This paper proposes the design of tunable dual-band
filters for multi-band multi-standard systems. The main
objective is to provide frequency tunability in the second band
while maintaining the first band fixed. To this end, the
capacitive-loaded stepped-impedance resonator is used. The
work is divided into two main parts. In the first part, a deep
analysis of the capacitive-loaded stepped-impedance resonator
structure is done, which provides analytical closed-form design
equations which eases their design in front of the several
approaches available in the literature up to now. The analysis is
also particularized for the case of the capacitive-loaded uniform
resonator. In the second part, resonators are integrated in two
dual-band tunable filters.Postprint (published version

Girbau Sala, David

Lázaro Guillén, Antoni

Martínez, Ester

Pradell i Cara, Lluís

Pérez, A.

Villarino Villarino, Ramón

English

This paper proposes the design of tunable dual-band

filters for multi-band multi-standard systems. The main

objective is to provide frequency tunability in the second band

while maintaining the first band fixed. To this end, the

capacitive-loaded stepped-impedance resonator is used. The

work is divided into two main parts. In the first part, a deep

analysis of the capacitive-loaded stepped-impedance resonator

structure is done, which provides analytical closed-form design

equations which eases their design in front of the several

approaches available in the literature up to now. The analysis is

also particularized for the case of the capacitive-loaded uniform

resonator. In the second part, resonators are integrated in two

dual-band tunable filters

UPCommons

Tunable dual-band filters based on capacitive-loaded stepped-impedance resonators

