Broadband and Compact Coupled Coplanar Stripline Filters With Impedance Steps

2874 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 55, NO. 12, DECEMBER 2007

Broadband and Compact Coupled Coplanar StriplineFilters With Impedance Steps

Ning Yang, Member, IEEE, Christophe Caloz, Senior Member, IEEE,Ke Wu, Fellow, IEEE, and Zhi Ning Chen, Senior Member, IEEE

Abstract—Novel broadband and compact stepped-impedancecoupled coplanar stripline bandpass filters are presented, an-alyzed theoretically, and demonstrated experimentally. Thesefilters are based on impedance step, capacitive gap, broadsidecoupling, and inductive shorted strip discontinuities, which aremodeled in terms of impedance ( -) and admittance ( -) in-verters. Broadside coupled coplanar stripline is analyzed for thefirst time by the even-/odd-mode decomposition technique usingthe finite-element method. The broadband and compact natureof the filters is explained from the discontinuities and the coupledline structures. Specifically, cross coupling is used to enhance theselectivity of both the low and high cutoffs, leading to triplet andquadruplet cross-coupled broadband filters with finite-frequencytransmission zeroes. Due to their broad bandwidth, compact size,differential configuration, and low fabrication cost, the proposedfilters represent excellent solutions for codesigned RF and mi-crowave systems such as ultra-wideband transceivers.

Index Terms—Broadband filter, coplanar stripline (CPS), cou-pled line, even mode, impedance step, inverter, odd mode.

I. INTRODUCTION

BROADBAND filters have recently drawn significant re-search interest due the current context of ever-increasing

data-rate requirements and time-domain electromagnetic appli-cations. Parallel coupled transmission-line half-wavelength fil-ters have found wide applications in microwave systems. Theirparallel strips provide large coupling for a small spacing be-tween resonators, and the filters using them can achieve up to50% bandwidth [1]. Such filters can be implemented in var-ious alternative configurations. Interdigital filters using foldedcoupled lines were presented in [2]. A broadband filter con-sisting of two parallel-conductor shorted-circuit spurline res-onators was proposed in [3]. A bandpass filter with increasedbandwidth obtained from resonator coupling enhancement withthree-line coupling sections was reported in [4]. Another ap-proach enhancing coupling based on apertures etched in the mi-crostrip ground plane was described in [5]. In [6], a double-layer coupled stripline resonator structure was applied to realize

Manuscript received May 2, 2007; revised July 9, 2007. This work wassupported in part by the Natural Sciences and Engineering Research Council(NSERC) of Canada.

N. Yang, C. Caloz, and K. Wu are with the Département de GénieÉlectrique, Poly-Grames Research Center, École Polytechnique de Mon-treal, Montreal, QC, Canada H3C 3A7 (e-mail: [email protected];[email protected]; [email protected]).

Z. N. Chen is with the Institute for Infocomm Research, Singapore 117674(e-mail: [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TMTT.2007.909609

miniature broadband bandpass filters. A multimode stepped-impedance microstrip line resonator, combined with tight edgecoupling in the input/output stages, was presented in [7] to de-sign a broadband filter. Moreover, impedance steps have alsobeen used in broadband filters and impedance matching sys-tems. For instance, half-wavelength resonator waveguide fil-ters based on impedance steps with very wideband performancewere extensively described in [1].

In addition to the broad bandwidth, another trend in modernmicrowave wireless systems is hardware miniaturization. Suchminiaturization requires a codesign approach, where differentelements in a complex circuit are designed specifically to meetthe requirements of their immediate environment, leading to op-timal overall performance of the circuit. In particular, many RFcomponents such as amplifiers, mixers, and antennas are de-signed with balanced inputs/outputs. In this context, balancedtransmission lines and filters are highly desirable in order tominimize transition or balun loss and maximize bandwidth per-formance.

Due to its balanced and uniplanar natures, the coplanarstripline (CPS) [8] represents an ideal platform for differen-tial circuits [9]–[12]. In [14], a matched wideband antennaintegrated with broadside and four edge-coupled coupled-linedifferential mode filters is demonstrated. CPS is useful fordesigning balanced circuits, such as mixers, differential am-plifiers, and feeds for printed antennas and modulators, and iswidely used as interconnects in high-speed digital circuits andintegrated electrooptic components. Thus, it exhibits severalfavorable characteristics for codesigned system, includingthe possibility of suppressing baluns and transitions, therebyreducing cost and enhancing performances. Like the CPW, theCPS is a uniplanar structure and, hence, offers flexibility in thedesign of planar microwave and millimeter-wave circuits, inwhich, the components and devices can be surface-mounted inseries and/or shunt without resorting to via holes. Furthermore,a CPS can achieve higher characteristic impedances than aCPW and a microstrip can by a simple increase of the distancebetween the two strips. Finally, it is convenient to implementbroadside-coupled transmission lines, which is very useful indesigning a wide-bandwidth filter, as will be shown in thisstudy. CPS may support surface-wave and leaky-wave modesif the substrate is electrically thick, however, these parasiticeffects, as in other open planar structures, can be avoided bysetting the cutoff frequency of these modes above the operationfrequency range [8].

However, most CPS filters reported to date have essentiallybeen narrowband [11]–[13]. Filling this gap is the purposeof this study. In our previous paper [19], we presented three

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YANG et al.: BROADBAND AND COMPACT COUPLED CPS FILTERS WITH IMPEDANCE STEPS 2875

novel designs of broadband and compact CPS filters, based onhigh-low impedance steps, edge-coupled CPS, and resonatorcross coupling. In this paper, the impedance-step, gap-coupling,and shorted-strip CPS discontinuities are analyzed theoreti-cally, and their equivalent - and -inverter parameters areextracted versus frequency and as the function of the layoutparameters. A novel broadside coupled-line CPS is investigatedfor the first time with even-/odd-mode analysis. Symmetricallycoupled CPS combined with the CPS discontinuities describedabove, is utilized to design broadband and high-selectivitybandpass filters. A detailed synthesis of these filters along withappropriate tuning procedures is presented. Compared with theedge-coupled CPS filter in [19], these filters, in addition to theirintrinsic balanced nature and broadband operation, providesharp wide stopband and compact circuit size.

