PARALLEL coupled-line filters are widely used in microwave

2812 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 9, SEPTEMBER 2005 Improved Coupled-Microstrip Filter Design Using Effective Even-Mode and Odd-Mode Characteristic Impedances Hong-Ming Lee, Student Member, IEEE, and Chih-Ming Tsai, Member, IEEE Abstract The transmission zeros of two coupled-microstrip circuits are studied in this research. Estimates for the frequencies of transmission zeros are given. A new concept of effective even- and odd-mode characteristic impedances is introduced. Useful design equations are given for improving the passband responses that are inherently distorted by different mode velocities. Finally, two filters with different topologies are designed for their zeros to be at the frequencies of the spurious harmonics and, thus, yield wide out-of-band rejections. Index Terms Coupled transmission lines, distributed parameter filters, microwave circuits. I. INTRODUCTION PARALLEL coupled-line filters are widely used in microwave circuits. Conventional design uses several quarter-wavelength coupled-line sections and the design procedures had been well established [1]. However, the frequencies of the transmission zeros created by these coupled lines are fixed at the multiplies of the fundamental frequency and, therefore, cannot be tuned. Filter design using nonquarter-wavelength coupled-stripline sections was also proposed [2], [3]. The transmission zeros can be tuned to desired frequencies and, therefore, are useful in achieving a steeper skirt or a wider stopband. However, in that research, only stripline filters were studied. It had been shown that two types of parallel coupled-stripline circuits, as shown in Fig. 1, can create transmission zeros when their electrical lengths equal and, respectively. However, they do not follow such rules when the circuit is realized in an inhomogeneous medium using microstrips. In order to estimate the frequencies of transmission zeros, impedance matrices are used to derive the design equations in this paper. For coupled-stripline filter designs, the circuits with desired passband responses can be exactly synthesized. The values of even-mode impedance, odd-mode impedance, and electrical length for each coupled-line section could be well determined. However, if the same design is realized by microstrips, the values of electrical length are different for even and odd modes. Usually, the average values are used and Manuscript received August 24, 2004. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 93-2213-E-006-075. The authors are with the Institute of Computer and Communication Engineering, Department of Electrical Engineering, National Cheng Kung University, Taiwan 70101, R.O.C. (e-mail: tsaic@mail.ncku.edu.tw). Digital Object Identifier 10.1109/TMTT.2005.854177 Fig. 1. (a) Parallel and (b) antiparallel coupled-line circuits. this approximation degrades the performance of filters. The passband bandwidth is usually decreased and the reflection increased. In order to study and reduce these effects, the coupled-microstrip filter is equivalent to a coupled-stripline filter with new effective parameters, as shown in Fig. 2(a). It can then be specifically compared with the original coupled-stripline filter. It is found that these parameters of the two filters are different, especially when nonquarter-wavelength coupled lines are used. Therefore, the design equations in [2] and [3] are not applicable to coupled-microstrip filter designs for the purpose of transmission zero control. In fact, to improve the passband responses of coupled- microstrip filters, these effective parameters should be forced to equal those of the original coupled-stripline filters, as shown in Fig. 2(b). The new parameters, as and, for a coupled-microstrip filter, can then be derived from these effective parameters. This proposed new design procedures compensate the effects of velocity difference and, therefore, can achieve an undistorted passband response. Two filter examples are designed to demonstrate the problems and the proposed solutions, with their second and third harmonics suppressed by the transmission zeros to yield wide stopband rejections. II. TRANSMISSION ZERO CONDITIONS Fig. 1 shows the two types of coupled-line circuits. In Fig. 1(a), the two diagonal ports are opened, whereas in Fig. 1(b), two ports on the same side are opened. They were called parallel and antiparallel coupled-line circuits in [3], respectively. They both can generate transmission zeros; however, under different conditions. The antiparallel coupled-line section is a special case of loaded coupled lines studied in [4] with its loads opened. The impedance parameters of parallel and antiparallel coupled-line circuits can be found as follows [5]. For parallel coupled-line circuits, (1) 0018-9480/$20.00 2005 IEEE

