244 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 17, NO. 4, APRIL 2007

Improved Design Equations of the Tapped-LineStructure for Coupled-Line Filters

Chih-Ming Tsai, Member, IEEE, and Hong-Ming Lee, Member, IEEE

Abstract—Coupled-line filters with tapped input and output arestudied in this letter. The tapped-line structures are used for trans-formation of the filter impedances to lower values. This makes thefilter more suitable for manufacture. However, the variation of thetransformed impedances around the passband central frequency,due to the tapped feeds, distorts the filter responses. In this letter,design equations are proposed to improve the design by taking intoaccount this variation. Finally, a filter design example is given tojustify the study.

Index Terms—Coupled transmission lines, distributed param-eter filter, microwave circuits.

I. INTRODUCTION

COUPLED-LINE filters are widely used in microwave cir-cuits and their design procedures had been well estab-

lished [1], [2]. Conventional filters were designed by severalsections of quarter-wavelength coupled lines. However, it wasfound that the coupled lines of the input/output sections usuallyhave tighter coupling than those of the others, especially for afilter with a wide bandwidth. Therefore, the gaps of edge-cou-pled lines might be very narrow and impractical. Increasing thecharacteristic impedances of coupled lines is helpful to reducethe coupling and widen the gaps [3], however, the conductor losswill also be increased due to the narrowed line widths.

In order to overcome the problem of tight coupling, tapped-line structures can be employed to replace the exterior cou-pled-line sections in the filter design [4]–[8]. Dishal [4] andWong’s methods [5] are straightforward, and in their studies thetapped-line filters were designed with the filter parameters suchas loaded ’s and coupling coefficients. However, experimentalmeasurement or electromagnetic field analysis is required to ad-just the distances between resonators, and the locations of feedpoints are also approximately determined. Although Cristal [6]provided an equivalent circuit of the tapped-line structure, hisdesign procedure might be somewhat complicated. The methodproposed in [7], [8] can simplify the design process since thereare explicit design equations for the coupled lines in the internalsections. However, the design equations of the tapped-line struc-ture in [8] are correct only for the central frequency of the pass-band and further optimizations with CAD tools are still required.

This letter starts from the review of coupled-line filter syn-thesis based on Richards’ transformation. The filter requires

Manuscript received May 31, 2006; revised December 6, 2006.The authors are with the Institute of Computer and Communication Engi-

neering, Department of Electrical Engineering, National Cheng Kung Univer-sity, Tainan 701, Taiwan, R.O.C. (e-mail: tsaic@mail.ncku.edu.tw).

Digital Object Identifier 10.1109/LMWC.2007.892933

Fig. 1. High-pass prototype filter.

additional impedance transformation networks at the outerstages to ensure a reasonable impedance level, which can beachieved by the tapped-line structures. Since the transformedimpedances are frequency-dependent, the impedance variationsshould be taken into account for the parameters of the adjacentresonators. Finally, a filter example was designed following theproposed design procedure, and its measurement results aregiven to verify the theoretical study.

II. COUPLED-LINE FILTER SYNTHESIS BASED

ON RICHARDS’ TRANSFORMATION

It is known that microwave bandpass filters (BPFs) can beachieved by the Richards’ transformation from the high-passprototype lumped filters, as shown in Fig. 1. For distributed fil-ters, the capacitors represent the quarter-wavelength open stubs,where their characteristic impedances are , and the termina-tion resistances are . Between the open stubs there are idealimpedance inverters.

Under Richards’ transformation, the cutoff frequency of thehigh-pass prototype filter is given by

(1)

where is the fractional bandwidth of the transformed BPF.The parameters of the filters can then be determined by the spec-ifications from the classical filter synthesis as

(2)

to (3)

where is the element value of the low-pass prototype filter.The coupled-line structure with the diagonal ports opened

shown in Fig. 2(a) is suitable to realize the internal sections ofthe filter, where and are the even- and odd-mode charac-teristic impedances and is the electrical length. The equivalentcircuit of the coupled-line structure is shown in Fig. 2(b).

For a BPF with a typical bandwidth of 10% or 20%, the char-acteristic impedances and derived by (2) will be several

1531-1309/$25.00 © 2007 IEEE

TSAI AND LEE: IMPROVED DESIGN EQUATIONS 245

Fig. 2. (a) Coupled-line structure and (b) its equivalent circuit.

hundred ohms for the 50- system terminations. The imped-ances are high and impractical for realization, and thus, addi-tional impedance transformation networks are required at theinput and output of the filter. This is done by the tapped-linestructure in this letter, and the details are discussed as follows.

III. TAPPED-LINE STRUCTURE

The tapped-line structure for the coupled-line filter designis shown in Fig. 3(a), where is the system impedance and

is the characteristic impedance of the tapped transmissionline. The electrical lengths and are defined at the centralfrequency of the BPF, . Let and

, the transformed impedance can be obtainedas

(4)

where is defined as the impedance ratio of , and andare the real and imaginary part of , respectively. andcan then be derived as

(5)

(6)

Since the transformed impedance is required to have only thereal part at , the electrical length can then be solved bysetting (6) to be zero as

(7)In order to avoid any distortion of passband responses, the vari-ation of the transformed impedance over the passband shouldalso be taken into account. The variation of around canbe described by its derivative with respect to frequency as

(8)

Fig. 3. (a) Tapped-line structure and (b) its equivalent circuit.

Fig. 4. Equivalent circuits of the tapped-line structure in the input of the cou-pled-line filter.

