Theory and Design of Octave Tunable Filters With Lumped Tuning Elements

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 12, DECEMBER 2013 4353

Theory and Design of Octave Tunable FiltersWith Lumped Tuning Elements

Akash Anand, Student Member, IEEE, Joshua Small, Member, IEEE,Dimitrios Peroulis, Member, IEEE, and Xiaoguang Liu, Member, IEEE

Abstract—This paper presents octave-tunable resonators and fil-ters with surface mounted lumped tuning elements. Detailed the-oretical analysis and modeling in terms of tuning range and un-loaded quality factor are presented in agreement with sim-ulated and measured results. Based on the models, a systematicdesign method to maximize tuning ratio and optimize of theresonator is suggested. A resonator tuning from 0.5 to 1.1 GHzwith ranging from 90 to 214 is demonstrated using solid-statevaractors. A two-pole filter with a tuning range of 0.5–1.1 GHzwith a constant 3-dB fractional bandwidth (FBW) of 4 0.1%and insertion loss of 1.67 dB at 1.1 GHz is demonstrated alongwith a three-pole filter with a tuning range of 0.58–1.22 GHz witha constant 3-dB FBW of 4 0.2% and insertion loss of 2.05 dBat 1.22 GHz. The measured input third-order intermodulation isbetter than 17 dBmover the frequency range for the two-pole filter.

Index Terms—Combline filter, combline resonator, evanes-cent-mode (EVA) design, filter design, filters, full-wave simulation,measurement and modeling, modeling, tunable filters, tunableresonators, waveguide filters.

I. INTRODUCTION

R ECENTLY, there has been a growing interest in tunableRF/microwave filters. The driving parameters for these

filters are low loss, wide tuning, low power consumption, smallsize, fast tuning, high power handling, and ease of fabrication ata low cost. Various demonstrated tunable filters excel in someparameters at the cost of sacrificing other parameters. For ex-ample, planer microstrip filters with lumped tuning componentsare easy to fabricate, but the unloaded quality factor suf-fers due to the low of the planer resonators [1]–[3]. To achievehigher than planer structures, highly loaded 3-D evanes-cent-mode (EVA) resonators integrated with various types oftuning technologies have shown promising results [4]–[8]. Forexample, a tunable filter with a piezoelectric actuator showedunloaded quality factor of 700–300 at 4.6–2.3 GHz [4].Two EVA resonators with RF microelectromechanical systems

Manuscript received July 05, 2013; revised October 07, 2013; accepted Oc-tober 16, 2013. Date of publication November 05, 2013; date of current versionDecember 02, 2013. This work was supported by the National Defense Scienceand Engineering Graduate Program and the University of California at Davisunder the Hellman Fellows Program. This paper is an expanded paper from theIEEE International Microwave Symposium, Seattle, WA, USA, June 2–7, 2013.A. Anand, J. Small, and X. Liu are with the Department of Electrical and

Computer Engineering, University of California at Davis, Davis, CA 95616USA (e-mail: [email protected]; [email protected]; [email protected]).D. Peroulis is with the School of Electrical and Computer Engi-

neering, Purdue University, West Lafayette, IN 47907 USA (e-mail: [email protected]).Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TMTT.2013.2287674

(RF-MEMS) tuners, one with a switched capacitor network andthe other with a silicon diaphragm, achieved of 500–300 at5.58–4.07 GHz [5] and of 1000–300 at 24–6 GHz, respec-tively [8]. While these technologies attain high , complexityarises in fabrication due to the precise assembly needed to eitheralign the tuners with the EVA cavity’s vertical gap (typically inmicrometer) or insert the RF-MEMS switching network insidethe cavity.To avoid complicated fabrication and yet maintain high ,

3-D cavities are integrated with commercially available surfacemount tuning components as an alternative medium betweenlow planer structures and high 3-D cavities. In [9], pack-aged RF-MEMS switches mounted on a substrate integratedwaveguide (SIW) are used to get of 132–93 at 1.6–1.2 GHz.However, the tuning range is limited to a few states. RF-MEMScapacitor banks mounted on a combline resonator resulted in

of 1300–374 at 2.50–2.39 GHz with a limited tuning ratio(TR) of 1.05:1 [10], [11]. A surface ring gap resonator structureloaded with solid-state varactors reports a of 160–40 withlimited tuning range of 3.1–2.6 GHz [12], [13].The authors of this paper demonstrated a continuous octave

