IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 4, APRIL 2006 1545 The Effects of Component Q Distribution on Microwave Filters Chih-Ming Tsai, Member, IEEE, and Hong-Ming Lee, Student Member, IEEE Abstract The effects of lossy components on the passband response of a bandpass filter are studied in this paper. It is found that the resonator has pronounced effects on the insertion losses, while its effects on the group delays are rather minor. The firstorder approximation is used to estimate the deviations of insertion loss and group delay from the ideal. The effects of a lossy resonator in each stage are individually analyzed to evaluate which components are more critical and should be paid more attention. Filters designed by the predistortion technique are also discussed with emphasis on the effects of component distribution on the transmission level. When resonators with different are used in a filter, a bell-shaped distribution is proposed to achieve the optimal passband response. Finally, three filters with different component distributions were designed by combining the dielectric and microstrip-line resonators, and the measured results agreed well with the theoretical predictions. Index Terms Filter distortion, lossy circuits, microwave filters, factor. I. INTRODUCTION MICROWAVE communication systems require numerous bandpass filters for functions such as preselection and suppression of mixer spurious products. Cost, size, and performance are the important considerations for the filter in the design of microwave communication systems. The design procedure of microwave filter starts from finding the low-pass prototype having the desired insertion loss response and then scaling the frequency and impedance to yield the parameters of resonators. However, the component losses, i.e., factors, are not taken into account in the traditional filter design, and thus the frequency responses may have serious deviations from the expected specifications due to the dissipation losses. The most evident deviation of the insertion loss is in the passband, especially around the band edges, and thus leads to a rounded passband response. In order to reduce this effect, high- resonators are usually required in high-frequency filter designs. However, as the frequencies go higher, the component losses and distortions become unnegligible. Therefore, a filter design that takes into account the finite component is desirable. Generally, all of the resonators in a filter are selected to be the same type, and therefore the component (unloaded) s of all resonators are equal, i.e., they are uniformly distributed Manuscript received September 22, 2005; revised January 6, 2006. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 93-2213-E-006-074. The authors are with the Institute of Computer and Communication Engineering, Department of Electrical Engineering, National Cheng Kung University, Tainan 70101, Taiwan, R.O.C. (e-mail: tsaic@mail.ncku.edu.tw). Digital Object Identifier 10.1109/TMTT.2006.871929 among the stages. The dissipated insertion loss of the filter can be estimated from its original lossless transmission function by shifting the zeros and poles a constant distance of away from the axis in the plane. The distance is given by [1], [5] (1) where and are the unloaded of the resonators and the fractional bandwidth, respectively. However, by using this method, the different effects of each resonator on the insertion loss are concealed, and it is unable to distinguish which resonator is more crucial. If this information is available, more attention could be paid to a few critical components. The cost and filter size could even be reduced if the others are replaced with low- alternatives. A simple formula was derived by Cohn to evaluate the insertion loss of a filter with resonators of nonuniform distribution. However, only the loss at the center of the passband could be determined [2]. In order to study and compare the loss effects over the entire passband caused by each lossy resonator, first-order approximations of insertion loss and group delay are given in this paper. The effect of each lossy resonator was analyzed individually and compared. The responses of the filters designed with the lossless parameters for the maximally flat or equal-ripple functions will be distorted by the presence of component losses. However, it is possible for the filters composed of low- resonators to have their responses approach these functions, if they are designed with the method of predistortion technique [3] [7]. Once the unloaded s of the resonators are known, one may use the predistortion technique to compute the circuit parameters. In order to flatten the insertion loss that is rounded at the band edges, the predistortion technique introduces proper mismatch in the band center, and, thus, maximally flat or equal-ripple responses can still be obtained. Fig. 1 shows a typical insertion loss response of a predistorted Chebyshev filter, which is compared to that of a lossy filter designed by the traditional lossless parameters. Although the unloaded s are the same, the passband transmission of the predistorted filter is much lower. The predistortion technique is applicable to the communication system that needs a low-variation transmission in the passband, such as satellite transponder input multiplexers [5], [6]. The high insertion loss may be compensated by a low-noise amplifier (LNA) followed by the filter [6]. In this paper, the predistorted filter design is also discussed with the effects of component distribution. Finally, three filters were designed to demonstrate the effects of component distribution. These filters were the combinations of high- dielectric resonators and relatively low- mi- 0018-9480/$20.00 2006 IEEE
