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PAPER

Special Section on 2004 International Symposium on Antennas and Propagation

Periodically Nonuniform Coupled Microstrip Lines with Equalized Even- and Odd-Mode Velocities for Harmonic Suppression in Filter Design Sheng SUN†a) , Student Member and Lei ZHU†b) , Member

SUMMARY Periodically nonuniform coupled microstrip line (PNCML) loaded with transverse slits is characterized using the fullwave method of moments and short-open calibration technique. Guided-wave characteristics of both even- and odd-modes are thoroughly investigated in terms of two extracted per-unit-length transmission parameters, i.e., phase constants and characteristic impedances. As such, frequency-dependent coupling between the lines of the finite-extended PNCML is exposed via two dissimilar impedances. Meanwhile, two phase constants try to be equalized at a certain frequency by properly adjusting the slit depth and periodicity, aiming at realizing the transmission zero. Further, equivalent J-inverter network parameters of this finite-length PNCML are derived to reveal the relationship between the transmission zero and harmonic resonance. By allocating this zero to the frequency twice the fundamental passband, one-stage and two-stage PNCML filters are then designed, fabricated and measured to showcase the advantageous capacity of the proposed technique in harmonic suppression. key words: coupled microstrip line, periodic structure, bandpass filter, even- and odd-mode, transmission zero

1.

Introduction

Parallel-coupled microstrip line has been gaining a wide application in the bandpass filter design due to its distributed and tightened coupling behaviors [1]. However, this type filter suffers from spurious harmonic passbands at the upper stopband. So far, much effort has been made to reject or attenuate these harmful harmonic passbands [2]–[12]. For this purpose, the two effective approaches have been explored by equalizing the phase velocities and differentiating the traveling routes of even- and odd-mode. In [2], an overcoupled resonator was constituted to extend the odd-mode phase length, thus compensating the difference in phase velocity between two modes. Recently, the strip-width modulation technique is developed to make up the “wiggle-line” bandpss filter with good harmonic suppression over a wide frequency range [3]. In parallel, the corrugated coupled microstrip line [4] is presented to extend the actual traveling path for the odd-mode such that the 1st-harmonic be suppressed by equalizing the two phase velocities of the dominant even and odd modes. Similarly, in order to achieve the Manuscript received October 1, 2004. Manuscript revised January 13, 2005. † The authors are with the School of Electrical & Electronic Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798. a) E-mail: [email protected] b) E-mail: [email protected] DOI: 10.1093/ietcom/e88–b.6.2377

same goal, the square grooves [5] are periodically etched on the parallel-coupled lines and the coupled meander lines are constituted [6]. In addition, the concerned harmonic passband can be effectively suppressed by suspending the dielectric layer above the ground plane or forming a backside aperture as discussed in [7]–[9]. On the other hand, the stepped-impedance resonator technique was proposed in [10] and it provides us with an alternative degree of freedom in controlling the frequency resonance responses, thereby effectively widening the stopband between the dominant and 1st spurious passbands in the filter design. By directly tapfeeding the half-wavelength transmission line resonators, the 1st spurious passband of the designed filter can also be suppressed with the emergence of the transmission zeros, as originally implemented in [11], [12]. The objective of this work is to explore the harmonic-suppressed parallelcoupled transmission line filters with periodical shape. In this aspect, it is commonly recognized as the most critical issue to characterize and/or extract the two per-unit-length transmission parameters of both even and odd modes propagating in the periodically nonuniform coupled microstrip line (PNCML) with varied unit shapes. As such, the exhibited frequency dispersion of these two per-unit-length parameters allows one not only to understand the operating principle of these filters but also to optimize their bandpass filtering behavior via simple and efficient synthesis procedure [1]. In [13], the spectral-domain approach is applied to analyze the effective dielectric constants of the even- and odd-mode in the PNCML. But, it does not take into account the frequency dispersion so that this approximate approach may not be suitable for CAD-based design of microwave circuits. Very recently, the finite difference time domain (FDTD) technique is employed to analyze such two frequency-dependent phase constants of PNCML [14]. But, unfortunately, no reported work to date has been carried out to calculate the even- and odd-mode effective characteristic impedances of the PNCML with any shaped configuration. In this work, our developed hybrid method of moments (MoM) and short-open calibration (SOC) technique [15], [16], namely, “MoM-SOC,” is extended to directly extract these two per-unit-length transmission parameters from the fullwave modeling of these PNCML with finite length. Obtained results of the uniform coupled lines are at first confirmed with the quasi-static ones [17]. These parameters are extracted to expose the basic guided-wave characteristics of

c 2005 The Institute of Electronics, Information and Communication Engineers Copyright