UPCommons. Portal del coneixement obert de la UPC

Tunable Dual-Band filters based on capacitive-loaded stepped-impedance resonators D. Girbau1, A. Lázaro1, A. Pérez2, E. Martínez2, L. Pradell2, R. Villarino1 1 DEEEA, Universitat Rovira i Virgili, Av.Paisos Catalans 26, 43007 Tarragona (Spain) david.girbau@urv.cat 2 TSC, Universitat Politècnica de Catalunya, C/ Jordi Girona 1-3, 08034 Barcelona (Spain)   Abstract— This paper proposes the design of tunable dual-band filters for multi-band multi-standard systems. The main objective is to provide frequency tunability in the second band while maintaining the first band fixed. To this end, the capacitive-loaded stepped-impedance resonator is used. The work is divided into two main parts. In the first part, a deep analysis of the capacitive-loaded stepped-impedance resonator structure is done, which provides analytical closed-form design equations which eases their design in front of the several approaches available in the literature up to now. The analysis is also particularized for the case of the capacitive-loaded uniform resonator. In the second part, resonators are integrated in two dual-band tunable filters.  I. INTRODUCTION The information society is rapidly approaching to a convergence between communication, multimedia and computing with the goal of a broad connection for all at any time and any place [1]. To operate in an effective and transparent way between all the different wireless access systems, terminals must be able to connect to different networks, driving the need for multi-radio platforms, front-ends and circuits. Moreover, the increasing demand in some wireless communication services requires unfolding the number of frequency bands assigned to a specific service. For instance, wireless local-area networks (WLAN) operate at 2.45 GHz but also at ISM 5 GHz bands. In this context, dual-band filter structures are emerging for simultaneous operation in multiple frequency bands in wireless communication systems. A number of topologies for dual-band filters have been proposed so far and some of them use the stepped-impedance resonator (SIR) [2,3] to fix its spurious response by adjusting the line characteristic impedances [4,5]. In addition, within the framework of reconfigurable filters, several works have recently appeared proposing frequency-tunable dual-band filters based on stepped-impedance resonators, by shunt-connecting a variable capacitor at its center [6,7]; the concept of tunable stepped-impedance resonators was first proposed in [8], but applied to monoband filters. Two are the objectives of this work. First, to propose a new analysis of the capacitive-loaded stepped-impedance resonator, leading to a closed-form analytical expression which greatly eases the design of this structure, in comparison to the analyses available in the literature up to now. Second, to present two dual-band frequency-agile filters which permit to tune the second band while maintaining the first fixed, based on the capacitive-loaded stepped-impedance resonator in an open-loop form in order to miniaturize the whole structure.  II. CAPACITIVE-LOADED STEPPED-IMPEDANCE RESONATOR Fig. 1 shows the capacitive-loaded stepped-impedance resonator. The structure is based on a symmetrical stepped-impedance resonator (a resonator with lines of two different characteristic impedance, Z1 and Z2) with a capacitor C shunt-connected at the midpoint; here the case Z1>Z2 is considered, since this topology permits f2>2f1 [2]. The input admittance of the resonator YIN  is:  ( ) ( )( ) ( ) ( ) θθθθθθtan11tantantan1tantan121322222132KjZZKKKKjZZKjYYIN+−⎟⎟⎠⎞⎜⎜⎝⎛−+−+−−+−=     (1) where Y2 =1/Z2 is the characteristic admittance of the outer line, Y1 =1/Z1 is the characteristic admittance of the inner line and Z3=1/jωC =-j|Z3| is the impedance of the shunt-connected capacitor. The case θ1=β1l1=θ2=β2l2=θ has been considered since it minimizes the total resonator length (being βi the propagation constant at each line section and li their respective lengths). The parameter K is defined as the ratio between characteristic impedances K=Z2/Z1.  Z2l2C2l1Z2Z1l2,θ2 ,2θ1,θ2YINFig. 1. Capacitive-loaded stepped-impedance resonator.   The resonance condition is |YIN|=0. From (1), the first resonance can be obtained with the condition:  0tan 2 =− θK        (2)  978-2-87487-011-8 © 2009 EuMA 29 September - 1 October 2009, Rome, ItalyProceedings of the 39th European Microwave Conference113Then, the fundamental resonance frequency is given by:  ( )π2tan 1111Klvf−=   (3)  where v1 is the propagation velocity in line 1. The second resonance is obtained from the numerator of (1), which after some re-ordering is:  ( ) 0tan12tan 1321 =−+− KZZKZ θθ  (4)  The solution to equation (4) will be:  ( ) ( ) KZZKZZK +⎟⎟⎠⎞⎜⎜⎝⎛+±+=213213 11tanθ         (5)  Since (4) depends on Z3, the second resonance will depend on the value of the capacitor C. If we consider C=0 (|Z3|= ∞ ), we would have a stepped-impedance resonator, and the ratio between second (f2) and fundamental (f1) frequencies would depend only on the ratio between characteristic impedances K:  ( )Kff 112 tan2 −=π   (6)  From now on, we will address the second resonance frequency that will be obtained (already considering C) as fr. Then, fr=f2 for C=0 (6) and fr<f2 for C ≠ 0. The ratio between resonance frequencies can be expressed as:  ( )Kff rrr 111 tan2 −==θθθ       (7)  where θr is the electrical length that corresponds to fr. By using (7), (5) can be expressed as:  KKCZKCZKffr121111 tan11tan−−⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛+⎟⎟⎠⎞⎜⎜⎝⎛ +++=ωω         (8)  Two resonators are considered along this work: K=1 (uniform resonator, which is considered a particular case of the SIR) and K=0.56 (SIR). For the case K=1 the following parameters are used: f1=2.45 GHz, f2=4.9 GHz and Z1=Z2=50Ω. For the case K=0.56 the following parameters are considered: f1=2.45 GHz, f2=6 GHz, Z1=60Ω and Z2=33.6Ω. The reason for the case K=0.56 is that it is a good example of K<1 and will permit to use the resonators in filters suitable for WLAN applications (with f1=2.45 GHz and f2 tuned between 5.75 and 5.25 GHz). The dependence of fr/f1 on Z3 (or C) is obtained numerically solving (8) and is plotted in Fig. 2 for the two resonators. It can be noticed in (2)-(3) that the first resonance frequency is independent of C, which only modifies f2. Then, it can be considered for this analysis ω=ωr=2πfr and expression (4) can be rewritten as:  ( ) 0tantan122tantan 111121 =⎟⎟⎠⎞⎜⎜⎝⎛+−⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛−− KffKCfKKffZ rrr π(9) Only the positive solution is considered, since the negative leads to an unphysical solution (fr<0 or fr>f2). Manipulating (9), the following expression is obtained:  ( )( ) ⎟⎟⎠⎞⎜⎜⎝⎛−+−⎟⎟⎠⎞⎜⎜⎝⎛=−−KffKKKffCfZffrrr1111111 tan2cos11tan2sin1π          (10)  Fig. 3 shows the graphical solution of (10) for the two resonators (Fig. 3.a for K=1 and Fig. 3.b for K=0.56). However, neither solving (8) nor (10) does not simplify the resonator design procedure; to this end, a closed-form analytical expression relating C and fr/f1 is obtained next. Isolating the capacitance C in (10) and manipulating, expression (11) is obtained:  ( )( ) ⎟⎟⎠⎞⎜⎜⎝⎛−+−⎟⎟⎠⎞⎜⎜⎝⎛=−−KffKKKfffZCrrr 11111 tan2cos11tan2sin1π             (11)  Finally, developing expression (11) by using (6) and manipulating, expression (12) can be obtained which directly relates f1, f2, fr, K (Z1 and Z2) and C:  ⎟⎟⎠⎞⎜⎜⎝⎛−⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛=22121 coscossin1fffffffZCrrr ππππ  (12)  By using (12) it is very simple to design any capacitive-load stepped-impedance resonator, on the contrary to the works available up to now in the literature [7,8]. In [8] an analysis based on the ABCD parameters is provided, leading to complicated expressions which do not make its design straightforward. On the other hand, the authors provide in [7] a numerical approximation. Both works are far from the versatility and simplicity that the closed-form analytical expression (12) provides to the designer. A three-step simple design procedure can now be defined: 1/ Fix f1 and f2, which correspond to the fundamental resonance and the maximum value for the second resonance (case C=0), respectively. This determines K. 2/ Fix the values of Z1 and Z2, according to line widths that can be physically fabricated. 3/ Fix the tuning range of the resonator; its limits determine two resonance 114frequencies fr (fr1 and fr2) which, in turn, determine the two extreme capacities C (C1 and C2). In order to validate the proposed analysis, Fig.4 shows a comparison between the analytical expression (12) and an AgilentTM ADS-based design for the two resonators. A subplot with a detail of the region of maximum interest (small C) is included. A good agreement is obtained.  10 0 10 1 10 2 103 10 411 .522 .5Z3 (Ω)f r/f 1f2/f1 =2 (K =1)f2/f1 =2 .45 (K =0.56)fr/f1 (K =1 )fr/f1 (K <1 )Fig. 2. Dependence of fr/f1 on Z3, obtained by numerically solving (8). Case K=1 and K<1 (0.56).  2.5 3 3 .5 4 4.5012345fr (G Hz)f r /f 1fr / f10.1 pF0.2 pF0.3 pF0.4 pF0.5 pF0.6 pF0.7 pF1.2 pF2.2 pF- a - 2 .5 3 3.5 4 4.5 5 5.5 6012345fr (G H z)f r /f 1fr / f10.1 pF0.2 pF0.3 pF0.4 pF0.5 pF0.6 pF0.7 pF1.2 pF2.2 pF- b - Fig.3. Graphical solution of (10) for the two resonators: K=1 (a) and K<1 (0.56) (b).  III. TUNABLE DUAL-BAND FILTERS DESIGN The capacitive-loaded stepped-impedance resonator has been integrated in two second-order Chebyshev filters, with K=1 and K=0.56. In order to miniaturize both filters, the resonators have been designed in its open-loop form. They have been manufactured on RO4003 substrate, with εr=3.55, thickness t=0.813 mm and loss tangent tgδ=0.0027. Photographs of the manufactured filters are given in Fig. 5.  2.5 3 3 .5 4 4 .5 5 5 .5 60246810fr (G H z)C (pF)4 4 .5 5 5.5 600.511 .5fr (G H z)C  (pF)f2  (K <1)f2 (K =1)f2  (K =1 ,K <1 )A D S , K=1A D S , K<1(12 ), K =1(12 ), K <1Fig. 4. Validation of the proposed closed-form analytical expression (12) for the capacitive-loaded stepped-impedance resonator for the cases K=1 and K<1 (0.56) by means of comparison to ADS simulation.   - a - - b - Fig. 5. Photographs of the manufactured filters. Tunable dual-band filter with K=1 (a) and with K=0.56 (b). (Not to scale).  The filter with K=1 (see Fig. 5.a) has its first pass-band at 2.45 GHz and the second at 4.9 GHz. Fig. 6 shows the measured insertion and return loss compared to ADS/Momentum simulation.   Frequency (GHz)S21 and S11 (dB)2 3 4 51 6-40-20-600S21 measured          simulatedS11 measured          simulatedFig. 6. Comparison between measurement and simulation of the tunable dual-band filter with K=1 (C=0pF).  