1546 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 4, APRIL 2006 To evaluate the deviation of insertion loss caused by the lossy resonators, the lossless transmission function of the prototype filter is first determined and given by the general form as where and. The rest of the coefficients are functions of the prototype values. The response of along the -axis is then transformed into the natural logarithmic form as (4) Fig. 1. Comparison of the insertion losses between the traditional and predistorted filters. (5) where and. The lossy conductances and resistances cause the deviation of by. The first-order deviation of is given by McDonald and Temes studies in [4] as Fig. 2. (a) Circuit model of a lossy bandpass filter and (b) its low-pass prototype. crostrip-line resonators. The passband transmissions of the filters with their high- resonators in the inner and outer stages are compared in order to verify the theoretical predictions. (6) This equation was extended here to derive the expressions of the insertion loss deviation. Letting and, the in (6) can then be obtained as II. FIRST-ORDER APPROXIMATIONS OF THE DEVIATION OF PASSBAND RESPONSES (7) The circuit model of lossy bandpass filter is shown in Fig. 2(a); the shunt conductances and series resistances attached to the resonators represent the lossy terms. It is rather complicated to directly estimate the loss effects of the bandpass filter. For simplicity, the low-pass prototype of the bandpass filter, as shown in Fig. 2(b), is used for the analysis of insertion loss deviation. The conductances and resistances in Fig. 2(b) are given by [2] Therefore, the deviation of can be written as for is odd (2) for is even (3) where is the element value of the filter prototype. The fractional bandwidth herein represents the 3-dB bandwidth for Butterworth filters and the equal-ripple bandwidth for Chebyshev filters. It is obvious that the filter is less lossy if the filter has a larger bandwidth or higher resonators. where is part of and is defined as the factor of insertion loss deviation. Equation (8) is the general form of for calculating the deviation of insertion loss. To recover the decibel scale of the deviation of, the following equation can be used: (8) db (9)
TSAI AND LEE: EFFECTS OF COMPONENT DISTRIBUTION ON MICROWAVE FILTERS 1547 FUNCTION D TABLE I OF THE THIRD- AND FOURTH-ORDER FILTERS The effect on the insertion loss due to the lossy component in each stage is characterized by the deviation factor.for small-ripple db Chebyshev filters, the deviation in the passband can be simply approximated by db (10) The functions for third- and fourth-order filters are listed in Table I. The expressions of for higher order filters are more complicated. The zeroth-order terms of for all of the filters are found to be or. For the filters that are matched at, i.e.,, one can calculate the deviation of as db (11) which coincides with the result of [2]. However, for the other filters with, (8) is more accurate in the estimate of the mid-band loss. Moreover, by using (8) or (10), the deviation of over the entire passband can be easily examined. As examples, of the third- to fifth-order Butterworth filters and Chebyshev filters with 0.1-dB ripple are plotted in Figs. 3 and 4, respectively. The frequency scale of the low-pass filters is used and the corresponding frequency of the bandpass filters can be calculated by the frequency transformation as (12) where is the central frequency of the bandpass filters. The frequency ranges of the curves are plotted from 1 to 1 and correspond to the bandwidths of the bandpass filters. Fig. 3. Deviation factor of insertion loss F in each stage for the: (a) thirdorder, (b) fourth-order, and (c) fifth-order Butterworth filter. It is obvious that, for all of the filters, the deviations caused by the lossy components in the first and last stages are less and more uniform than those by the others. This is because the lossy terms of the input/output resonators are directly attached to the terminations. The loss of a filter caused by the outer-stage resonators can be estimated by the power division between the lossy terms and terminations. Therefore, the insertion loss deviation is uniform over the passband. However, it is not easy to intuitively explain the loss effects of the inner-stage resonators. As shown
1548 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 4, APRIL 2006 caused by the losses could be sim- deviation of the phase of ilarly derived as (13) where can be obtained as (14) Therefore, can be expressed by (15) The deviation of group delay can then be yielded by the derivative of with respect to as (16) Fig. 4. Deviation factor of insertion loss F in each stage for the: (a) thirdorder, (b) fourth-order, and (c) fifth-order Chebyshev filter with 0.1-dB ripple. in these figures, when the lossy elements are present closer to the middle stage, the filter will suffer from more loss at the band edges. Therefore, in order to avoid the rounded band edges and maintain a flat passband insertion loss, the resonators which are closer to the middle of the filter should have higher unloaded s. The component losses will also affect the group delay of filter transmissions. As the expression of in (6), the first-order where is defined as the factor of group delay deviation. Similarly, of the third-order to fifth-order Butterworth filters and Chebyshev filters with 0.1-dB ripple, which are normalized to their original group delay, are plotted in Figs. 5 and 6, respectively. The group delay has the largest deviations around the band edges due to the losses, and the negative values of could actually help to lower the peaks of group delay. The group delay deviations in the band center are minor. For a typical filter with and, the group delay deviations caused by the losses in each stage are less than 2%. Furthermore, is either positive or negative in the band center, and thus the group delay deviations due to the losses could offset each other. This results in a group delay that is close to the ideal in the band center. Therefore, the loss effects on the group delay are not as severe as those on the magnitude. The variation of group delay over the passband could, in fact, be slightly improved with the introduction of lossy components. As examples, two fifth-order Chebyshev filters with 0.1-dB ripple and 10% bandwidth are given to demonstrate the effects of component distributions. They are denoted filter A and filter B, with the distributions given by and, respectively. Fig. 7(a) gives the passband insertion losses of the two filters, which are compared with the responses of a lossless filter. It shows that the midband loss of filter B is better since its average is larger. However, in filter B, the unloaded s of the resonators are decreased with their positions closer to the middle stage, which are contrary to those in filter A, and, therefore, the insertion loss of filter B at band edges is larger and rounded. It is apparent that filter A has a flatter insertion loss and still keeps sharp band edges, even though its average is lower. For comparison, three filter responses with uniformly distributed,, and