Although also varies with frequency, its effect on the pass-band responses is much minor and can be neglected. Therefore,the equivalent circuit of the tapped-line structure can be simpli-fied by the resistance at with the series reactancewhose slope with respect to frequency is , as shown inFig. 3(b).

The equivalent circuit of the coupled-line filter with thetapped-line structure in its input is shown in Fig. 4(a), whereis the average of the even- and odd-mode characteristic imped-ances of the coupled lines in the second section. The reactance

in series with the adjacent open stub can be includedin a single open stub with a characteristic impedance ofapproximately around , as shown in Fig. 4(b). By equatingtheir derivatives of reactance with respect to frequency at ,the characteristic impedance of the open stub, , in Fig. 4(b)can be obtained as

(9)

Therefore, the coupled-line filter with tapped input and outputfinally becomes the classical configuration of a BPF, as shownin Fig. 1, and the design equations follow those discussed above.

IV. FILTER DESIGN EXAMPLE

A fifth-order Chebyshev filter with 0.1-dB ripple and 20%bandwidth is presented here as an example. The passband cen-tral frequency is at 5 GHz. The filter is symmetrically config-ured and its equivalent circuit (only half the circuit is plotted)is shown in Fig. 5, where the system terminations are 50 .The open stubs in the internal sections are divided into two parts,which represent the elements for the two adjacent coupled-linesections. The characteristic impedance of the tapped transmis-sion line and the electrical length can be freely selectedfor a proper transformed impedance level. However, low ispreferred because the realizations of the resulted coupled-linesections are easier. In this example they are chosen to be

246 IEEE MICROWAVE AND WIRELESS COMPONENTS LETTERS, VOL. 17, NO. 4, APRIL 2007

Fig. 5. Equivalent circuit of the fifth-order coupled-line filter with tapped input/output.

Fig. 6. Comparison between the circuit simulation results. Solid-lines: designwith considering of X (proposed method in this letter); short-dashed lines:design without considering the variation ofX ; long-dashed lines: design withconsidering of X (compared to the responses with solid-line, fixed R isassumed).

32 and 42 . The electrical length is determined by(7) and is calculated to be 41.4 . All the circuit parame-ters in the internal sections of the filter are then calculated andthey are also given in Fig. 5.

The open stubs and impedance inverters in the equivalent cir-cuit of the filter can be replaced with the coupled-line sections,and they are found to be 62.5 and 40.3 forthe coupled lines in the second/fifth sections and 61.3and 41.5 for those in the third/fourth sections. Thedesigned filter is simulated using the circuit model in Fig. 4 forthe tapped-in structure, which does not include the variationand the results are shown in Fig. 6 as long-dashed lines. They arewell with the specifications. If the variation of is includedin the simulation, the results (shown as solid lines) still meet thespecifications. This proves that the effect of variation onthe filter is minor and negligible. For comparison, the simula-tion results of the filter design without considering the variationof for the tapped-line structure are also given in Fig. 6 asshort-dashed lines and, apparently, the passband responses arehighly distorted. Therefore, taking into account the variation of

for the tapped-line filter design is important and can effi-ciently improve the passband responses.

Finally, the filter design example was realized on a RogersRO3003 substrate, with a thickness of 0.51 mm, a dielectricconstant of 3, and a loss tangent of 0.0013. It was analyzedby electromagnetic simulations to include the discontinuity ef-fects, and finally it was optimized. Fig. 7 shows the photographof the filter and the electromagnetic simulation and measure-

Fig. 7. (a) Photograph of the filter and (b) the electromagnetic simulation andmeasurement results of the passband responses of the optimized circuit.

ment results. The insertion loss in the band center is about 1 dB.The coupled-line filter design with tapped input/output has beenachieved and successfully verified the theoretical study.

V. CONCLUSION

Coupled-line filter design with tapped input/output has beenstudied in this letter. The tapped-line structure is treated as animpedance matching network. It was found that the variationof the transformed impedance around the passband centralfrequency significantly affects the passband responses, andit should also be included in the filter design. Finally, a filterexample of 20% bandwidth is given to demonstrate the designprocedure, and the measurement results show good agreementwith the theoretical predictions.

REFERENCES

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[2] G. L. Matthaei, “Design of wide-band (and narrow-band) band-passmicrowave filters on the insertion loss basis,” IRE Trans. Microw.Theory Tech., vol. MTT-8, no. 11, pp. 580–593, Nov. 1960.

[3] D. Ahn, C.-S. Kim, and M.-H. Chung, “The design of parallel coupledline filter with arbitrary image impedance,” in IEEE MTT-S Int. Dig.,1998, vol. 2, pp. 909–912.

[4] M. Dishal, “A simple design procedure for small percentage bandwidthround-rod interdigital filters,” IEEE Trans. Microw. Theory Tech., vol.MTT-13, no. 9, pp. 696–698, Sep. 1965.

[5] J. S. Wong, “Microstrip tapped-line filter design,” IEEE Trans. Microw.Theory Tech., vol. MTT-27, no. 1, pp. 44–50, Jan. 1979.

[6] E. G. Cristal, “Tapped-line coupled transmission lines with applica-tions to interdigital and combline filters,” IEEE Trans. Microw. TheoryTech., vol. MTT-23, no. 12, pp. 1007–1012, Dec. 1975.

[7] C.-Y. Ho and J. H. Weidman, “Improved design of parallel coupled linefilters with tapped input/output,” Microw. J., pp. 127–128, Oct. 1983.

[8] K.-W. Kim, C.-H. Park, and S.-J. Han, “A new design procedure oftapped coupled-line filters,” in IEEE Antennas Propag. Soc. Int. Symp.Dig., 2004, vol. 3, pp. 2863–2866.