The organization of this paper is as follows. Section IIrecalls the fundamental characteristics of CPS and describesthe impedance-step, gap-coupling, and inductive shorted-stripdiscontinuities in terms of their equivalent impedance andadmittance inverters. In Section III, a novel broadside cou-pled CPS is proposed and analyzed with the even-/odd-modeapproach based on the quasi-static finite-element method(FEM). In Section IV, two wideband CPS filters are designedwith double high–low impedance transitions, gap coupling, andbroadside coupled CPS sections. A complete theoretical anal-ysis followed by synthesis procedures is given. In Section V,the gap coupling between the CPS resonators is applied toform triplet and quadruplet cross-coupled broadband filterswith finite-frequency transmission zeroes, which is used toenhance the selectivity in the lower and upper cutoffs. Thetheory and synthesis procedures of these filters are validated bythe simulations and measurements.

II. CHARACTERISTIC OF CPS

Since the CPS filters have inherent discontinuities in the de-sign, their analysis will first be presented here, and their usewill become apparent in subsequent sections. After a brief re-call on CPS, the three types of discontinuities used will be dis-cussed: impedance step, capacitive gap coupling, and inductiveshorted strip.

A. Transmission-Line Characteristics

CPS offers several advantages over microstrip fabrication.In planar fabrication technology, process limitations imposecertain restrictions on the conductor’s minimum width andseparation. This usually confines the range of the transmissionline’s characteristic impedance. For microstrip fabrication, thehighest attainable impedance is approximately 110 , limitedby the narrowest conductor width [8]. However, CPS canachieve higher characteristic impedances by varying not onlythe conductor width but also the separation between the strips,where the highest impedance can reach approximately 250[8]. Fig. 1 shows a cross-sectional view of a CPS structure,with the - and -field distributions. CPS can be regarded asthe complementary structure of a coplanar waveguide (CPW).By using a conformal mapping method, the expression of the

Fig. 1. Field distributions in the cross section of a CPS.

Fig. 2. Equivalence between an impedance step and an impedance inverter.

effective dielectric constant and characteristic impedance of aCPS have been derived in closed form and are available in [8].

B. Discontinuities

First of all, the impedance step as shown in Fig. 2 is discussed.This structure, along with its induced transmission-line discon-tinuity, is equivalent to an ideal impedance -inverter. They arerelated by junction VSWR, where both circuits exhibit the samecoupling level at the transition junction [1]. The junction VSWRfor the impedance step of Fig. 1 from a low impedance of toa high impedance of is given by

(1)

where represents the reflection coefficient at the discontinuityjunction.

On the other hand, the junction VSWR for the idealimpedance -inverter with the characteristic impedance of ,as shown in Fig. 2, is obtained by

(2)

where , and stand for the input impedance lookinginto the inverter and the characteristic impedances of the con-necting transmission lines, respectively, and choose when

and choose when , which enforcesthe VSWR always to be bigger than one. Equating (1) and (2)yields

(3)

In (3), the high-to-low impedance ratio is defined as theimpedance ratio. Since the electrical length of an impedancestep is zero, it operates as an impedance inverter over all fre-quencies, while an ideal 90 impedance inverter is only validover a small bandwidth. However, (1) does not consider thediscontinuity electromagnetic effects in the step impedanceand only gives an approximation corresponding to an idealizedsituation. For practical filter design, it is suggested to use afull-wave simulator to select the step impedance for the same


Fig. 3. Equivalence between a capacitive gap and an admittance inverter.

Fig. 4. Normalized J-inverter admittance versus frequency extracted by full-wave analysis for different gap sizes g with w = 1 mm, s = 0:3 mm, and" = 10:2; h = 1:27 mm.

Fig. 5. Equivalence between an inductive shorting strip and an impedanceinverter.

insertion loss as the original inverter circuit, where the couplingcoefficient is inversely proportional to the impedance ratio.

The second discontinuity investigated is the capacitive gap-coupled CPS structure, as shown in Fig. 3, which can be likenedto a -inverter. The -admittance values are extracted fromfull-wave simulations by equating the ABCD matrix of the gapat ports – with an ideal -inverter [14]. Its normalized valuefor various strip separations is plotted in Fig. 4 versus fre-quency and for different gap widths. As increases, the charac-teristic impedance of the coupled line increases too, along withthe equivalent normalized value.

The third type of discontinuity is the strip discontinuity shownin Fig. 5. The shorted strip between the CPS conductors, mod-eled by a shunt inductance, can be used as a -inverter. The

-inverter impedance values are extracted in a similar way asin the case of the gap discontinuity. Its normalized values versusfrequency for various shapes and lengths of the shorting stripsare plotted in Fig. 6. As and the length of the shorting stripincrease, the equivalent value also increases.

III. BROADSIDE-COUPLED CPS STRUCTURE

To the best of the authors’ knowledge, the broadside-coupledCPS has not been covered in any literature. The even-/odd-modeapproach will now be used to analyze coupled CPS for the caseof symmetrically coupled lines, that is, identical lines of equalcharacteristic impedances. The problem of asymmetric coupledlines may be solved by using the more complex - and -modes

Fig. 6. Normalized K-inverter impedance versus frequency extracted by full-wave analysis for different gap sizes g and length of the shorting strip, withw = 1 mm and " = 10:2; h = 1:27 mm.