TSAI AND LEE: EFFECTS OF COMPONENT DISTRIBUTION ON MICROWAVE FILTERS 1549 Fig. 5. Normalized deviation factor of group delay F = in each stage for the: (a) third-order, (b) fourth-order, and (c) fifth-order Butterworth filter. for the resonators are given in Fig. 7(b). It is obvious the response of the filter with is highly distorted, and the others with higher resonators are much better and close to the ideal. Therefore, increasing the component can help to improve the filter response, and the distribution should be taken into account for filter design that consists of different kinds of resonators. The resonators at the first and last stages have the least effect on the passband responses and might be replaced with lower alternatives for the consideration of Fig. 6. Normalized deviation factor of group delay F = in each stage for the: (a) third-order, (b) fourth-order, and (c) fifth-order Chebyshev filter with 0.1-dB ripple. cost and circuit size reduction. Moreover, the resonators that are closer to the middle stage are more critical to the insertion loss and should be carefully manufactured. The group delays of filters A and B are shown in Fig. 7(c). It was found that they have the largest deviations from the ideal at the band edges, but in a positive sense, because the group delay variations over the entire passband are reduced. Moreover, in the band center, the group delays are undistinguishable from those of the ideal.
1550 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 4, APRIL 2006 [3] [6]. The presence of component losses will distort the passband responses of the traditional filter design. By using the predistortion technique, mismatch is introduced in the band center to compensate for the rounded band edges such that the entire passband response could still be flat. Therefore, the predistorted filter will experience larger reflection and insertion loss. However, it is still acceptable in some particular applications such as the satellite transponder [5], [6]. It is known that, for a filter with uniform s, its zeros and poles of the transmission function are shifted left in the plane by a distance of as shown by (1). For a predistorted filter, its poles are moved to the right in advance to offset their shifts due to the losses. However, for the predistorted filters having zeros, their zeros cannot be moved and need to be fixed on the -axis for the feasibility of filter synthesis. In this paper, only the allpole (ladder) filters are discussed, and their lossless predistorted transmission functions are given by (17) Fig. 7. (a) Passband insertion losses of filters A and B, which are compared with the response of a lossless filter and (b) three S responses of the filters with uniformly distributed Q = 100, 1000, and 5000 for comparison. (c) Group delays of filters A and B and the ideal lossless filter. This example shows that the resonator has more pronounced effects on the filter shape than the group delay. III. EFFECT OF COMPONENT DISTRIBUTION ON THE TRANSMISSION LEVEL OF THE PREDISTORTED FILTERS Here, the effect of component distribution is discussed for the filter designed with a specific predistortion technique where is the original transmission function and is a constant to ensure the maximum of is unity. It is expected that has two peaks at its band edges to overcome the rounding effect caused by the losses. The design procedure was discussed in detail in [7]. This design method cannot be applied to filters with nonuniform distributions; for these, the method proposed by McDonald and Temes [4] might be used. Low-pass prototype filters are also used to simplify the predistortion procedure. Several sixth-order filters, which are denoted filters C H, are given as examples, and they are all predistorted to have a Chebyshev filter response with a 10% bandwidth and 0.1-dB ripple. The unloaded s of the resonators in filters C and D are uniformly distributed, and they are given 1000 and 100, respectively. On the other hand, filters E H consist of resonators with both the s of 1000 and 100. The distributions and the low-pass prototype parameters of these predistorted filters are given in Table II, with. The passband insertion losses of these filters are shown in Fig. 8. It is apparent that all of the filters have Chebyshev-like insertion losses, and the differences between them are their transmission levels, which are also summarized in Table II. Since these predistorted filters have the same pole distributions as that of a lossless filter, their group delay responses are also the same. Table II shows that the predistorted filters will suffer from significant loss on their transmission when the number of low- resonators is increased. Filters E G have the same composition of resonators but different distributions, and it was found that filter E, which has its low- resonators at the first and last stages, has the least insertion loss. Generally, when the s of the predistorted filters are not uniformly distributed, they should be arranged to have a bell-shaped distribution for maximizing the transmission level. This result is similar to that discussed in the previous section, which states that the traditional filter should have higher for the resonator closer to the middle stage to decrease the loss at the band edges.
