1164 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

Low-Reflection Bandpass FiltersWith a Flat Group Delay

Antonije R. Djordjevic and Alenka G. Zajic

Abstract—Recently, low-reflection low-pass filters with alinear phase characteristic have been reported. This paper is anextension of this study on bandpass filters, including their realiza-tions with lumped-element and distributed-parameter networks.A topology particularly suitable for microstrip techniques is pro-posed. It yields a compact design that enables miniaturization andis easy for manufacturing and tuning.

Index Terms—Bandpass filters, impedance matching, linearphase filters.

I. INTRODUCTION

RECENTLY, a class of low-pass filters has been proposedthat has a quasi-Gaussian amplitude characteristic and a

linear phase (i.e., a flat group delay), as well as a low reflec-tion both in the passband and stopband [1]. The filters werederived from a hypothetical distributed-parameter network. Abrief summary of these low-pass filters is given in Section IIand the work from [1] is extended to include numerical data fora practical design.

Similar bandpass filters have certain applications [2]. Hence,in this paper, we investigate low-reflection bandpass filters witha flat group delay. Section III presents a design procedure forbandpass lumped-element and distributed-parameter networksthat have a linear phase, along with a low reflection at allfrequencies. Section IV presents experimental results for amicrostrip implementation and compares them with simulationdata.

II. LOW-PASS PROTOTYPES

Fig. 1 shows a topology of the basic network for low-passprototypes. The network in Fig. 1 consists of identical cascadedcells, where only the first and last series branch are differentfrom other series branches. Resistors can exist in all branches(complete filter) or only in series branches (incomplete filter).1

We assume that the nominal impedances of both ports are .

Manuscript received February 24, 2004, revised May 1, 2004. This work wassupported in part by the Serbian Ministry of Science, Technology, and Devel-opment.

A. R. Djordjevic is with the School of Electrical Engineering, Uni-versity of Belgrade, 11120 Belgrade, Serbia, and Montenegro (e-mail:edjordja@etf.bg.ac.yu).

A. G. Zajic is with the School of Electrical and Computer Engineering,Georgia Institute of Technology, Atlanta, GA 30325-0250 USA (e-mail:alenka@ece.gatech.edu).

Digital Object Identifier 10.1109/TMTT.2005.845725

1This topology is dual to the one presented in [1]. The topology presented in[1] begins and ends with shunt branches, and the incomplete filter has resistorsonly in shunt branches. The topology shown in Fig. 1 is chosen because it resultsin a compact printed-circuit design of bandpass filters.

Fig. 1. Low-pass prototype ofRLC filters. (a) Complete filter. (b) Incompletefilter (N = 3).

The incomplete filter has fewer resistors than the complete filter,a higher attenuation at very high frequencies in the stopband,and a worse reflection coefficient.

For both the complete and incomplete filters, ,while and satisfy the relation

(1)

The number of cells ( in Fig. 1) is selected to controlthe phase characteristic up to a required frequency limit or,equivalently, up to a required insertion loss level .

We introduce the standard normalization of filter ele-ments [3, p. 139]. A normalized inductance is given by

(where is the 3-dBangular frequency), a normalized capacitance is given by

, and a normalized resistance is .Filter optimization can be performed with respect to various

criteria. One can control the linearity of the phase characteristic,group delay, matching, etc. For example, for the complete filter,the normalized resistances of the optimal cell that has a maxi-mally flat group delay are and , where

in an infinite array of cells. For the incomplete filter,the optimal cell has , where in an infinitearray of cells. For a finite number of cells , the optimal pa-rameters and are somewhat smaller.

A set of practical values, obtained after a combined opti-mization of the phase characteristic and group delay, is givenin Table I for various .2 The second column shows the nor-malized capacitance . Due to (1), , whereas

2In the optimization, all cells have identical element values.

0018-9480/$20.00 © 2005 IEEE

DJORDJEVIC AND ZAJIC: LOW-REFLECTION BANDPASS FILTERS WITH FLAT GROUP DELAY 1165

TABLE IOPTIMAL LOW-PASS FILTERS

. The normalized resistances in the first and last branchesare . All these data are identical for the complete andincomplete filters. The optimal parameters that define the nor-malized resistances and for the complete and incom-plete filters are given in the third and fourth columns of Table I,respectively.

The next column lists the normalized frequency up to whichthe phase characteristic is linear within an arbitrarily chosenerror of 3 . The following column shows the insertion lossat this frequency. The next column gives the normalized groupdelay at low frequencies and it is followed by the normalizedtolerance of the group delay up to . Finally, the last columnshows the worst case return loss for the complete/incom-plete filters.

As an example, the phase characteristic of an incomplete filteris practically linear ( 3 maximal deviation) up to an insertionloss of 17 dB for (i.e., up to ). The returnloss at any frequency is better than 14 dB.