tuning substrate-integrated coaxial resonator with of 86–206and tuning range of 0.5–1.2 GHz using solid-state varactors andof up to 240 at 6.6 GHz using RF-MEMS varactors [14]. It is

the intention of this paper to further investigate this surface ringgap combline cavity. Compared to previous works [12]–[14],this paper presents an in-depth theoretical analysis of the reso-nant frequency, TR, and . Effects of the surface gap capac-itance and surface inductance are considered to show the com-promise between TR and . A design method is suggestedto maximize TR and optimize . This method demonstratestunable resonators and filters with higher and tuning rangethan the state-of-the-art with similar technologies. A two-polefilter with tuning range of 0.5–1.1 GHz and measured insertionloss of 1.67 dB at 1.1 GHz is demonstrated. This two-pole filtermaintains a constant 3-dB fractional bandwidth (FBW) of 4%.A three-pole filter with tuning range of 0.58–1.22 GHz with aconstant 3-dB FBW of 4 0.2% and measured insertion lossof 2.05 dB at 1.22 GHz is also demonstrated. The measuredinput third-order inter-modulation is between 17–30 dBm overthe frequency range for the two-pole filter.

II. SURFACE GAP COMBLINE RESONATOR

Fig. 1(a) and (b) schematically compares the cross section ofa vertical gap resonator with the cross section of the proposedsurface ring gap resonator. In the surface ring gap design, thecenter post is extended to short the bottom and top of the cavityand a ring gap is created on the surface to isolate the center

0018-9480 © 2013 IEEE

4354 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 61, NO. 12, DECEMBER 2013

Fig. 1. (a) Vertical gap resonator and (b) surface ring gap resonator. Both havethe same lumped equivalent circuit model.

post from the rest of the cavity’s ceiling. To tune the resonantfrequency, tuning elements are placed across the gap to varythe capacitance instead of having to change the physical gap.The figures show that the surface ring gap resonator still hasthe same lumped equivalent circuit model as the vertical gapresonator. The surface ring gap resonator in Fig. 1(b) resemblesa shorted coaxial transmission line and is commonly referred asa combline resonator.This design has several advantages compared to the vertical

gap. First, the structure is not limited to a particular tuning tech-nology, but allows for various types of tuning components, suchas RF-MEMS or solid-state varactors. Second, precise assemblyis not needed since the tuning components are surface mounted.This makes high-volume-manufacturing possible and the reso-nant frequency and tuning range independent of fabrication andassembly tolerance. Third, this structure is easily implementedon a low-cost printed circuit board (PCB), which can be in-tegrated with other RF components. However, as shown later,this type of tuning may not yield in the order of 1000 ifsolid-state varactors are used as the tuning components. Con-sequently, solid-state varactors are appropriate for applicationsthat need in the order of 100–500 without the added com-plexity of vertically aligned tuners.

III. THEORETICAL ANALYSIS AND DESIGN

A. Resonant Frequency

Fig. 2(a) shows the dimensions of the resonator where isthe height of the cavity, is the radius of the inner post, and isthe radius of the outer conductor. In Fig. 2(b), the resonator isapproximated with lumped elements where is the effectiveinductance of the cavity and capacitance in series withresistance models a varactor. The packaging and assembly,depending on the type of varactor (solid-state, RF-MEMS,etc.), can introduce other parasitics. To keep the model generic,package parasitics are excluded and the varactor is assumedto operate well below its self resonance so the impedanceresembles a series RC circuit. The conductors of the cavity,such as the inner post, outer wall, and the top and bottom walls,have resistances of , , and , respectively. A simplifiedmodel for the resonator is shown in Fig. 2(c), where andare the equivalent capacitance and equivalent resistance of allvaractors and is the sum of all other resistors.For an L–C resonator, the angular resonant frequency is

given as . Thus, from Fig. 2(c), and are neededto find the resonant frequency of the resonator. The capacitance

Fig. 2. (a) Cavity resonator with dimensions. (b) Cross-sectional lumpedmodel. (c) Simplified LC resonator.

is extracted from the equivalent impedance of the varactors, which is approximated as

(1)

(2)

(3)

where is the number of varactors placed in parallel on thering gap. Since the resonator resembles a shorted coaxial trans-mission line, is extracted from the input impedance ofa shorted transmission line