TSAI AND LEE: EFFECTS OF COMPONENT DISTRIBUTION ON MICROWAVE FILTERS 1551 TABLE II LOW-PASS PROTOTYPE PARAMETERS OF THE PREDISTORTED FILTERS Fig. 8. Passband insertion losses of filters C H. IV. FILTER DESIGN EXAMPLES To experimentally verify the theoretical studies, three fourth-order filters were designed to demonstrate the effects of component distribution. These filters, denoted filters I III, are composed of dielectric and microstrip-line resonators, and they were designed to have central frequencies around 10 GHz. Filter I consists of dielectric resonators only, and its circuit structure is shown in Fig. 9(a). The filter is fed by half-wavelength open stubs, which were fabricated on the Rogers RO3003 substrate with a relative dielectric constant of 3, a thickness of 0.51 mm, and a loss tangent of 0.0013. The two outer resonators are located near the centers of the open stubs to maximize the magnetic coupling. In order to increase the bandwidth, low-impedance transmission lines were used to transform the 50- termination to a lower impedance for a smaller loaded. The unloaded of the dielectric resonators was measured to be about 2500. A housing is needed for the dielectric resonator filter, as shown in Fig. 9(b). The couplings between the resonators are controlled by the distances between them, and their resonant frequencies can be fine tuned with the screws. The filter design followed the time-domain tuning method [8], and the measured results are given in Fig. 9(c). Filters II and III consist of both the dielectric and microstrip-line resonators, and their circuit structures are shown in Figs. 10(a) and 11(a), respectively. The transmission lines were also fabricated on the Rogers RO3003 substrate. The half-wavelength transmission lines were used as the resonators, and their unloaded s were measured to be 150. The dielectric resonators were placed in the inner and outer stages for filters Fig. 9. (a) Photograph of filter I. (b) Complete structure with a housing. (c) Measured and simulation results. II and III, respectively. These two filters need to be housed as well. The time-domain tuning method was also used to optimize their passband responses, and their measured results are shown in Figs. 10(b) and 11(b). Among these three filters, additional transmission zeros were found at the lower or upper side of the passband. This is because the nonadjacent resonators are not fully isolated, and there exists a cross-coupling effect between them. The bandwidth of filter I was measured to be 1.8%, and those of filters II and III were intentionally designed to be similar and about 2.5%. The comparisons between the simulation and measured results of filters I III are also shown in Fig. 9 11. All of the simulated filters are given as Chebyshev filters with 0.1-dB ripple. Since there exist cross-coupling effects in the fabricated filters, the measured out-of-band responses are different from the simulations of the all-pole filters. Also, because the filters are manually tuned, the return losses in the passbands are somewhat different.