Fig. 7. Perspective view of symmetrically broadside-coupled CPSs (1–1’ and2–2’ are the two ports of the CPS on the top layer, and 3–3’ and 4–4’ are thetwo ports of the CPS on the bottom layer).

[16], in contrast to the even and odd modes in typical symmet-rically coupled lines.

Fig. 7 shows a 3-D transparent view of a pair of symmetri-cally coupled CPS structure. 1–1’ and 2–2’ are the ports of theCPS on top of the substrate, while 3–3’ and 4–4’ are the otherCPS ports on the bottom of the substrate, where broadside cou-pling takes place between the top and bottom layers. Fig. 8(a) isthe cross-sectional view with -field distribution of the coupledCPS when the two CPS lines are evenly excited, correspondingto an equivalent magnetic wall along the center of the substrate.Its equivalent capacitance network is shown in Fig. 8(b). Theodd-mode excitation case is shown in Fig. 8(c), correspondingto an electrical wall placed along the center of the substrate,thereby resulting in an equivalent capacitance network shownin Fig. 8(d).

In order to electrically characterize the broadside-coupledCPS structure, the even- and odd-mode capacitances have to bedetermined by employing a capacitance matrix that describesthe coupling phenomena occurring between the various con-ductors. The capacitance matrix can be generated with a 2-Dquasi-static field solver or by conformal mapping analysis.In this paper, Ansoft Maxwell 2-D, which solves the Laplaceequation based on a 2-D FEM, is used to extract the capacitancematrix of the coupled pairs. By using the model shown in Fig. 9,the problem is simplified with an electric wall placed in the


Fig. 8. Even- and odd-mode excitations and electrical field distributions ofthe broadside-coupled CPSs and equivalent circuits. (a) Even-mode excitationand field. (b) Corresponding capacitance network. (c) Odd-mode excitation andfield. (d) Corresponding capacitance network.

Fig. 9. Simplified configuration for 2-D FEM analysis.

symmetrical plane of the coupled structure. The FEM utilizesa discrete calculation space to solve the potentials at any pointwith given boundary conditions (e.g., a specified potential atthe strips and zero potential at the ground, electric wall, andboundary of the space). Finally, the 2 2 capacitance for asystem of two conductors

(4)

defined as

(5)

where and are the per-unit-length charges on the twoconductors and and are the potential on each conductor,is directly obtained from the stored electrostatic energy. For thesymmetrically coupled lines, both the per-unit-length self-ca-pacitances and mutual capacitances are equal, i.e.,and .

Equation (5) may be rewritten as

(6)

(7)

Therefore, the self-capacitance and mutual capacitance have thefollowing relationships with the capacitances shown in Figs. 8and 9:

(8)

(9)

where represents the capacitance between one of the stripconductors and the virtual ground (0-V potential by symmetry)in the absence of the other strip conductor, while representsthe capacitance of the two strip conductors in the absence ofthe ground conductor. With even-mode excitation, as shown inFig. 8(a) and (b), the even-mode capacitance is obtained as

(10)

For odd-mode excitation as shown in Fig. 8(c) and (d), the odd-mode capacitance is obtained as

(11)

For TEM transmission lines, the electrical characteristics ofthe coupled lines can be determined from the above obtainedcapacitances and the velocity of the propagation in the line.However, the propagation for CPS is related to a scenario ofa quasi-TEM because the medium is inhomogeneous and thephase velocities of the even and odd modes are therefore dif-ferent. If the substrate of the coupled CPS structure is replacedby air , it becomes a pure TEM line, and we obtain

(12)

where is the speed of light, and are the per-unit-length even- and odd-mode capacitances of the coupled lines,which can be extracted from the above-mentioned quasi-staticmethod, and and are the per-unit-length even- and odd-mode inductances, which are equal for any nonmagnetic dielec-tric materials ( . By solving (12) forand , the even- and odd-mode characteristic impedances areobtained as

(13)


Similarly, the effective propagation constants are expressed as

(14)

from which the effective dielectric constant of the coupled CPSis obtained as

(15)

Since the medium of the CPS structure considered here is inho-mogeneous and only quasi-TEM, a coupled line of length ofhas dissimilar even- and odd-mode electrical lengths given by

(16)

and are responsible for nonperfect isolation in a coupled-linecoupler application. Finally, the voltage coupling coefficient,which corresponds to the maximum coupling level, is given by[16]

(17)

The even- and odd-mode characteristic impedances, along withthe effective dielectric constants of the coupled CPS are plottedin Fig. 10(a) and (b) versus the normalized strip widthand gap between the strips.

If the CPS structure is perfectly symmetric, it supports onlythe differential mode described above. Therefore, only this dif-ferential mode is considered in this paper. As in any symmetricdifferential structure, a weak parasitic common mode may beexcited in practical implementations.

In order to complete the electrical characterization of thebroadside-coupled CPS structure, the impedance and admit-tance matrices are to be derived. A simple approach is torepresent a pair of symmetrically coupled lines in an inhomo-geneous dielectric medium for the structure shown in Fig. 7 bya four-port coupler, as shown in Fig. 11. The correspondingimpedance parameters are derived in [15] and given as

(18a)

(18b)

(18c)

(18d)

IV. BROADBAND CPS FILTERS

This section presents and analyzes three impedance-step CPSfilters based on gap-coupled and broadside-coupled resonators.

Fig. 10. Even- and odd-mode parameters for coupled CPS. (a) Characteristicimpedances: " = 10:2; to w=h =0.3 to 1.6, and g=h =0.15 to 1.25.(b) Effective dielectric constants " = 10:2; w=h = 0.3 to 1.6, and g=h =

0.15 to 1.25.

Fig. 11. Symmetrically coupled transmission lines.

Fig. 12. Layout of the edge-coupled impedance-step CPS filter. From [19].