(4)

where

(5)

where is the characteristic impedance of the coaxial trans-mission line, is the dielectric constant of the material insidethe cavity, and is the phase constant. The above equations as-sume very low cavity losses, which is typically the case for thedimensions considered in this work. It has been shown that nearresonance, the reactance of is inductive [15] and the effec-tive inductance is approximated as

(6)

where is the speed of light. Assuming low losses, resonantfrequency can be solved numerically in

(7)

ANAND et al.: THEORY AND DESIGN OF OCTAVE TUNABLE FILTERS WITH LUMPED TUNING ELEMENTS 4355

TABLE ICAVITY DIMENSIONS USED IN HFSS SIMULATION

Fig. 3. Simulated and theoretical resonant frequency versus for the res-onator. The difference between theory and simulation (up to 9%) is addressedin Fig. 10.

Equation (6) shows that depends on , but the change isis small (up to 3% for the cavity parameters and frequency

range considered in this paper). Thus, neglecting the variationin , the change in or TR, is determined by the tuning factor,

, of the varactor

(8)

Ansys HFSS [16] is used to compare (7) with full-wavesimulations with the cavity parameters given in Table I. Thesurface gap radius and surface gap width is explained inSection III-C. In Fig. 3, resonant frequencies from (7) andHFSS simulations are plotted versus . Resonant frequenciesfrom theory is between 3%–9% higher primarily because (7)does not include the effects of the gap capacitance and surfaceinductance (discussed later in Sections III-C and III-D).

B. Quality Factor

The resonator’s is dependent on the quality factor of thevaractors and the quality factor of the cavity . Sinceis the resistance of the varactors, the series combination ofand [see Fig. 2(c)] gives

(9)

Fig. 4. Theoretical versus frequency for three fixed capacitances. As thecapacitance is tuned, increases with frequency.

Fig. 5. Simulated and theoretical versus frequency for the resonator. Theconductors of the cavity are simulated as perfect conductors to extract .

Substituting (7) into (9) and neglecting the variation in givesratio as

(10)

In Fig. 4, versus frequency for three different capacitancesare plotted. For a fixed capacitance, decreases as frequencyincreases. However, as the varactor is tuned to a lower capaci-tance, larger , (10) and (8) show that both and frequencywill increase. Fig. 4 also shows versus frequency as istuned from 2.67 to 0.63 pF: increases from 124 at 0.6 GHzto 248 at 1.2 GHz. In Fig. 5, (9) is compared to results fromHFSS simulations. To extract from simulation, the conduc-tors (post and walls of the cavity) are simulated as perfect con-ductors so that is approximately . The reason for higherin simulation is investigated in Section III-C.Since is the resistance of the cavity conductors, the series

combination of with [see Fig. 2(c)] gives

(11)


Fig. 6. Simulated and theoretical versus frequency for the resonator. Inorder to extract from HFSS, the resistance of the varactors are set to zero.

which is essentially the quality factor of the cavity with losslessvaractors. The sum of , , and in Fig. 2(b) is . From[17],

(12)

in which is the surface resistance. Surface current densityon the top and bottom of the cavity varies radially from theouter conductor to the inner post. Thus, integrating in the radialdirection gives

(13)

Summing (12) and (13) gives ,

(14)

Substitution (6) and (14) into (11) gives

(15)

HFSS is used to compare (15) with simulation with cavity pa-rameters from Table I. In order to get only, in simula-tion is set to zero and is varied from 2.67 to 0.63 pF. Fig. 6shows a plot of theoretical from (15) compared to simulated. The simulated is about 3%–4% lower than the theoret-

ical one. Fig. 6 also shows that radiation from the ring gap hasa negligible effect by comparing simulation results from a res-onator with a shielding cap covering the ring gaps and varactorson the surface. Note that the ring gap width is about 100–200times smaller than the wavelength range considered in simu-lation. Part of the 3%–4% loss in simulation comes from theresistance associated with the ring gap, which is considered inSection III-C and Fig. 12.The of a coaxial resonator is optimized when the ratio

is 3.6 [18]. This optimum ratio is again verified by plotting (15)in Fig. 7. The outer radius and height of a copper air cavity arefixed to mm and mm, while is varied from

Fig. 7. Theoretical with respect to . Optimal is achieved at.

Fig. 8. Simulated and theoretical versus frequency for the resonator. Sinceis much larger than , .