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 e,o   A + De,o 1 nπ + cos−1 L 2

the PNCML loaded with transverse slit in a wide frequency range. Further, the equivalent J-inverter network [18] of PNCML is formed and its network parameters are derived to demonstrate how the transmission zero and harmonic resonance vary with the slit depth and periodicity. In final, oneand two-stage PNCML filter design examples are given and the predicted S-parameters are evidently confirmed by the experimental results.

where Z0e,0o and βe,o are the even- and odd-mode characteristic impedances and phase constants, while n is an integer number.

2.

3.1 Per-Unit-Length Transmission Parameters

MoM-SOC Modeling of PNCML

Figure 1(a) depicts the cross-section of the uniform coupled microstrip line (CML), in which the two parallel-strip conductors are transversely spaced by the gap width (s). Figure 1(b) indicates the geometrical sketch of the finiteextended PNCML loaded with transverse slits in periodical interval. This PNCML is fed at two sides by the two uniform coupled-line feeders. In the MoM algorithm, the left- and right-side CML feeders are simultaneously driven at ports by a pair of delta gap sources [15]. In order to excite the even- and odd-mode separately, the even- or odd-polarity of voltage sources are impressed between the two separated strip lines at both left- and right-side ports. By applying the Galerkin’s technique, the current densities over the two strip conductors are numerically solved as a response of impressed port voltages. As detailed in [15], the two-port network parameters at ports are accordingly determined as the two port currents become known. Next, the two sets of even- and odd-mode short- and open-end standards are formulated in the same MoM, as described in [15], to carry out the SOC procedure. As such, the network parameters at two ports can be converted to those at the two terminals of the finite-length PNCML section to be considered, i.e., reference plane R1 and R2 , as shown in Fig. 1(b). Following [16], this PNCML section can be readily modeled in terms of a two-port ABCD matrix with the four elements of Ae,o , Be,o , C e,o and De,o for the even and odd modes, respectively. Thus, the two sets of effective perunit-length transmission parameters of this PNCML can be derived in the closed form [16]
Be,o Z0e,0o = (1) C e,o

βe,o =

3.

(2)

Extracted Parameters of PNCML

Based on the above MoM-SOC technique, the finiteextended PNCML line with varied depth (∆) of transverse slits in periodical interval is characterized. The two per-unit-length transmission parameters are numerically extracted to expose its guided-wave characteristics in comparison to those of the uniform coupled microstrip line (CML). Figure 2(a) and 2(b) depict the extracted characteristic impedance and normalized phase constant of the finite-length CML with ∆=0 mm and the PNCML line with ∆=0.4 mm under the fixed periodicity of T =1.0 mm, respectively. As shown in Fig. 2(a), the even- and odd-mode characteristic impedances of this PNCML rise up from 62.4 to 67.9 Ω and from 34.5 to 44.2 Ω in the non-synchronous

(a)

(a)

(b) (b) Fig. 1 (a) Cross-section of uniform coupled microstrip line (CML). (b) Top-view of periodically nonuniform coupled microstrip line (PNCML) with finite length.

Fig. 2 Extracted even- and odd-mode per-unit-length transmission parameters of uniform CML and PNCML (εr =10.8, h=0.635 mm, w=0.6 mm, s=0.2 mm T =1.0 mm, t=0.2 mm). (a) Characteristic impedances. (b) Normalized phase constants.

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way, respectively, as ∆ increases from 0.0 mm (uniform) to 0.4 mm. Meanwhile, it can be observed in Fig. 2(b) that the normalized odd-mode phase constant (βo /k0 ) increases significantly from 2.48 to 3.02 at 2.0 GHz, whereas its evenmode counterpart (βe /k0 ) slightly rises up from 2.82 to 2.98. In fact, for the uniform CML, a large portion of energy is mainly concentrated in the central gap region for the oddmode and moves to be distributed in the dielectric layer underneath the two strips for the even-mode [4]. Thus, in PNCML, as the odd-mode traveling path is enlarged, the effective phase velocity of the odd-mode becomes low and its corresponding effective phase constant increases more dramatically while the even-mode phase constant is almost stably unchanged. In addition, the results with circle markers, in Fig. 2, are derived in [17] for the uniform CML case (∆=0.0 mm) and they are found well matched to those from our MoM-SOC.