A. Stepped-Impedance Gap-Coupled Filter

Fig. 12 shows the layout of the first proposed gap-coupledCPS filter. This filter includes a double transition from alow-impedance port to a high-impedance transmission-linesection and a gap in the center of the high-impedance line


Fig. 13. Equivalent circuits for the high-impedance-section resonator inFig. 12. (a) Global. (b) Even (e). (c) Odd (o). From [19].

section. A standing wave, which is a result of any discon-tinuity along a transmission line, is subsequently generatedin the high-impedance section sandwiched between the twolow-impedance sections. The external coupling of the filter isrealized by the stepped impedance coupling. As explained inSection II-B, the impedance step is equivalent to a -inverter.Compared with other kinds of inverters, such as capacitorsand inductors, the coupling of this inverter very strong, be-cause the transmission level is high, having values around0.5–1.2 dB for an impedance ratio of 2:1 to 3:1. To determinethe resonant frequency of the high-impedance section, a loosecoupling mechanism, which does not greatly load the circuitto yield influence on the resonant frequency, is required forexternal excitation. From Section II-B, the coupling level isproportional to the inverse of the impedance ratio. Thus, aninfinite impedance ratio is required in the ideal case, indicatingthat the low-impedance section should have zero impedance,i.e., a short circuit. Therefore, the high-impedance section isterminated with a short circuit, as shown in Fig. 13(a). Theequivalent even and odd circuits are shown in Fig. 13(b)and (c), respectively.

For the even mode, the input impedance looking into theopen-ended circuit is

(19)

where is the phase constant of the CPS line. Thus, the cor-responding even-mode resonance frequency is given by thecondition , i.e.,

(20)

which indicates that the line is a quarter-wavelength resonator,irrelevant to its characteristic impedance. For the odd mode, theinput impedance looking into the short-ended circuit is

(21)

Fig. 14. Variation of the normalized odd-mode resonance frequency (f =f )with C and Z . From [19].

where is the capacitance of the coupling gap. The corre-sponding odd-mode resonance frequency is given by the con-dition

(22)

which indicates that the odd-mode resonance frequency de-pends on both the characteristic impedance of the line andon the value of the capacitance. Fig. 14 plots the variation ofodd-mode resonant frequency versus the high-impedance-sec-tion impedance and coupling capacitance .

In the case of the structure of Fig. 12, the even-mode reso-nance given in (20) is constant (quarter wavelength), while theodd-mode resonance varies with both coupling capacitance andtransmission-line impedance, as seen in (22). The two effectsare seen in Fig. 14. First, as the coupling capacitance increasesfor given impedance, is shifted to lower frequencies. Second,for a given coupling capacitance, is shifted to lower frequen-cies as impedance is increased.

From these analyses, the high-impedance section can be re-garded as a dual-mode resonator, and the filter is a structurewhere the resonator is externally excited with two impedancesteps. The even- and odd-mode reflection coefficients of thefilter are given by

(23)

respectively, where is the port’s impedance. This yields thefollowing expression for the return loss [16]:

(24)

showing that matching is achieved under the condition

(25)


Fig. 15. Equivalent circuit for the prototype of the filter in Fig. 12.

Substituting (19) and (21) into this expression and defining theimpedance ratio yields the frequency of perfectmatching for this filter in Fig. 12, which is given as

(26)

This transcendental equation has zero, one, or two solutions,depending on the parameters and , and represents a tradeoffbetween in-band insertion loss and bandwidth.

The filter can be synthesized with traditional filter designtechniques, where, as illustrated in Section II-B, the impedancesteps and capacitive coupling gap work as - and - inverters,respectively. Thus, the filter in Fig. 12 can be represented byalternative - and -inverters along with quarter-wavelengthresonators, as shown in Fig. 15.

The synthesis equations are then as follows [1]:

(27a)

(27b)

(27c)

where is the fractional bandwidth defined byand are element values of the low-pass filter pro-

totype, and, since the two resonators are quarter-wavelength,we have the susceptance slope parametersand reactance slope parameters . Substi-tuting (3) into (27a) and (27c), the characteristic impedancesof high-impedance sections are obtained; the normalized -in-verter admittance is achieved directly from (27b). Then, thecoupling gap parameters are decided from full-wave simulationand the methods explained in Section II-B.

From (27), it may be inferred that three unique characteris-tics make the proposed filter structure wideband. First, the re-actance and susceptance slope parameters are half of the valuesof those for a corresponding half-wavelength resonator. For thisreason, with the same coupling, the bandwidth of this filter dou-bles that of a filter with half-wavelength resonators. Second, theCPS resonators can take very small susceptance slope parame-ters by increasing the characteristic impedance, so that couplingrequirement for -inverter admittance can be greatly eased forthe same bandwidth. Finally, high values can be achieved bythe stepped impedance coupling, which directly leads to a widebandwidth.

However, the synthesis method based on - (or -) invertersis only valid for a relatively narrowband range (smaller than30%) when compared to with wideband range of this filter.To design a wideband filter over 50% fractional bandwidth,

Fig. 16. (a) Prototype of an edge-coupled CPS filter (ruler unit: cm).(b) S-parameters by circuit model, full-wave simulation, and measurement.W = 3 mm, W = 1:15 mm, L = 7:85 mm, S = 0:2 mmS = 0:2 mm, and S = 3:9 mm.

circuit simulations and full-wave simulations are requiredto account for the effects of discontinuities and to optimizethe whole structure. The fractional bandwidth of this filteris , using a two-order 0.5-dB-equal-ripple (Cheby-shev) low-pass filter prototype, and the element values are

, and . Thisfilter is designed with RT/Duroid 6010 substrate with dielectricconstant of 10.2 and thickness of 1.27 mm. The prototype isshown in Fig. 16(a), and the circuit simulated results (ADSSchematic) and full-wave simulations (IE3D) along with mea-sured -parameters are shown in Fig. 16(b). The measurementswere performed on a probe station with G-S probe and TRLcalibration. The measured results display some ripples in thepass band. These spikes are not present in the theoretical andfull-wave simulation results. They are probably produced bycalibration errors or by common-mode excitation in the G-Sprobe due to the proximity of metallic mechanical supports ofthe probe station.