LEE AND TSAI: IMPROVED COUPLED-MICROSTRIP FILTER DESIGN USING EFFECTIVE EVEN- AND ODD-MODE CHARACTERISTIC IMPEDANCES 2813 Fig. 2. (a) Concept of effective stripline parameters of coupled-microstrip-line filters and (b) proposed new design procedures. (2) and for antiparallel coupled-line circuits, where and are the electrical lengths defined at the central frequency of the filter. The input impedances of the two-port network under even and odd excitations can then be obtained from their T-type equivalent circuits, and they are found to be and. The condition for a transmission zero is [4], which means or both and are infinite. The last condition is possible when the electrical lengths are for coupled striplines. However, it could never be possible for coupled-microstrip circuits because is not equal to. For example, when, it is found that and has a finite value for both the parallel and antiparallel coupled lines. Also, it can be found that is finite and for. For coupled microstrips, the zero condition can only be achieved when, i.e., for parallel coupled-microstrip circuits and for antiparallel coupled-microstrip circuits, where is the ratio of the zero frequency to the central frequency. It is then clear that the zero conditions for circuits in an inhomogeneous medium will not only depend on the length of coupled lines, as they do for circuits in homogeneous media, but also on the evenand odd-mode impedances and velocities. In other words, the linewidths and gapwidth of coupled microstrips are also important factors in the analysis of the frequencies of transmission zeros. For antiparallel coupled lines, it was found in [4] that the lowest transmission zero occur at the frequency when. Similarly, the lowest transmission zero (3) (4) (5) (6) Fig. 3. Approximate equivalent circuit for parallel and antiparallel coupled-line sections. for the parallel coupled lines can be found at the frequency when. III. FILTER DESIGN PROBLEMS For the purpose of filter design, the two types of coupled-stripline circuits with electrical length are approximately equivalent to an impedance inverter having a 90 or 90 phase shift, and on both of its sides, there is a transmission-line section with characteristic impedance and electrical length, as shown in Fig. 3. From the studies in [2] and [3], the characteristic impedances of the nonquarter-wavelength coupled lines were found as for parallel coupled lines and (7) (8) (9) (10) for antiparallel coupled lines. The electrical lengths of the coupled lines in each section are usually designed shorter than. It should be noted that (7) (10) were derived based on the assumption that the inverters have a 90 phase shift. However, in order to maintain the relation of for the antiparallel coupled lines with electrical lengths longer than 90

2814 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 9, SEPTEMBER 2005 Fig. 4. Equivalent circuit of a coupled-line filter. inverters should be chosen in their equivalent circuits and all the positive and negative signs in (9) and (10) should be interchanged. In Cohn s design procedure [1], the parallel coupled-line filter is equivalent to several impedance inverters separated by transmission-line sections with characteristic impedances of and electrical lengths of, as shown in Fig. 4. The transmissionline section with electrical lengths of can then be approximately equivalent to a simple lumped parallel resonator with and, which has a resonant frequency at. Therefore, an original direct-coupled-resonator filter is derived [6]. The value of inverter impedance is then determined from the specifications by the classical filter synthesis as (11) to (12) where is the relative bandwidth, and is the element value of the low-pass prototype filter. As examples, two third-order Chebyshev filters, with central frequency at 2.45 GHz, 5% bandwidth, and 0.1-dB ripple, were designed using substrates with a relative dielectric constant of 3 and a thickness of 0.51 mm. The values of were found to be 181.2 for the first and fourth sections, and 692.6 for the second and third sections. The first example, shown in Fig. 5(a), is denoted filter A and uses the