1 to 25 at four fixed frequencies: 0.1, 0.5, 1, and 2 GHz. Theoptimized ratio is evident at all frequencies.Using (9) and (15) and the series L–C resonator model in

Fig. 2(c), of the resonator is approximated as

(16)

The resonator’s is simulated in HFSSwith cavity parametersfrom Table I. Fig. 8 shows that simulation results are larger thantheory because was larger in simulation previously (Fig. 5).In this case, is much larger than due to the relatively large

(Figs. 6 and 5) so . In Fig. 8, is initially limitedby at lower frequencies (larger ), but starts to exceedat higher frequencies (smaller ). The limitations ofis analyzed in more detail in Section III-C.

C. Surface Ring Gap Capacitance

In Fig. 3, the theoretical resonant frequency based on thelumped model in Fig. 2(c) was up to 9% higher than the sim-ulated results. Part of this difference is due to the surface ringgap capacitance . As shown in Fig. 9(a), this capacitance isdependent on the ring radius , gap width , depth of the gap ,and the dielectric constant . Since is in parallel with ,from (7) is modified as

(17)


Fig. 9. Capacitance exists in parallel with the tuning components (varac-tors). (a) is dependent on the radius of the ring , width of the gap , depthof the gap , and the dielectric constant . (b) There is resistance due to thewall plates (dark color region) around the surface ring gap.

Fig. 10. Simulated and theoretical resonant frequency versus varactor capaci-tance when the effects of is included. The difference in frequency is reducedto within 1%.

where is given by (23). The value of can be extracted fromsimulated at two different values ( and ) for a fixedcavity structure from

(18)

where is the resonant frequency for and is the reso-nant frequency for . is extracted for simulated data fromFig. 3 to be pF. Compared to Fig. 3, whenpF is included in (17), Fig. 10 shows the difference in resonantfrequency is within 1% of simulation. The extraction of hereassumes that is constant as or varies. The variation

in (23) and how to minimize the variation is considered inSection III-D.Since is in parallel with , the total electromagnetic en-

ergy will be distributed between and . Previously, if theconductors were lossless, then . However, if islarge enough, then a significant portion of the electromagneticenergy will be stored in . Since the of is expected to behigher than (resistance associatedwith is small comparedto ), should exceed . This is, however, accomplishedat the cost of reduced tuning range. The equivalent impedance,

in parallel with , is given by

(19)

where

(20)

and is the resistance associated with shown in Fig. 9(b)and is initially neglected . of this impedance thenbecomes

(21)

and from (16) is modified as

(22)

Note that if is small and negligible, then (22) reduces to (16).A cavity with the parameters from Table I is simulated, expectwith , so that the effects of are more significant com-pared to . Fig. 11(a) plots (22) versus frequency whenvaries from 1 to 11 mm to show the effects of on and TR.As becomes larger, TR decreases and increases. Alsoplotted in the figure is theoretical at pF (max-imum ). When is larger, maximum for each curveis not limited by maximum theoretical (dotted line) and farexceeds , but as decreases, maximum for each curveapproaches maximum .In the case of ideal varactors or when ,

such as when no varactors are mounted in a static surface ringgap resonator [19], if . Fig. 9(b) shows thatdepends on the two thin layers of metal wall (dark regions

labeled as ring gap resistance in figure) that border the surfacering gap. These two thinwalls are simulated in HFSS as a perfectelectric conductors (PECs) or , while the rest of thecavity surface is still copper and with parameters fromTable I. Fig. 12 compares of this PEC wall’s simulation withthe previous case in Fig. 6 when all of the cavity surface wascopper (no PEC or ). As expected, Fig. 12 shows that

with is higher than with . Before, inFig. 6, was 3%–4% lower than theoretical (15), but with

, simulation is within 1% of theoretical .In Fig. 13, resistance of the varactors is in-

cluded in HFSS simulation to compare when and. Since dominates both and , Fig. 13 shows

that effects of is negligible for the parameters in Table I.Thus, the previous assumption of made in Fig. 11 is


Fig. 11. (a) Theoretical (solid lines), simulated (dashed lines), andtheoretical maximum (dotted line) versus frequency as is changed.(b) Extracted from simulation versus ring radius with cavity dimensionsfrom Table I, except with .