B. Broadside-Coupled Filter

The bandwidth of the filter in the previous section is mainlyrestricted by the coupling level of the two quarter-wavelengthresonators. In order to yield a much wider bandwidth, a tightcoupling between the resonators is desired. The symmetricallycoupled CPS analyzed in Section III can be applied to the broad-band CPS filter design. Fig. 17(a) shows a top-view of a steppedimpedance coupled CPS section, along with its equivalent cir-cuit in Fig. 17(b).

The coupling structure in Fig. 17(b) can be analyzed byimposing an open circuit on ports 2 and 3 of the symmetri-cally coupled CPS structure of Fig. 11. From the definition of


Fig. 17. (a) Layout of a two-order symmetrically broadside-coupled CPS filter.(b) Equivalent circuit.

impedance matrix, the four-port -matrix parameter formu-lated in (18a)–(18d) is reduced to a 2 2 matrix given as

(28)

From the reduced -matrix, the frequency behavior of thefilter in Fig. 17 can be obtained by using the image impedancemethod. Thus, the image impedance in term of the -param-eters is given in (29), shown at the bottom of this page [16].The frequency range in which is a real number indicates apassband of the filter, while the range in which is imaginaryindicates a stopband. The calculated image impedances forboth the homogeneous medium and inhomogeneous mediumare displayed in Fig. 18. For homogeneous medium, whenthe coupled lines are long ( ) at , thisindicates a passband delimited by two cutoff frequenciesand ; however, when becomes imaginarywith an infinite value, which indicates a maximum attenuationfrequency at 2 (attenuation pole). However, the coupledCPS is not perfectly TEM, since from (16). Thus,as shown in Fig. 18, approaches at a frequencyfar below . Therefore, the broadside-coupled CPS filter ofFig. 17 displays a high selectivity in the upper cutoff. Thistransmission zero frequency is obtained by making asfollows:

(30)

Substituting 18(a)–(d) into (30) yields

(31)

Fig. 18. Image impedance of the coupled CPS section.

Fig. 19. Equivalent circuit for the prototype of filter shown in Fig. 17.

which shows that a transmission zero exists at a frequency wherethe even- and odd-mode electrical lengths satisfy the condition

, since is always larger than .Fig. 18 also shows a spurious passband ( is real) betweenand the harmonic passband . However, from the defi-

nition of image impedance, we know that the filter is matchedto port impedance if . At a frequency close to ,and with a coupled section of approximately long, canbe approximately equivalent to the inhomogeneous situation:

. Thus, the filter is matched at the bandif is satisfied. However, the filter is mis-matched at this band, since the image impedance in the spuriouspassband is much higher than in the first passband.Thus, the filter still exhibits a wide stopband up to , but withonly a slight degradation in attenuation levels.

As shown in the equivalent circuit of the filter in Fig. 19, thecoupled-line section acts simultaneously as two quarter-wave-length resonators and one admittance inverter, similar totraditional parallel coupled-line half-wavelength resonator fil-ters. The difference is that the traditional parallel coupled-linefilter employs half-wavelength resonators, while this filter is de-signed with quarter-wavelength resonators, thus reducing thefilter size by half. The resonators are alternately coupled withexternal ports by - and - inverters. The even- and odd-modeimpedances are related with the -inverter admittance by [16]

(32a)

(32b)

(29)


Fig. 20. Simulated results by both circuit simulator and full-wave simulation.W = 1:6 mm, W = 0:45 mm, L = 7:8 mm, S = 0:15 mm, andS = 2:4 mm.

The design equations are the same as (27a)–(27c) with thereactance slope and susceptance slope parameters expressed by

and , respectively.This gives an initial design, and then a circuit simulator is usedfor optimizing the filter performance. The Chebyshev responsefilter is designed with RT/Duroid 6010 substrate with a dielec-tric constant of 10.2 and thickness of 1.27 mm. Fig. 20 givesthe ADS simulated -parameters and those by full-wave simu-lation. As expected, a transmission zero exists at around 6 GHz.The -parameters by full-wave simulation are shifted to a lowerfrequency compared with that from circuit model analysis dueto some discontinuity effects of the impedance steps; in addi-tion, open end effects of the coupled lines are not consideredin the circuit simulations. The measurement is not conductedsince, from Fig. 17, it is impossible to directly probe the cir-cuit’s input and output ports as they are on different layers. Forhigher order (odd number) filters, the port can be arranged onthe same side of the printed circuit board (PCB), which givessome flexibility for system design.

V. COMPACT AND ENHANCED-SELECTIVITY CONFIGURATIONS

A. Third-Order Filter With Enhanced Higher Cutoff Selectivity

A third-order CPS bandpass filter is proposed in Fig. 21(a),which is composed of two stepped-impedance sections and twobroadside-coupled CPS sections. The detailed equivalent circuitis shown in Fig. 21(b). The addition of the broadside-coupledstrips along the high-impedance section of the filter in Fig. 17generates one additional transmission pole, which transformsthe filter into a triplet cross-coupled filter [18], as represented inFig. 21(c), where the direct and cross-coupled paths are repre-sented by solid and dashed lines, respectively. In Fig. 21(b), thecross coupling between resonators 1 and 3 is through the edgegap coupling, while the direct coupling between resonators 1, 3,and 2 is through the broadside coupling. Due to the capacitivenature of the cross coupling, an additional transmission zero ap-pears towards the upper side of the band and may therefore bedesigned to enhance the selectivity of the higher cutoff.