parallel coupled lines for all the sections. The second example, denoted filter B, uses parallel and antiparallel coupled lines for the first/fourth and second/third sections, respectively, as shown in Fig. 5(b). Equations (7) (10) were used for these filter designs. The lengths of all coupled-line sections were not limited to a quarterwavelength. They were selected with the helps of (5) and (6) for the transmission zeros to be at the frequencies of the second and third spurious passbands. The connecting lines with characteristic impedance between each coupled-line section should be added to ensure that the electrical length of the transmission line in the equivalent circuit, as shown in Fig. 4, is between impedance inverters. In filter A, only parallel coupled-line sections were used. The length of a coupled-line section with zero at combined with the length of a coupled-line section with zero at are longer than and, therefore, they cannot be adjacent to each other. Thus, filter A was configured as Fig. 5. Circuit configurations of the filter design examples. (a) Filter A. (b) Filter B. TABLE I CIRCUIT PARAMETERS OF FILTER A TABLE II CIRCUIT PARAMETERS OF FILTER B an asymmetric filter. Alternatively, both parallel and antiparallel coupled lines were used in filter B. The total lengths of the two coupled-line sections with zeros at and were less than and, therefore, the filter could be symmetrically configured. Their circuit parameters were summarized in Tables I and II. These parameters could be exactly realized using striplines. However, when microstrips are used, the velocities are different in even and odd modes. The parameters of the electrical lengths in TablesIandIIcouldonly beapproximated,usuallybytheaverage values of the two modes. Therefore, the designs are no longer ideal. The simulation results are compared in Fig. 6. It is clear that the bandwidths of the microstrip filters become narrower, and the return losses are also distorted. Filter B has much more

LEE AND TSAI: IMPROVED COUPLED-MICROSTRIP FILTER DESIGN USING EFFECTIVE EVEN- AND ODD-MODE CHARACTERISTIC IMPEDANCES 2815 Fig. 7. Passband S of filters A and B modified by the method in [7] compared to the coupled-stripline filter response. (14) The effective electrical length is free to be selected between and. In order to avoid any singularity in (13) and (14), only when (15) Fig. 6. Comparisons between the passband responses of the filters realized using microstrip lines and striplines. (a) Filter A. (b) Filter B. distortion than filter A. Apparently, the performances of the filters are degraded by the different even- and odd-mode velocities. IV. EFFECTIVE AND The effect of different mode velocities on quarter-wavelength coupled-microstrip filters had been studied [7]. An equivalent relative permittivity was defined, and minor correction of line lengths was proposed to improve the return loss in the passband. However, it is not effective for nonquarter-wavelength coupled-microstrip filters. As examples, the responses of filters A and B modified by the method in [7] are shown in Fig. 7. It was found that there is no significant improvement and they are almost the same as those shown in Fig. 6. In order to study and overcome the degradation of the filter responses, a new concept of effective parameters is proposed. The impedance parameters and at the central frequency of the parallel coupled microstrips are forced to equal those of their stripline equivalents with and. After solving the simultaneous equations, the effective parameters can be derived as should be set to equal. To fulfill this requirement, the general expression for is proposed as (16) This expression is equivalent to the definition of an equivalent relative permittivity in [7]. Using (13), (14), and (16), coupled-microstrip circuits could be converted into their stripline equivalents and compared to the original filter designs, as shown in Fig. 2(a). The definition of the effective electrical lengths is associated with the characteristic impedances instead of those that are originally defined by the average electrical lengths. However, the slight variations of the electrical lengths are not the cause of the distorted filter passband responses such as the decreased bandwidths. Actually, the distortion is caused by the deviations of the characteristic impedances due to the different mode velocities. The effective characteristic impedances and will be close to and, respectively, only as approaches. That means the distortions would be minor only for filters based on quarter-wavelength coupled lines. Similarly, the effective parameters of the antiparallel coupled microstrips can be found as (17) (18) (13) could not be used for the filter design since infinite values of the characteristic impedances will be obtained from

