:5 GHz,
N = 5, R = 0:2, 1 = 10%, and passband ripple = 0.1 dB.
Fig. 8. Simulation and measurement responses for caseC of Fig. 7.
onators. This issue is outside of the scope of this paper, and will be discussed in another form.
Fig. 8 compares the simulation and measured responses for case of Fig. 7. It can be observed that both responses have a good agreement. Their worst rejection levels in the entire upper stopband are 45 dB before the spurious response goes up.
From the results shown in Fig. 7, it is possible to exploit the two transmission zeros to cancel the two leading spurious res-onances in the upper stopband with a tradeoff of the attenua-tion level. If this is done for the case of , the stop-band can be further extended to , the third spurious har-monic, as predicted in Fig. 3. To this end, we have to choose and move the tapping to the positions that creates trans-mission zeros whose frequencies equal the first and second spu-rious resonances of the SIR at and , respectively. Fig. 9(a) plots the simulated and measured responses of the bandpass filter. A minimal rejection level of 30 dB is obtained in the stopband before the third spurious response at rises. Fig. 9(b) shows the photograph of the filter.
It is to be noted that, in our experience of measuring the re-sponses of filters of Figs. 6, 8 and 9, the results in the stopband can be quite sensitive to the flatness of the circuit board. Since the RT/Duroid 5880 substrate is a soft board, a metallic carrier is suggested to support the circuit during the measurement.
(a)
(b)
Fig. 9. (a) Simulation and measured responses of the fabricated filter. (b) Photograph of the circuit. The passband specification of the filter is identical to that of Fig. 8.
VII. CONCLUSION
Given a characteristic impedance ratio of the high- and low- segments, an SIR is shown to have a maximally wide stopband for certain length ratios of the segments. Filters with SIRs of lower impedance ratios are found to have higher spurious resonant frequencies and better rejection levels at , twice the passband frequency. The singly loaded for a tapped SIR is derived. It is shown that proper tappings at both the input and output resonators can create two independent tunable transmission zeros in the stopband, which can be used to improve the attenuation and selectivity of the filters. Bandpass filters with stopbands up to , , and are designed and fabricated. A very good agreement between the simulation and measurement is obtained.
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Jen-Tsai Kuo (S’88–M’92) received the Ph.D.
de-gree from the Institute of Electronics, National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 1992.
Since 1984, he has been with the Department of Communication Engineering, NCTU, as a Lecturer in both the Microwave and Communication Electronics Laboratories. He became a Professor in 2000. From 1995 to 1996, he was a Visiting Scholar with the University of California at Los Angeles. His research interests include the analysis and design of microwave circuits, high-speed interconnects and packages, field-theoretical studies of guided waves, and numerical techniques in electromagnetics.
Eric Shih was born in Taoyuan, Taiwan, R.O.C., on
April 12, 1976. He received the B.S. degree in engi-neering and system science from the National Tsing Hua University (NTHU), Hsinchu, Taiwan, R.O.C., in 1998, and is currently working toward the Ph.D. degree in communication engineering at the National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C.
His research interests include the design of microwave planar filters and associated RF modules for microwave and millimeter-wave applications.