Fig. 12. is higher when the ring gap resistance is simulated with PECcompared to when ring gap resistance is simulated with copper. Simulation with is within 1% of theoretical and

simulation with is within 3%–4% of theoretical .

valid. Also, compared previously to Fig. 8, theory and simula-tion are in better agreement, within 1%, now that the effects ofare included in theory. Moreover, this figure shows that

is limited by , which reduces to only if is negligible.In Fig. 11(a), the ring radius was changed to vary .

Fig. 11(b) shows that is strongly dependent on .increases with increasing (increasing circumference) sincethe surface area of the capacitance increases. Even though

has an air gap, the dielectric material of the cavity willpartly change due to fringing electromagnetic fields. Fig. 14confirms this by showing the effects of various on resonantfrequency and TR. The results in the figure are based on HFSSsimulation with cavity parameters from Table I as rangesfrom 2.67 to 0.63 pF. Simulation shows that when

Fig. 13. Effects of is negligible on for the cavity parameters consideredin Table I since . Additionally, theory and simulation arewithin 1% since the effects of is included and is limited by .reduces to if .

Fig. 14. Plot of frequency versus capacitance at various dielectric constants.As the dielectric constant increases, increases and reduces TR.

and when , a reduction of about 11%in TR. Based on (18), pF for , pFfor , and pF for .

D. Surface Inductance

The current that flows on the top surface from the inner postto the outer cavity wall results in a surface inductance . Thiscurrent flow depends on the ratio of and , and on . Ifcurrent flow is concentric, will be minimal. The arrangementof the varactors can alter the current path, thus changing .1) Consider the case of . When is large and thevaractors are spatially distributed, current flows equally inall directions through the varactors and current flow is con-centric. Fig. 15(a) illustrates this with . Fig. 15(b)shows current flow when is small. Current is forced toflow through the varactor resulting in a longer cur-rent path. Thus, increases as decreases, which is ver-ified in Table II. In simulation, as is varied, is alsochanged to keep constant for different values.

2) Consider the case of . Since most of the cur-rent is distributed in the surface ring gap, current flow isconcentric regardless of . Fig. 15(c) and (d) shows thatcurrent flow is concentric when or when .

3) When and are comparable, then current is dis-tributed between the varactors and surface ring gap. If


Fig. 15. HFSS simulation showing surface currents on the varactor loadedsurface. (a) Current is equally distributed in eight varactors and is small.(b) Current flows through just one varactor, which increases current path and. (c) When is dominant, current flows concentrically through regard-

less of or (d) .

TABLE IIVERSUS

is large and the varactors are equally spaced on the sur-face ring gap, current flow is concentric. For small , cur-rent flow depends on ring radius and . As gets larger,gets larger and leads to a more concentric current flow,

decreasing . Also, however, as gets larger, the pathof the current flow through the varactor gets longer, in-creasing . Additionally, as is tuned, changes andthe impedance ratio of and changes, making varywith . Thus, when and are comparable and issmall, is highly dependent on , ratio, and .

The surface inductance is included in (17) by replacingwith

(23)

resulting in

(24)

Though not discussed earlier, the effects of is included in allthe theoretical analysis and plots in Section III-C. Even thoughis dependent on various parameters, based on the above dis-

cussion, is always minimal as long as is sufficiently large.The effects of is further reduced by increasing . As men-tioned previously and shown in (6), varies slightly with .For , the variation in is less than 0.4% and for ,the variation in is less than 3.5% over the frequency rangewith parameters from Table I.

E. Design Methodology

Based on the theoretical analysis, the compromise in andtuning range is highly dependent on . With the aid of simula-tion software, appropriate is designed by changing the sur-face ring gap dimensions or the dielectric constant (Figs. 11(b)and 14). For example, a resonator is designed for high ,160–40, but with a limited TR of 1.2 in [12]. This is analogousto the curve in Fig. 11 with pF, which has a TR of1.27.Alternatively, to design a resonator with maximum tuning

range, has to be minimized so that it is small relative to .To maximize , and can be increased up to the size limi-tations and can be set by the optimal ratio. From(7) and (2), an appropriate number of varactors can then bemounted to lower the frequency to the desired range, resultingin an optimized for a maximum TR design.This design, however, may result in that is too large. For

example, a cavity with parameters from Table I, except withmm for optimal ratio, requires varactors on the