Fig. 7 shows the measured frequency tuning range of the filter by varying the shunt-connected capacitance from 0 to 0.5 pF. The maximum design frequency f2 for this filter is 2f1 (when C=0 pF). This limit can be pushed up by using the SIR resonator with K<1, as it has been demonstrated. A second filter has been designed with f1=2.45GHz and f2=6GHz (see Fig. 5.b). Since f2 is very different from 2f1, a dual-band 115input/output matching network is designed based on using a stepped-impedance transmission line (nonsyncronous alternating-impedance transformer) [9,10]. The filter measured insertion and return loss are compared to simulation in Fig. 8. Fig. 9 shows its measured tuning range for C between 0 and 0.2 pF.   Frequency (GHz)S21 and S11 (dB)C= 0 pF2.0 3.0 4.0 5.01.0 5.5-40-20-600-10-30-50C= 0.5 pFC= 0.4 pFC= 0.3 pFC= 0.2 pFC= 0.1 pF Fig. 7. Measurement of the tunable dual-band filter response with K=1 for C between 0 and 0.5 pF, with a measured tuning range between 4.9 and 4.12 GHz, insertion loss 1.8 and 2.4 dB and relative bandwidth 8.1% and 6.3%, respectively. The measured first pass-band at 2.42 GHz has 0.63 dB insertion loss and 11.5% relative bandwidth.  2 3 4 5 61 7-30-10-500Frequency (GHz)S21 measured          simulatedS11 measured          simulated-20-40S21 and S11 (dB) Fig. 8. Comparison between measurement and simulation of the dual-band filter with K=0.56 (C=0pF).  2 3 4 5 61 7-30-10-500Frequency (GHz)S21 and S11 (dB)-20-40C=0.1 pFC=0.2 pFC=0 pF Fig. 9. Measurement of the tunable dual-band filter response with K=0.56 for C between 0 and 0.2 pF, with a measured tuning range between 5.97 and 5.45 GHz, insertion loss 2.1 and 2.9 dB and relative bandwidth 7.2% and 6.1%, respectively. The measured first pass-band at 2.47 GHz has 0.83 dB insertion loss and 10.5% relative bandwidth.  A faster degradation in the filter performance has been experienced in the case K=0.56, since it has been proved that it is very sensitive to the precision in placing the variable capacitors. On the other hand, it has been demonstrated with this filter that f2 can be placed beyond 2f1, which allows to design filters reconfigurable between the different frequency specifications of WLAN. It is also important to note that when resonators are integrated into a filter the optimum position for the capacitor is no longer the physical midpoint and now this parameter is optimized by means of electromagnetic simulation. If the capacitor is not properly placed, a rapid degradation of the filter response in terms of ripple, losses and bandwidth is experienced while tuning the filter.  IV. CONCLUSIONS This paper has proposed the design of tunable dual-band filters. They provide frequency tunability in the second band while maintaining the first band fixed. The capacitive-loaded stepped-impedance resonator is used. An analysis of the capacitive-loaded stepped-impedance resonator structure has been done, providing analytical closed-form design equations, which ease their design in front of the several approaches available up to now in the literature. The resonator has been integrated in two tunable dual-band filters which demonstrate good tuning range. ACKNOWLEDGMENT This work was supported by the Spanish Government Projects TEC2008-06758-C02-02/TEC and TEC2007-65705/TCM. REFERENCES [1] R. Castello, “Introduction to the Special Issue on Wireless Reconfigurable Terminals”, IEEE Circuits and Systems Magazine, Vol.6, Nº1,pp.7. 2006. [2] M. Makimoto, S. Yamashita, “Bandpass filters using parallel coupled stripline stepped impedance resonators”, IEEE Trans. on Microwave Theory and Techniques, Vol.28, Nº12, pp.1413-1417. Dec. 1980. [3] M. Sagawa, M. Makimoto, S. Yamashita, “Geometrical structures and fundamental characteristics of microwave stepped-impedance resonators”, IEEE Transactions on Microwave Theory and Techniques, Vol.45, Nº7, pp. 1078-1085. July 1997. [4] M. Mokhtaari, J. Bornemann, S. Amari, “New reduced-size step-impedance dual-band filters with enchanced bandwidth and stopband performance”, IEEE MTT-S International Microwave Symposium Digest, pp. 1181-1184, June 2006. [5] Y.P. Zhang, M. Sun, “Dual-band microstrip bandpass filter using stepped-impedance resonators with new coupling schemes”, IEEE Transactions on Microwave Theory and Techniques, Vol.54, Nº10, pp.3779-3785. October, 2006. [6] X.Y. Zhang, Q. Xue, “Novel Centrally Loaded Resonators and Their Applications to Bandpass Filters”, IEEE Transactions on Microwave Theory and Techniques, Vol. 56, Nº4, pp.913-921. April, 2008. [7] D. Girbau, A. Lázaro, E. Martínez, D. Masone, L. Pradell, “Tunable Dual-Band Bandpass Filter for WLAN Applications”, accepted to Microwave and Optical Technology Letters. December, 2008. [8] B. Kapilevich, R. Lukjanets, “Modelling varactor tunable microstrip resonators for wireless applications”, Applied Microwave and Wireless, Vol.10, Nº7, pp.32-44. September, 1998. [9] H.-M. Lee, C.R. Chen, C.-C. Tsai, C.-M. Tsai, “Dual-band coupling and feed structure for microstrip filter design”, IEEE MTT-S International Microwave Symposium Digest, Vol.3, pp. 1971-1974, June 2004. [10] C.-M. Tsai, C.-C. Tsai, S.-Y. Lee, “Nonsyncronous alternating-impedance transformers”, Asia-Pacific Microwave Conference ’01, Vol.1, pp. 310-313. 2001.  116

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Tunable dual-band filters based on capacitive-loaded stepped-impedance resonators

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