The filter characteristics can be further optimized by lettingall element values vary independently. The tuning can increasethe bandwidth in which the group delay remains flat or it canimprove the matching. The spread of element values remainsvery low, even for high filter orders.

III. BANDPASS FILTER REALIZATIONS

We want to synthesize bandpass filters that have a flat groupdelay up to a prescribed attenuation level and within a spec-ified frequency band. We start from low-pass prototypes de-scribed in Section II and implement the standard low-pass-to-bandpass filter transforms [3, p. 157]. Each coil of the low-pass filter of normalized inductance is replaced by a seriesLC resonant circuit with and ,where is the normalized 3-dB bandwidth,

is the 3-dB bandwidth, and is the central angular fre-quency of the bandpass filter. (The normalization for the band-pass filter is similar as for the low-pass filter, except thatis used instead of .) Each capacitor of the low-pass filterof normalized capacitance is replaced by a parallel reso-nant circuit with and . Denor-malized values of the elements of the bandpass filter are fi-nally

and .Here, we follow an example of an incomplete filter withand . By transforming the low-pass filter shown in

Fig. 1(b), we obtain the bandpass filter shown in Fig. 2. Normal-ized elements of this filter are given in Table II.

We also develop a transmission-line realization of theincomplete filter. We replace each resonant circuit by

Fig. 2. Bandpass lumped-element incomplete filter (N = 3).

Fig. 3. Bandpass transmission-line incomplete filter (N = 3).

a quarter-wavelength transmission line. A series circuitis replaced by an open-circuited line whose

characteristic impedance is(i.e., the normalized char-

acteristic impedance is ). A parallelresonant circuit is replaced by a short-circuited linewhose characteristic impedance is

(i.e., the normalizedcharacteristic impedance is ). An in-complete bandpass filter of this kind is shown in Fig. 3 (for

). For the same bandwidth as for the lumped-elementfilter shown in Fig. 2, the normalized characteristic impedancesin Fig. 3 are and . (Note that

.) The normalized resistances are the same as inTable II.

Fig. 4 shows the corresponding scattering parameters andgroup delay for the lumped-element and transmission-line re-alization of the incomplete bandpass filter, as computed usingMicrowave Office.3 All capacitors, inductors, and transmissionlines are assumed lossless. The characteristics of the lumped-el-ement filter are asymmetric on the linear frequency scale. Thecharacteristics of the transmission-line filter are symmetric, butat the expense of having parasitic passbands because the scat-tering parameters are periodic functions of frequency.

3Microwave Office 2001, Applied Wave Research Inc., El Segundo, CA,2001.

1166 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 53, NO. 4, APRIL 2005

TABLE IINORMALIZED ELEMENTS OF THE INCOMPLETE BANDPASS FILTER

Fig. 4. Bandpass filter characteristics. (a) Scattering parameters. (b) Groupdelay. LE refers to the lumped-element filter. TL refers to the transmission-linefilter.

IV. MICROSTRIP IMPLEMENTATION

AND EXPERIMENTAL RESULTS

Our next goal is to design a compact microstrip bandpassfilter and exploit advantages of printed-circuit techniques.One problem with the structure shown in Fig. 3 is that thetransmission lines in series branches have a very high char-acteristic impedance . Hence, we replace each line bytwo open-circuited lines (of two times lower characteristicimpedance ) connected in series, as shown in Fig. 5. Thisresults in wider printed traces for these lines. The short-cir-cuited transmission lines have a low characteristic impedance

, i.e., their conductors are very wide.The lower conductor of each transmission line whose char-

acteristic impedance is can be grounded at all points. Suchlines are directly produced as wide microstrip lines. We note

Fig. 5. Modification of the structure shown in Fig. 4.

Fig. 6. Cross section of multilayer structure.

Fig. 7. Simplified layout of microstrip implementation (not drawn to scale).

that the three transmission-line conductors joined by the dashedlines in Fig. 5 are at the same potential at the junction point.These three conductors can be merged. Hence, we can produceeach transmission line of characteristic impedance as anarrow microstrip line whose ground conductor coincides withthe top (“hot”) conductor of the short-circuited transmissionline. This results in a multilayer microstrip structure that isshown in Fig. 6. The bottom metallization layer is ground. Thelow-impedance microstrip lines are printed on a thin substrate(middle metallization layer). The high-impedance microstriplines are printed on a thick substrate (upper metallizationlayer). The top view of the structure is shown in Fig. 7. Vias areused to bring up connections for surface mount device (SMD)resistors, which are soldered on the top of the printed structure.The wide traces in the middle layer are connected to the groundby another set of vias.

To properly take into account couplings among resonators,the filter was analyzed as a multiconductor transmission lineusing programs from [4] and [5]. These couplings reduce thebandwidth of the filter. This is advantageous as the span of char-acteristic impedances is lowered for a given bandwidth.