surface ring gap to tune from 0.6 to 1.2 GHz. If fewer varactorsare to be used, then the cavity dimensions may be changed to getthe desired frequency range. The height and radius of the cavityare typically limited by the constraints onmaximum device size.Varactors with larger capacitances are not recommended sincethe increase in capacitance comes at the cost of reduced ,which directly limits . The reduction in tuning range andcost of high dielectric constant material restricts as a flexibleparameter for design.The optimum ratio may need to be sacrificed to

get the desired frequency range for a design with less than 20varactors. From (7) and (6), the resonant frequency depends onthe ratio. From Fig. 16, increasing from the optimumratio of 3.6 to 25 can decrease the frequency from to .The previous example where varactors were neededto tune from 0.6 to 1.2 GHz can alternatively be designed byincreasing the ratio from 3.6 to 24. Fig. 17 shows a reductionof 5% to 7% in for compared to .

IV. EXPERIMENTAL VALIDATION

A. Resonator

ASIW combline resonator with solid-state varactors as tunersis designed and fabricated to validate the theoretical derivations.Fig. 18(a) shows the designed resonator. In this design, metallicvias are inserted in a PCB substrate to create the outer wall of thecavity. The diameter of the vias ( 1 mm) and the spacing be-tween vias ( 2.5 mm) are designed with the recommendationsgiven in [25] to keep lossesminimal. Another center metallic via


Fig. 16. Plot of ratio versus normalized frequency , where is theresonant frequency for the optimum ratio. The resonant frequency decreasesby as is changed to 25.

Fig. 17. Resonator can be designed with with optimal ratio equalto 3.6. Alternatively, the optimal can be compromised to designa resonator with the same frequency range where fewer varactors are needed

with reduced .

Fig. 18. (a) Designed resonator’s top view and (b) bottom view showingCPW feed lines. (c) Rogers TMM3 substrate with vias and SkyworksSMV1405 varactors. (d) Close-up view of the two surface ring gaps showingthe varactors (not soldered on yet) and the dc bias point.

with radius of shorts the bottom and top ceiling of the cavity.Fig. 18(b) shows the coplanar-waveguide (CPW) feed lines forthis design with length . Since solid-state varactors are used,a structure with two ring gaps is needed to create a bias point for

TABLE IIICAVITY DIMENSIONS USED IN FABRICATION

Fig. 19. Comparison of measured, simulated, theoretical (with and withoutand ) versus frequency.

the varactors. The two rings also allows for back-to-back var-actor placement for improved linearity [20]. Since the additionalring is in series with the original ring, (2) and (3) are modifiedto

(25)

(26)

In order to keep the same and thus the same frequency range,needs to be doubled for a two-ring gap design compared to a

one-ring gap design.The first part of the theoretical analysis was based on the

lumped model presented in Fig. 2(c). In order for (7) and (16)to be valid, and had to be negligible. A SIW resonatoris fabricated on a Rogers TMM3 substrate with the dimensionsgiven in Table III. By choosing , the effects of andis minimized. Fig. 18(c) and (d) shows the fabricated res-

onator and a close-up of the arranged varactors with the isolatedbias point for the varactors (not yet soldered on). A 10-M re-sistor is included in the dc bias line to reduce RF losses. Anadditional lumped resistor was used during the measurement[see Fig. 24(c)] to further reduce RF losses. Fig. 19 comparesmeasured, HFSS, and theory. is plotted versus frequency asthe varactors are biased from 0 to 30 V. This figure validates


Fig. 20. Measured versus frequency for to along withmaximum theoretical (solid line) at pF. For larger ,and is limited by ; as decreases, becomes comparable to and

exceeds .

TABLE IVSUMMARY OF MEASURED PERFORMANCE OF FABRICATED

RESONATORS WITH VARYING

that the simplified lumped model from Fig. 2(c), withoutand , is reasonable in predicting and . The extracted

pF and nH from measured data are then in-cluded in theory and plotted in the same figure. The slope of thecurve with and is even closer to measured and simulationdata. The measured TR is about 2.1, simulated TR is about 1.94,and theoretical TR including the effects of is 2.01. Measured

is lower due to losses in dielectric material (loss tangent is0.002), fabrication, and assembly.Another set of resonators, with similar dimensions as

Table III, were fabricated with ranging from 1 to 20 to showthe effects of on and TR. To compare the measuredwith , Fig. 20 includes a plot of theoretical at(maximum ). In the figure, is limited by when islarge, but exceeds for or less: has become sig-nificant compared to . In fact, in the extreme case of ,

is 0.315 pF, which is less than pF. When thenumber of varactors decreases, ratio increases andTR decreases. Table IV summarizes the results of frequencyrange and TR. Table IV shows that the TR decreases from 2.17when to 1.78 when . Table IV also summarizesthe results of and ratio at various ’s. When ,and thus, dominates, (10) states that ,which is close to the measured ratio of 2.38.It should be noted that of the tunable resonator depends

heavily on the tuner technology. In this case, solid-state varac-tors have higher at lower frequencies. RF-MEMS tuners canbe used to get high at higher frequencies. For example, theauthors have demonstrated a surface ring gap resonator withof 240 at 6.6 GHz [14].