The equivalent circuit with alternative - and -inverters andquarter- and half-wavelength resonators is shown in Fig. 22.

Fig. 21. Layout and equivalent circuit model of the third-order filter.(a) Layout. (b) Transmission-line model. (c) Corresponding trisectioncross-coupled representation, where each node represents a resonator.

Fig. 22. Equivalent circuit for the cross-coupled third-order filter of Fig. 14.

Since the cross coupling is weak, we can start the design fromthe filter without considering the cross coupling. The synthesisequations are [1]

(33a)

(33b)

(33c)

(33d)

Since the first and third resonators are quarter-wavelength,while the second resonator is half-wavelength, we have the sus-ceptance slope parameters andreactance slope parameters . Combined with(3) and (32a) and (32b), the filter can be synthesized from thelow-pass filter prototype. This only gives approximate designparameters of the filter; a circuit optimization by ADS is re-quired to include the effects of the cross-coupling capacitance

and the three sections of short high-impedance transmis-sion lines as shown in Fig. 21(b). The cross coupling by the


Fig. 23. (a) Top view (ruler unit: cm) and (b) bottom view of the prototype ofthe designed cross-coupled third-order CPS filter. (c) Simulated and measuredS-parameters. W = 1:8 mm, W = 0:8 mm, L = 6:5 mm, S =

0:15 mm, S = 1:1 mm, and S = 3:9 mm.

capacitance introduces a transmission zero in the upper stop-band, which can be adjusted by the capacitive coupling value.In practical applications, a lumped capacitor can be surface-mounted on the circuit to freely control the transmission zerofrequency. The three short transmission-line sections also adddesign flexibility in the optimization process. The filter is againdesigned with RT/Duroid 6010 substrate with a dielectric con-stant of 10.2 and thickness of 1.27 mm. A Chebyshev responsefilter with 60% fractional bandwidth centered at 4.2 GHz is de-signed, simulated, and fabricated. The picture of the prototypeand full-wave simulated and measured -parameters are shownin Fig. 23. Full-wave simulations were carried out by using themoment of methods (IE3D), while the measurements were per-formed on a probe station with G-S probe and TRL calibra-tion. An additional transmission zero is observed at 8 GHz inthe results in addition to the one caused by the cross coupling(6.3 GHz). This transmission zero is due to the unequivalentphase velocities of the even and odd modes, which has alreadybeen explained in Section II-B. Around 9 GHz, a spurious pass-band occurs. This is because the second resonator of the filteris half-wavelength, and its first harmonic is at twice the funda-mental resonant frequency.

B. Fourth-Order Filter With Enhanced Lower andHigher Cutoffs Selectivity

To further improve the performance of the filter without in-creasing the form factor of the whole structure, a fourth-orderfilter is proposed in Fig. 24(a). In the previous third-order de-sign, a half-wavelength transmission line is coupled to the othertwo quarter-wavelength resonators by broadside-coupled line

Fig. 24. Layout and equivalent circuit model of a fourth-order filter. (a) Layout.(b) Transmission-line model. (c) Corresponding quad-section cross-coupledrepresentation, where each node represents a resonator.

sections. In this design, this resonator is split into two quarter-wavelength resonators by a shunt quasi-lumped inductor. Thisshunt inductance can easily be implemented by a chip induc-tance due to the uniplanar structure of the CPS or it can be real-ized by a printed narrow strip shorting the two signal lines. Theequivalent circuit of the entire circuit is shown in Fig. 24(b).The filter has been transformed into a quadruplet cross-coupledfilter [18], as represented in Fig. 24(c). Due to the additional de-gree of freedom, this structure can provide both upper and lowercutoff selectivity enhancement by a proper design. In addition,it has a higher order than the filter in Fig. 23, without occupyinga larger footprint. Consequently, this filter is extremely compactcompared with typical bandpass filters.

The equivalent circuit with alternative - and -inverters andquarter-wavelength resonators is shown in Fig. 25. The initialdesign of this fourth-order filter follows the same synthesis pro-cedure as shown in Section V-A, except that an additional -in-verter is required and all of the resonators are quarter-wave-length. First, the circuit parameters are found without consid-ering the capacitive coupling effect, since this coupling is gen-erally weak compared with the direct path. Then, circuit opti-mization is required to consider the cross coupling and the inter-connecting transmission-line sections [see Fig. 24(b)]. Finally,full-wave simulation is carried out to fine tune the circuit, con-sidering the discontinuity effects. To design a broadband filter,the inverter impedance of in Fig. 24, which is equivalent toa shorting strip from Section II-B, needs to be large to providesufficient coupling between resonators 2 and 3. This inverter isrealized by a meandered shorting strip as shown in Fig. 24(a).The extraction of the equivalent -inverter impedance of this


Fig. 25. Equivalent circuit for the cross-coupled fourth-order filter of Fig. 24.

Fig. 26. (a) Top view (ruler unit: cm) and (b) bottom view of the prototype ofthe designed cross-coupled fourth-order CPS filter. (c) Simulated and measuredS-parameters. W = 1:75 mm, W = 1 mm, L = 8:15 mm, S =

0:2 mm, S = 1:9 mm, and S = 1:7 mm.

kind of discontinuity has already been numerically studied inSection II-B.

The Chebyshev response filter with 80% fractional bandwidthcentered at 3.2 GHz is designed and fabricated with RT/Duroid6010 substrate [see Fig. 26(a) and (b)]. The -parameters ob-tained by both full-wave simulation and measurement are shownin Fig. 26(c). Since all of the resonators are quarter-wavelength,the first spurious passband occurs only at three times the centerfrequency. Along with the three finite frequency transmissionzeros generated, wide and deep stopband is observed.