Fig. 3 Frequency-dependent graphs for adaptive allocation of the transmission zero to the desired frequency. (εr =10.8, h=0.635 mm, w=0.6 mm, s=0.2 mm, L=7.0 mm, T =1.0 mm, t=0.2 mm)

3.2 Transmission Zero of PNCML Following the pioneered work [2], the coupling zero between the two coupled lines with the finite length (L) in the uniform CML or PNCML may be produced at a certain frequency, i.e., transmission zero. Its location can be explicitly determined in the closed-form equation (3), in which the two sets of per-unit-length parameters for even and odd modes are extracted above. sin (βe L) Z0e = Z0o sin (βo L)

(3)

Figure 3 plots the frequency-dependent graphs of the two functions, Z0o sin (βe L) and Z0e sin (βo L), for adaptive allocation of the transmission zero by adjusting the periodicity (T) and/or slit depth (∆). Graphically, the transmission zero can be solved as the intersection point of the two relevant curves. As ∆ increases from 0.0 (uniform CML) to 0.4 mm, the transmission zero is found to reduce from 9.92 to 6.95 GHz to a great extent. This property will be utilized later on to effectively cancel the 1st-harmonic passband of the CML filter by properly allocating the transmission zero towards the 2nd-resonance of a half-wavelength microstrip line resonator [2]–[6]. Figure 4 shows the normalized phase constants as a function of the aspect ratio t/T at the frequency of f =6.95 GHz that is the intersected point of the two solid curves in Fig. 3. As the slit with ∆=0.4 mm is periodically etched out, both phase constants are simultaneously raised due to the well-known slow-wave property. As shown in Fig. 4, the odd-mode βo /k0 rises up with t/T much more quickly than its even-mode βe /k0 , thereby producing the intersection point of these two curves or equalizing these two phase constants around t/T =0.2. 3.3 J-Inverter Network of PNCML Now, let’s consider to characterize the PNCML section with N=7 finite slit cells that is driven by the two single

Fig. 4 Normalized phase constants versus ratio t/T . (εr = 10.8, h = 0.635 mm, w = 0.6 mm, s = 0.2 mm, f = 6.95 GHz)

(a)

(b) Fig. 5 Two-port microstrip-fed PNCML. (a) Schematic. (b) Equivalent J-inverter network.

microstrip lines. As shown in Fig. 5(a), the two coupled PNCML microstrip lines are oppositely cascaded with the external feed line at one port and open-circuited at the other port. As the two per-unit-length parameters are derived, the two-port impedance matrix of such a symmetrical PNCML structure with the length (L) can be derived as below. −j (Z cotβe L + Z0o cotβo L) 2 0e −j Z12 = Z21 = (Z0e cscβe L − Z0o cscβo L) 2

Z11 = Z22 =

(4) (5)

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As usual, the coupling property of this PNCML prefers to be characterized using an equivalent J-inverter network. As illustrated in Fig. 5(b), This network consists of a Jsusceptance (J) and two equal electrical lines (θ/2). As all the parameters of the above impedance matrix are calculated using equations (4) and (5), the two J-inverter network parameters can be obtained as comprehended in our previous work [18]. Figure 6(a) and 6(b) depict the derived normalized ¯ and equivalent electrical length J-inverter susceptance ( J) (θ/2) of the PNCML section with the slit depths of ∆=0 and 0.38 mm as well as the length of L=7.0 mm. As can be well seen in Fig. 6(a), the parameter varies as a nonmonotonic function of the frequency over the range of 2.0 to 14.0 GHz. In addition to the peak, it can be observed that this has the null frequency, in which the coupling degree between the two lines becomes zero [18]. Furthermore, this null is found to move down from fb0 =9.92 to fa0 =7.15 GHz as ∆ varies from 0 (uniform CML: case A) to 0.38 mm (case B). Figure 6(b) illustrates us that the electrical length (θ/2) increases from 48◦ to 360◦ as a quasilinear function of the frequency. In fact, the frequency of θ/2 = 180◦ indicates the 2nd resonant frequency or 1stharmonic passband in the design of CML or PNCML bandpass filter [2]–[6]. As ∆=0.38 mm and t=0.2 mm are se-