2816 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 9, SEPTEMBER 2005 TABLE III EFFECTIVE PARAMETERS OF THE FILTERS A AND B TABLE IV EFFECTIVE PARAMETERS OF THE MODIFIED FILTERS A AND B Fig. 8. Passband responses of the modified microstrip filters and their stripline counterparts. (a) Filter A. (b) Filter B. (9) and (10). Thus, the effective electrical length of the antiparallel coupled lines does not have the restriction as that of the parallel coupled lines, and can be simply defined to be the average of the even- and odd-mode electrical lengths, i.e., (19) The effective parameters of the previous microstrip filter examples are summarized in Table III. From and, the effective impedance inverter and characteristic impedance and for the equivalent circuit in Fig. 3 can be obtained by solving (7) (10). The corresponding bandwidth can then be calculated by (11) and (12). Since the corresponding bandwidths are quite different from the original filter design, the performances of the filters are degraded. In filter A, the corresponding bandwidths of the first and the last coupled lines, which have the electrical lengths longer than, are larger than 5%, whereas those of the others are smaller than 5%. In filter B, however, all the coupled lines have their corresponding bandwidths smaller than 5% and, thus, it ends up with more distortion. It may also conclude that the coupled lines in a microstrip filter design may not have the required couplings. The coupled lines shorter then a quarter-wavelength are under coupled and those longer than a quarter-wavelength are over coupled. V. DESIGN EQUATIONS FOR THE RIGHT AND Equations (13), (14), and (16) are used to derive the effective parameters of coupled microstrips. Once the circuits are designed, these parameters could be derived and used to evaluate the deviations, due to the different mode velocities, from the ideal filter designs. On the other hand, one may try to find the right parameters of microstrip circuits to yield the correct effective stripline equivalents so the performance of the filter could be close to ideal, as shown in Fig. 2(b). Instead of doing a lot of trial and error, design procedures are developed as follows. Firstly, given the required and, the ratio of and is determined by transmission-line design tools, although it actually depends on and. This is valid because the ratio is not sensitive to the changes of characteristic impedances. With this ratio, and, the physical lengths of the parallel-coupled lines are then chosen to satisfy the equation (20) The design equations for and can now be rewritten from (13) and (14) as (21)

LEE AND TSAI: IMPROVED COUPLED-MICROSTRIP FILTER DESIGN USING EFFECTIVE EVEN- AND ODD-MODE CHARACTERISTIC IMPEDANCES 2817 Fig. 9. (a) Filter A. Measured results of: (b) passband and (c) out-of-band responses. (22) Similarly, for antiparallel coupled lines, the physical lengths are chosen by satisfying (23) The design equations for and can be easily derived as (24) (25) In the final stage of the design procedure, the transmission zero frequency is checked as to whether it is shifted due to the modifications of characteristic impedances. However, the frequency shifts are usually minor and can be neglected. Fig. 10. (a) Filter B. Measured results of: (b) passband and (c) out-of-band responses. As examples, the filters in Section III are modified by the design equations to make their effective parameters closer to the intended design, and they are summarized in Table IV. The frequency shifts of the transmission zeros due to the modifications of characteristic impedances are small. The antiparallel coupled lines have the largest frequency shift, which is from to ; however, they are still capable of suppressing the spurious harmonic. The passband responses of the modified filters are shown in Fig. 8, where it can be seen that the bandwidths and return losses have been recovered and, thus, the effects of different mode velocities are eliminated. VI. FILTER DESIGNS AND MEASUREMENTS The modified filters A and B in Section V were fabricated, and the photographs and measurement results are presented in Figs. 9 and 10, respectively. The connecting lines of filter B between each coupled-line section were bent to make the circuit more compact. Figs. 9(b) and 10(b) show the electromagnetic