VI. CONCLUSION

Several broadband and compact stepped-impedance cou-pled CPS filters have been presented, analyzed theoretically,and demonstrated experimentally. These filters are based onimpedance-step, capacitive-gap, and inductive shorted-stripdiscontinuities, which have been modeled in terms of - and

-inverters. Broadside coupled CPSs have been analyzed forthe first time by the even-/odd-mode decomposition approachusing a quasi-static method. The broadband and compact na-ture of the filters has been explained from the discontinuitiesand the coupled line structures utilized. Specifically, crosscoupling has effectively been used to enhance the selectivityof both the low and high cutoffs. In addition to their broadbandwidth and compact size, the proposed CPS filters havea differential input/output configuration, which provides anattractive solution for the codesign with other differential RFcomponents and printed antennas, which require differentialline interface. Therefore, these filters should find applicationsin various broadband systems requiring high integration andcompact footprint, such as ultra-wideband (UWB) transceivers.

ACKNOWLEDGMENT

The authors would like to thank S. Abielmona, Poly-GramesResearch Center, École Polytechnique de Montreal, Montreal,QC, Canada, for assistance in the preparation of the man-uscript, R. Brassard, Poly-Grames Research Center, ÉcolePolytechnique de Montreal, for fabrication of the proto-types, and A. Patrovsky, Poly-Grames Research Center, ÉcolePolytechnique de Montreal, or help with the probe-stationmeasurements. The authors would also like to thank ZelandSoftware, Fremont, CA, for donating IE3D licenses and theAnsoft Corporation, Pittsburgh, PA, for providing the Maxwellsolver.

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Ning Yang (M’03) was born in Taixin, JiangsuProvince, China, in December 1976. He received theB.Eng. and Ph.D. degrees from Southeast University,Nanjing, China, in 1998 and 2004, respectively.

He began his career as a Research Engineerwith the Institute for Communications Research,Singapore, in November 2001, and later became anAssociate Scientist with the Institute for InfocommResearch (I R), Singapore, where he conductedresearch on RFID systems, small and widebandantennas/circular polarized array for RFID and mo-

bile devices, electromagnetic bandgap structures, and planar integrated UWBdevices. In 2004, he was with the Center for Wireless Communications (CWC),National University of Singapore, Singapore, where he completed his doctoralthesis. From 2005 to 2006, was with the PCS Section, Motorola, as a SeniorRF Engineer to develop mobile devices. Since October 2006, he has been aPost-Doctoral Research Fellow with the Département de Génie Électrique,Poly-Grames Research Center, École Polytechnique de Montreal, Montreal,QC, Canada. His current research interests include substrate integrated meta-materials waveguides, leaky-wave antenna arrays, distributed amplifiers, andthin-film ferroelectric/ferromagnetic components.

Christophe Caloz (S’99–M’03–SM’06) receivedthe Diplôme d’Ingénieur en Électricité and Ph.D.degree from the École Polytechnique Fédérale deLausanne (EPFL), Lausanne, Switzerland, in 1995and 2000, respectively.

From 2001 to 2004, he was a Post-DoctoralResearch Engineer with the Microwave ElectronicsLaboratory, University of California at Los An-geles (UCLA), where he conducted research onmicrowave devices, antennas and systems, photonicbandgap (PBG) structures, and electromagnetic

metamaterials. In June 2004, he joined the Département de Génie Électrique,École Polytechnique de Montreal, Montreal, QC, Canada, where he is currentlyan Associate Professor, a member of the Poly-Grames Research Center, andthe Holder of a Canada Research Chair (CRC) entitled “Future IntelligentRadio-frequency Metamaterials” (FIRMs), associated with a novel CanadianFoundation for Innovation (CFI) infrastructure. He is also the Holder of theNatural Sciences and Engineering Research Council of Canada (NSERC)Strategic Project Grant “Novel Ultra-Wideband (UWB) Front-End TransceiverSystems.” He has authored or coauthored 200 technical conference, letter, andjournal papers, among which 35% were invited papers (over 45% of conferencepapers). He holds several patents. He authored the first unified textbook onmetamaterials, entitled Electromagnetic Metamaterials: Transmission LineTheory and Microwave Applications (IEEE Press, 2005). He has also authoredthree book chapters. He was the Guest Editor of the March–April 2006 “SpecialIssue on Metamaterials” of the International Journal for Numerical Methods(IJNM). He is a member of the Editorial Board of the IJNM, the InternationalJournal of RF and Microwave Computer-Aided Engineering (RFMiCAE),and Metamaterials. He also serves as a reviewer for Applied and WirelessComponents Letters, Electronic Letters, the Journal of Applied Physics,Applied Physics Letters, the Journal of Optics, the New Journal of Physics,

and other international periodicals. His current research interests include novelmetamaterials for millimeter-wave and optical applications, nonlinear andactive devices, thin-film/bulk ferroelectric and ferromagnetic components,UWB systems, terahertz technology, and numerical methods.

Dr. Caloz is a member of the IEEE Microwave Theory and Techniques So-ciety (IEEE MTT-S) Technical Coordinating Committee (TCC) MTT-15 andthe chair of the Commission D (Electronics and Photonics) of the CanadianUnion de Radio Science Internationale (URSI). He serves as a reviewer formany journals including the IEEE TRANSACTIONS ON MICROWAVE THEORY AND

TECHNIQUES, the IEEE MICROWAVE AND WIRELESS COMPONENT LETTERS,andthe IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has partici-pated in 15 courses, tutorials, and workshops around the world over the pastthree years. He has organized several focused sessions and workshops at inter-national conferences. He was the Technical Program Committee (TPC) chairof the International Symposium of Signals, Systems and Electronics (ISSSE),École Polytechnique de Montréal, July 30–August 2, 2007. He was the recip-ient of the 2007 IEEE MTT-S Outstanding Young Engineer Award. In March2004, he was the recipient the University of California at Los Angeles (UCLA)Chancellor’s Award for Postdoctoral Research.