1552 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 4, APRIL 2006 Fig. 12. Comparison among the insertion losses of filters I III. Fig. 10. (a) Photograph of filter II. (b) Measured and simulation results. their passband responses are quite different due to their component distributions. It should be noted that the low- resonators will not only degrade the filter transmission but also the reflection response. Their midband transmissions were measured to be approximately 4 db. However, the high- resonators of filter II were placed in the inner stages, and therefore it has a uniform insertion loss in the passband. It is almost a shift in transmission level from the ideal. On the contrary, filter III has low- resonators in the middle stages, and its passband transmission is highly distorted and rounded. The measured midband transmission is about 0.5 1 db lower than the estimations, which is supposed to be caused by the losses of the connectors and the input/output transmission lines. These measured results show good agreements with the theoretical predictions and these experiments have demonstrated the importance of component distribution. Fig. 12 gives the comparison among insertion losses of the three filters. Fig. 11. (a) Photograph of filter III. (b) Measured and simulation results. However, the trends of the measured and simulated responses in the passbands of filters I-III are similar. Filter I, which is composed of all-high resonators, has the smallest bandwidth and highest transmission, which is measured to be approximately 1.1 db. Although filters II and III have the same components, V. CONCLUSION The effects of component distribution on the passband responses of microwave filters have been studied in this paper. It was found that the resonator has pronounced effects on the insertion losses, while its effect on the group delays are rather minor. In order to find the critical components which are sensitive to component losses, the first-order approximation has been derived in this paper to estimate the deviation of passband transmission. For the traditional filter design, it was found that the resonators in the first and last stages have the least effect on the insertion loss, and they might be replaced with low- alternatives for cost or circuit-size reduction. The resonators closer to the middle stage are more crucial to the passband insertion loss, especially at the band edges, and therefore they should be carefully manufactured. The filters designed with the predistortion technique were also discussed with different component distributions. Similarly, the lossy resonators in the first and last stages cause the least loss on the transmission level. When the lossy resonators are placed in the inner stages, the predistorted filter will suffer from more severe insertion loss. Generally, for both the traditional and predistorted filters designed with different kinds of resonators, a bell-shaped distribution should be chosen to optimize the passband responses. Three filters had
TSAI AND LEE: EFFECTS OF COMPONENT DISTRIBUTION ON MICROWAVE FILTERS 1553 dis- been designed to demonstrate the effects of component tribution and verify the theoretical studies. REFERENCES [1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filter, Impedance-Matching Networks, and Coupling Structures. Norwood, MA: Artech House, 1980, ch. 4. [2] S. B. Cohn, Dissipation loss in multiple-coupled-resonator filters, Proc. IRE, vol. 47, no. 8, pp. 1342 1348, Aug. 1959. [3] M. Dishal, Design of dissipative bandpass filters producing desired exact amplitude-frequency characteristics, Proc. IRE, vol. 37, no. 9, pp. 1050 1069, Sep. 1949. [4] J. MacDonald and G. Temes, A simple method for the predistortion of filter transfer functions, IEEE Trans. Circuit Theory, vol. CT-10, pp. 447 450, Sep. 1963. [5] A. Williams, W. Bush, and R. Bonetti, Predistortion techniques for multicoupled resonator filters, IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 5, pp. 402 407, May 1985. [6] M. Yu, W.-C. Tang, A. Malarky, V. Dokas, R. Cameron, and Y. Wang, Predistortion technique for cross-coupled filters and its application to satellite communication systems, IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2505 2515, Dec. 2003. [7] I. Hunter, Theory and Design of Microwave Filters. London, U.K.: IEE Press, 2001, ch. 8. [8] Simplified filter tuning using time domain, Agilent Technol., Palo Alto, CA, Applicat. Note 1287-8, 2001. Chih-Ming Tsai (S 92 M 94) received the B.S. degree from the National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1987, the M.S. degree from the Polytechnic University, Brooklyn, NY, in 1991, and the Ph.D. degree from the University of Colorado at Boulder, in 1993, all in electrical engineering. From 1987 to 1989, he was a Member of the Technical Staff with Microelectronic Technology Inc., Taiwan, R.O.C., where he was involved with the design of digital microwave radios. In 1994, he joined the Department of Electrical Engineering, National Cheng Kung University, Tainan, Taiwan, R.O.C., where he is currently an Associate Professor. His research interests include microwave passive components, high-speed digital design, and measurements. Hong-Ming Lee (S 03) was born in Nantou, Taiwan, R.O.C. He received the B.S. and Ph.D. degrees in electrical engineering from the National Cheng Kung University, Tainan, Taiwan, R.O.C., in 2002 and 2006, respectively. He is currently a Post-Doctoral Research Fellow with the Institute of Computer and Communication Engineering, Department of Electrical Engineering, National Cheng Kung University. His research interests include microwave passive components and measurements.