lected, this harmonic frequency is lowered from fbh =8.15 to fah =7.15 GHz. The latter one is the exactly same as the null frequency, i.e., fa0 =7.15 GHz. As a consequence, it implies us that the 1st-harmonic resonance can be completely suppressed if fa0 = fah =7.15 GHz is exactly valid. 4.

Harmonic Rejected Bandpass Filter Design

In this section, the above transmission-zero allocation technique is implemented to explore the harmonic-rejected PNCML bandpass filter. It is realized by suitably allocating the transmission zero to the 1st-harmonic passband as demonstrated in the above sections. Figure 7(a) and 7(b) show the layouts of the one-stage CML/PNCML bandpass filters, which are made up of the two cascaded CML/PNCML sections. Relevant S-parameters of these filters are simply calculated via cascaded transmission line theorem, as shown in Fig. 8. In the uniform CML filter, the transmission zero of fb0 =9.92 GHz is higher than its 2nd transmission pole of fbh =8.15 GHz, thus suffering from the 1st-harmonic passband. But, the 1st-harmonic passband of the PNCML filter is completely canceled since the transmission zero ( fa0 ) and 2nd pole ( fah ) are tuned to the same frequency, i.e., 7.15 GHz. Next, the one- and two-stage bandpass filters are optimally designed to have the same fundamental passband at the frequency of 3.6 GHz. Figure 9 illustrates the photograph and frequency responses of the fabricated one-stage

(a)

(b) Fig. 7 Layouts of the two one-stage bandpass filters. (a) Uniform CML filter. (b) PNCML filter. (εr =10.8, h=0.635 mm, w=0.6 mm, s=0.2 mm, L=7.0 mm, T =1.0 mm and t=0.2 mm) (a)

(b) Fig. 6 Frequency-dependent J-inverter network parameters of the PNCML structure in Fig. 5(a) with different slit depths. (a) Normalized J-susceptance. (b) Electrical line length.

Fig. 8 Predicted S-parameters of the one-stage CML and PNCML filters. (εr =10.8, h=0.635 mm, w=0.6 mm, s=0.2 mm, L=7.0 mm, T =1.0 mm and t=0.2 mm)

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(a)

stage bandpass filter example is further designed and fabricated. Figure 10(a) is the photograph of this filter and Fig. 10(b) shows the comparison between the predicted and measured results over a wide frequency range. Again, both them are well matched with each other. In particular, the measured insertion loss is really raised beyond 45 dB at the 1st harmonic passband around 7.20 GHz that is existed in the traditional uniform CML filter. 5.

(b) Fig. 9 Comparison between the predicted and measured S-parameters of the one-stage PNCML filter. (a) Photograph. (b) S-parameters. (εr =10.8, h=0.635 mm, w=0.6 mm, s=0.2 mm, T =1.0 mm and t=0.2 mm, ∆=0.38 mm, L=7.0 mm)

Conclusion

In this paper, guide-wave characteristics of the PNCML structure with multiple transverse slits in periodical intervals are thoroughly investigated in terms of two effective per-unit-length transmission parameters. Our effort is primarily made to predict the location of transmission zero of the finitely-extended PNCML section and further allocate it to suppress the 1st-harmonic passband in the design of microstrip bandpass filters. Optimized results with good harmonic suppression are evidently verified over a wide frequency range. Moreover, this PNCML filter has the compact size due to slow-wave propagation of PNCML even and odd-modes. References

(a)

(b) Fig. 10 Comparison between the predicted and measured S-parameters of the two-stage PNCML filter. (a) Photograph. (b) S-parameters. (εr =10.8, h=0.635 mm, 1st-stage PNCML: w=0.6 mm, s=0.2 mm, T =1.0 mm, t=0.2 mm, ∆=0.375 mm and L=7.0 mm, 2nd-stage PNCML: w=1.0 mm, s=0.6 mm, T =1.0 mm, t=0.2 mm, ∆=0.665 mm and L=7.0 mm)

filter. The measured results are in good agreement with the network-derived results, thus evidently verifying that the proposed technique can really suppress the 1st-harmonic passband that is harmfully existed in the traditional parallelcoupled microstrip line bandpass filter [1]. Finally, a two-