Ke Wu (M’87–SM’92–F’01) is Professor of elec-trical engineering, and Tier-I Canada Research Chairin RF and millimeter-wave engineering with theÉcole Polytechnique de Montréal, Montréal, QC,Canada. He also holds a Cheung Kong endowedchair professorship (visiting) with Southeast Uni-versity, and an honorary professorship with theNanjing University of Science and Technology,Nanjing, China, and the City University of HongKong. He has been the Director of the Poly-GramesResearch Center. He has authored or coauthored

over 515 referred papers and several books/book chapters. He has served onthe Editorial/Review Boards of numerous technical journals, transactions,and letters, including being an Editor and Guest Editor. His current researchinterests involve substrate integrated circuits (SICs), antenna arrays, advancedcomputer-aided design (CAD) and modeling techniques, and development oflow-cost RF and millimeter-wave transceivers and sensors. He is also interestedin the modeling and design of microwave photonic circuits and systems.

Dr. Wu is a Fellow of the Canadian Academy of Engineering (CAE) andthe Royal Society of Canada (The Canadian Academy of the Sciences and Hu-manities). He is a member of the Electromagnetics Academy, Sigma Xi, andURSI. He has held key positions in and has served on various panels and in-ternational committees including the chair of Technical Program Committees,International Steering Committees, and international conferences/symposia. Heis currently the chair of the joint IEEE Chapters of the Microwave Theory andTechniques Society (MTT-S)/Antennas and Propagation Society (AP-S)/Lasersand Electro-Optics Society (LEOS), Montréal, QC, Canada. He is an electedIEEE MTT-S Administrative Committee (AdCom) member for 2006–2009 andserves as the chair of the IEEE MTT-S Transnational Committee. He was the re-cipient of many awards and prizes including the first IEEE MTT-S OutstandingYoung Engineer Award.

Zhi Ning Chen (M’99–SM’05) received the B.Eng.,M.Eng., and Ph.D. degrees from the Institute of Com-munications engineering (ICE), Nanjing, China, in1985, 1988, and 1993, respectively, and the Do.E. de-gree from the University of Tsukuba, Tsukuba, Japan,in 2003.

Since 1988, he has been with the Institute for Com-munications Engineering, Southeast University, CityUniversity of Hong Kong, with teaching and researchappointments including Teaching Assistant, Lecturer,Associate Professor, Research Fellow, and Post-Doc-

toral Fellow. From 1997 to 1999, he was with the University of Tsukuba as aResearch Fellow through a fellowship awarded by the Japan Society for Promo-tion of Science (JSPS). In 2001 and 2004, he visited the University of Tsukubaagain, this time sponsored by an Invitation Fellowship Program (senior level) ofJSPS. In 2004, he conducted research at the Thomas J. Watson Research Center,IBM, Yorktown, NY, as an Academic Visitor. In 1999, he joined the Institute forInfocomm Research, Singapore, as Member of Technical Staff (MTS) and thenbecame Principal MTS. He is currently a Principal Scientist and DepartmentManager for Radio Systems, Institute for Infocomm Research, Singapore. He is


concurrently appointed an Adjunct Associate Professor with the National Uni-versity of Singapore and Nanyang Technologies University, a Concurrent Pro-fessor with Nanjing University, and an Adjunct Professor with the EM Academyat Zhejiang University and Southeast University. He submitted the first pro-posal on UWB technology to Thematic Strategic Research Program of Agencyof Science, Technology and Research (A*Star), Singapore, in 2002, and now isleading one of projects of the program UWB Technologies and Pervasive Com-puting. He is coordinating the joint projects of A*Star and DSTA in the fieldsof Microwave and Electromagnetics. Since 1990, he has authored or coauthoredover 190 technical papers published in international journals and presented atinternational conferences. He holds three patents with 11 patents pending. Heauthored Broadband Planar Antennas (Wiley, 2005), coedited UWB Communi-cations (Wiley, 2006), and edited Antennas for Portable Devices (Wiley, 2007).In addition, he has contributed chapters to two books about antennas. He is theEditor for Field of Microwaves, Antennas and Propagation for the InternationalJournal on Wireless and Optical Communications (IJWOC). He has edited spe-cial issues for IJWOC, the International Journal of Antennas and Propagation,and the IEICE Transaction on Communications and has also reviewed manypapers for prestigious journals and conferences. His main research interests in-

clude applied electromagnetics as well as antenna theory and design. Particu-larly, he focuses on small and broadband antennas and arrays for wireless com-munications systems, such as WLAN/WiFi/WiMAX, multiinput multioutput(MIMO) and UWB systems, and RF imaging systems.

Dr. Chen founded the IEEE International Workshop on Antenna Technology(IEEE iWAT) and, as general chair, organized the IEEE iWAT: Small Antennasand Novel Metamaterials, 2005, Singapore. He is chairing the iWAT SteeringCommittee for future iWAT events. He was invited to give keynote speechesabout UWB and antennas at Loughborough Antennas and Propagation, U.K.,in 2005 and 2007, as well as at the China–Ireland International Conferenceon Information, Communication and Electronics in 2007. He is a chair for theSubcommittee of Emerging Technology at the 2006 IEEE Radio and WirelessSymposium. He was invited to give talks at UWB technology workshops of2003–2004 IEEE Radio and Wireless Conference. He has also organized andchaired many special sessions at many international events and served many in-ternational conferences as a member of the Technical Program Committee orInternational Committee.

Documents

Broadband and Compact Coupled Coplanar Stripline Filters With Impedance Steps