As an example, we present results for a filter whose cen-tral frequency is MHz, the relative bandwidth is

, and the nominal impedance is . The

DJORDJEVIC AND ZAJIC: LOW-REFLECTION BANDPASS FILTERS WITH FLAT GROUP DELAY 1167

Fig. 8. Computed (Th) and measured (Exp) response of a microstrip filter.

thickness of the lower substrate is 0.254 mm. The thickness ofthe upper substrate is 0.508 mm. The relative permittivity ofboth substrates is 2.33. The width of each high-impedance traceis 0.43 mm and the spacing between each pair of adjacent tracesis 0.8 mm. The width of each low-impedance trace is 3.0 mmand the spacing is 2.0 mm. Standard metal-film SMD resistors(1206 size) are used with and .

The low- and high-impedance lines have slightly differenteffective permittivities. To compensate for the different elec-trical lengths, the high-impedance lines should be terminated attheir open ends by small capacitances (0.4 pF). In addition, toequalize velocities of modal propagation, we place a compen-sating mutual capacitance (0.04 pF) between each pair of adja-cent high-impedance lines at their open ends. Hence, each pairof adjacent high-impedance lines is terminated in a -networkof capacitors. These capacitors can be technically created in avariety of ways. We use the standard technique of increasing thewidth of the traces and reducing the spacing between them nearthe open ends of high-impedance lines.

Fig. 8 shows the simulated and measured response of thefilter. Very good agreement has been achieved between the twosets of results. The filter has an insertion loss of approximately1.8 dB at the central frequency due to the conductor and dielec-tric losses. (The conductor losses dominate.) The first parasiticpassband of the filter is at approximately twice the central fre-quency (1.8 GHz). This passband can be further suppressed bymore carefully equalizing velocities of the guided modes. Thenext parasitic passband is at three times the central frequency(2.7 GHz). (This passband can be attributed to the periodicityof the scattering parameters of the filter shown in Fig. 5.) Thefilter is fairly insensitive to manufacturing tolerances and easyfor tuning.

ACKNOWLEDGMENT

The authors thank Dr. D. Tosic, School of Electrical Engi-neering, University of Belgrade, Belgrade, Serbia, and Mon-tenego, for his useful suggestions during the preparation of thispaper’s manuscript.

REFERENCES

[1] A. R. Djordjevic, A. G. Zajic, A. S. Stekovic, M. M. Nikolic, Z. A.Maricevic, and M. F. C. Schemmann, “On a class of low-reflection trans-mission-line quasi-Gaussian low-pass filters and their lumped-elementapproximations,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp.1871–1877, Jul. 2003.

[2] J. D. Rhodes, “Prototype filters with a maximally flat impulse response,”Int. J. Circuit Theory Applicat., vol. 17, pp. 421–427, 1989.

[3] A. I. Zverev, Handbook of Filter Synthesis. New York: Wiley, 1967.[4] A. R. Djordjevic, M. B. Bazdar, T. K. Sarkar, and R. F. Harrington,

LINPAR for Windows: Matrix Parameters for Multiconductor Trans-mission Lines, Software and User’s Manual. Norwood, MA: ArtechHouse, 1999.

[5] A. R. Djordjevic, M. B. Bazdar, G. M. Vitosevic, T. K. Sarkar, andR. F. Harrington, MATPAR: Scattering Parameters of Microwave Net-works with Multiconductor Transmission Lines, Software and User’sManual. Norwood, MA: Artech House, 1989.

Antonije R. Djordjevic was born in Belgrade,Serbia, on April 28, 1952. He received the B.Sc.,M.Sc., and D.Sc. degrees in electrical engineeringfrom the University of Belgrade, Belgrade, Yu-goslavia, in 1975, 1977, and 1979, respectively.

In 1975, he joined the School of Electrical En-gineering, University of Belgrade, as a TeachingAssistant. He became an Assistant Professor, As-sociate Professor, and Professor, in 1982, 1988,and 1992, respectively. In 1983, he was a VisitingAssociate Professor with the Rochester Institute of

Technology, Rochester, NY. Since 1992, he has also been an Adjunct Scholarwith the School of Electrical Engineering, Syracuse University, Syracuse, NY.In 1997, he was elected a Corresponding Member of the Serbian Academy ofSciences and Arts. His main area of interest is numerical electromagnetics, inparticular applied to fast digital signal interconnects, wire, and surface antennas,microwave passive circuits, and electromagnetic-compatibility problems.

Alenka G. Zajic was born in Belgrade, Yugoslavia,on April 28, 1977. She received the B.Sc. andM.Sc. degrees from the University of Belgrade,Belgrade, Serbia, in 2001 and 2003, respectively,and is currently working toward the Ph.D. degree atthe Georgia Institute of Technology, Atlanta.

From 2001 to 2003, she was a Design Engineerwith Skyworks Solutions Inc., Fremont, CA. Sheis interested in solving numerical problems inelectromagnetics applied to antennas and passivemicrowave components, design of RF and passive

microwave components, and communications.