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[11] K. Wada, K. Nakagawa, O. Hashimoto, and H. Harada, “Technique for improving out-of-band characteristics of planar microwave filters using tapped resonators,” IEICE Trans. Electron., vol.E85-C, no.2, pp.391–399, Feb. 2002. [12] K. Wada, T. Ohno, K. Nakagawa, and O. Hashimoto, “Realization of low spurious responses by various bandpass filters using openended λ/2 resonators,” IEICE Trans. Electron., vol.E86-C, no.12, pp.2394–2402, Dec. 2003. [13] F.-J. Glandorf and I. Wolff, “A spectral-domain analysis of periodically nonuniform coupled microstrip lines,” IEEE Trans. Microw. Theory Tech., vol.36, no.3, pp.522–528, March 1988. [14] J.-N. Hwang and J.-T. Kuo, “FDTD analysis of periodically nonuniform coupled microstrip lines,” 2003 IEEE AP-S Int. Symp., vol.1, pp.741–744, June 2003. [15] L. Zhu and K. Wu, “Unified equivalent-circuit model of planar discontinuities suitable for field theory-based CAD and optimization of M(H)MIC’s,” IEEE Trans. Microw. Theory Tech., vol.47, no.9, pp.1589–1602, Sept. 1999. [16] L. Zhu, “Guided-wave characteristics of periodic coplanar waveguides with inductive loading: Unit-length transmission parameters,” IEEE Trans. Microw. Theory Tech., vol.51, no.10, pp.2133–2138, Oct. 2003. [17] Transmission Line Calculator, free software from Applied Wave Research, Inc., CA, USA, 2001. [18] L. Zhu, H. Bu, and K. Wu, “Broadband and compact multi-pole microstrip bandpass filters using ground plane aperture technique” IEE Proc. Microw. Antennas Propag., vol.149, no.1, pp.71–77, Feb. 2002.

Sheng Sun received the B.Eng. degree in information engineering from the Xi’an Jiaotong University, Xi’an, China, in 2001, and is currently working toward the PhD degree in microwave engineering at Nanyang Technological University, Singapore. His research interests include the study of full-wave modeling of planar integrated circuits and antennas, as well as numerical de-embedding techniques. He was awarded the Nanyang Technological University Scholarship Award for his PhD research (2002– 2005) and the Young Scientist Travel Grant (YSTG) from the 2004 International Symposium on Antennas and Propagation (ISAP’04) in Japan.

Lei Zhu received the B.Eng. and M.Eng. degrees in radio engineering from the Nanjing Institute of Technology (Now: Southeast University), Nanjing, Jiangsu, China, in 1985 and 1988, respectively, and PhD degree in electronic engineering from the University of ElectroCommunications, Tokyo, Japan, in 1993. From 1993 to 1996, he was with the MatsushitaKotobuki Electronics Industries, Ltd. in Japan as a research engineer. From 1996 to 2000, he was with the Ecole Polytechnique, University of Montreal, Canada, as a research fellow. Since July 2000, he has been with the School of Electrical & Electronic Engineering, Nanyang Technological University, Singapore, as an associate professor. His current research works/interests include the study of planar integrated filters, broad-band interconnects, planar periodic structures, planar antenna elements/arrays, as well as fullwave method-of-moments (MoM) modeling of planar integrated circuits and antennas, numerical de-embedding or parameter-extraction techniques, field-theory CAD synthesis and optimization design procedures. Dr. Zhu currently serves as an associate editor of the IEICE Transactions on Electronics and a member of editorial board of the IEEE Transactions on Microwave Theory Techniques. He received the Japanese Government (Monbusho) Graduate Fellowship from 1989 to 1993, the First-Order Achievement Award in Science and Technology from the National Education Committee in China in 1993, the Silver Award of Excellent Invention from the Matsushita-Kotobuki Electronics Industries, Ltd. in Japan in 1996, the Asia-Pacific Microwave Prize Award at the 1997 Asia-Pacific Microwave Conference in Hong Kong in 1997.