2818 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 9, SEPTEMBER 2005 simulation and measurement results of the passband responses. The losses of the conductor and substrate had also been taken into account. The measured results were in good agreement with those of simulations. The desired bandwidths were achieved and the reflections were low and without distortion. Note that the small coupling between the coupled lines of the first and fourth sections of filter A causes the cross-coupled effect and, thus, an additional transmission zero is created at 2.2 GHz. Therefore, the shape factor of filter A is smaller than that of filter B. Figs. 9(c) and 10(c) give the measured results of the out-ofband transmission of the filters. In Fig. 9(c), the response of filter A is compared to that of a conventional filter using all quarter-wavelength parallel coupled microstrips. The measured transmission zero frequencies of the designed filter were found at 4.9, 7.67, 7.33, and 5.04 GHz, which are due to the first to fourth coupled-line sections, respectively. Since the transmission zeros were intentionally designed at the frequencies where the spurious harmonics occur, the transmission of the spurious harmonics was reduced. In comparison with filter A, the transmission zeros of the coupled lines of filter B were all designed at the frequencies of the second and third harmonic, and they were measured at 4.9 and 7.29 GHz. Therefore, the suppression of the harmonics is better than that of filter A, and a very wide stopband rejection below 35 db was obtained. VII. CONCLUSION Two types of coupled microstrips have been analyzed in this paper. The transmission zero conditions have been derived. This paper has also demonstrated the problems of coupled-microstrip filter design due to the inherent different mode velocities. It has been shown that the problems cannot be solved by simply changing the coupled-line lengths. A new concept of effective parameters is proposed to study the effects such as increased reflection and reduced bandwidth. Design equations have also been given to rectify the circuit parameters for improving the passband responses. Two filters with their second and third harmonics suppressed by the transmission zeros due to the coupled microstrips have been designed. With the help of the effective parameters, the filters have successfully achieved good passband responses and wide out-of-band rejections. REFERENCES [1] S. B. Cohn, Parallel-coupled transmission-line-resonator filters, IRE Trans. Microw. Theory Tech., vol. MTT-6, no. 4, pp. 233 231, Apr. 1958. [2] M. Makimoto and S. Yamashita, Bandpass filters using parallel coupled stripline stepped impedance resonators, IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 12, pp. 1413 1417, Dec. 1980. [3] M. Matsuo, H. Yabuki, and M. Makimoto, The design of a half-wavelength resonator BPF with attenuation poles at desired frequencies, in IEEE MTT-S Int. Microwave Symp. Dig., vol. 2, 2000, pp. 1181 1184. [4] C.-M. Tsai, S.-Y. Lee, and H.-M. Lee, Transmission-line filters with capacitively loaded coupled lines, IEEE Trans. Microw. Theory Tech., vol. MTT-51, no. 5, pp. 1517 1524, May 2003. [5] G. Zysman and A. Johnson, Coupled transmission line networks in an inhomogeneous dielectric medium, IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 10, pp. 753 759, Oct. 1969. [6] S. B. Cohn, Direct-coupled-resonator filters, Proc. IRE, vol. 45, no. 2, pp. 187 196, Feb. 1957. [7] D. Kajfez and S. Govind, Effect of difference in odd- and even-mode wavelengths on a parallel-coupled bandpass filter, Electron. Lett., vol. 11, pp. 117 118, Mar. 1975. Hong-Ming Lee (S 03) was born in Nantou, Taiwan, R.O.C. He received the B.S. degree in electronic engineering from National Cheng Kung University, Tainan, Taiwan, R.O.C., in 2002, and is currently working toward the Ph.D. degree at National Cheng Kung University. His research interests include microwave passive components and measurements. Chih-Ming Tsai (S 92 M 94) received the B.S. degree in electrical engineering from the National Tsing Hua University, Taiwan, R.O.C., in 1987, the M.S. degree in electrical engineering from the Polytechnic University, Brooklyn, NY, in 1991, and the Ph.D. degree in electrical engineering from the University of Colorado at Boulder, in 1993. 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.