Fig. 21. (a) Two resonators with inter-resonator coupling and weak externalcoupling. Both the width of the coupling iris and post-to-post distancecan be changed to change . (b) One-port resonator structure used to extractby varying the coupling angle when the feed line and coupling gap

size are constant.

Fig. 22. Simulated design curves for: (a) versus post distance and (b)versus coupling angle .

B. Filter

To demonstrate the application of the resonator, a two- andthree-pole tunable filter are designed and fabricated with the res-onator parameters from Table III. A similar procedure as out-lined in [21] is used to get the external quality factor andinter-resonator coupling from HFSS simulation. HFSS sim-ulation is used to simulate various values of by changing thewidth of the coupling iris and the post distance while thefeed line is kept far from the post [see Fig. 21(a)]. Fig. 22(a)shows versus when mm is simulated around1.1 GHz. A resonator with one port shown in Fig. 21(b) is usedto simulated . As seen in the figure, another opening is cre-ated at the end of CPW feed lines with a coupling gap sizeand coupling angle of . Fig. 22(b) shows versus


Fig. 23. (a) Designed and (b) fabricated octave tunable two-pole filter withmeasured (c) and .

when mm and cm is simulated around1.1 GHz.The required values of and for a two-pole Butterworth

filter response are

(27)

(28)

where %, , , and .Fig. 23(a) and (b) shows the designed and fabricated two-polefilter with cm, , and mm ina 5 3 0.5 cm volume. The filter tunes from 0.5 to 1.1 GHzwith measured insertion loss from 4.46 to 1.67 dB andmeasuredreturn loss from 8.3 to 27.8 dB [see Fig. 23(c)]. A 4 0.1%FBW is maintained through out the tuning range. An AgilentPNA-X is used to measure the input third-order intermodulationintercept point (IIP3) with the two tones separated by 100 kHz.The IIP3 ranged from 17 to 30 dBm when the varactors werebiased at 0–30 V.The required values of and for a three-pole Butter-

worth filter response are

(29)

(30)

where %, , , and . Fig. 24(a)and (b) shows the designed and fabricated three-pole filterwith cm, , and mm ina 5 3 0.5 cm volume. This filter has a tuning range of0.58–1.22 GHz with a constant 3-dB FBW of 4 0.2%.

Fig. 24. (a) Designed and (b) fabricated octave tunable three-pole filter withmeasured (c) and .

Fig. 24(c) shows that the insertion loss varies from 6.2 dB at0.58 GHz to 2.05 dB at 1.22 GHz.Since both the two- and three-pole filters were designed at

1.1 GHz, the simulated and values are best matched to thedesired values at 1.1 GHz. As seen in Figs. 23(c) and 24(c), thereturn loss degrades as the filter is tuned to lower frequencies.In [26], a varactor is mounted on the CPW feed line to changeand another varactor mounted between the two resonators to

change . Though [26] had a different resonator technology, thesame concept of tunable and can be implemented in thepresented filter design without added fabrication complexity.This allows and to be tuned as the frequency responseof the filter is tuned to improve return loss away from 1.1 GHz.However, adding more varactors will introduce more insertionloss.Table V compares some of the recent tunable filters. As men-

tioned earlier and summarized in Table V, the complexity ofthe presented surface ring gap resonator (and other planar var-actor tuned filters) is reduced compared to some of the verticallyaligned piezoelectric or RF-MEMS-based filters. Moreover, tothe best of authors’ knowledge, the presented surface ring gapresonator exceeds the TR of all other solid-state varactor based


TABLE VCOMPARISON OF TUNABLE FILTERS

filters for a given varactor capacitance ratio : measured TRexceeds [see (8)] for this work.

V. CONCLUSION

This paper has presented the modeling and design of an oc-tave tunable combline filter using surface mount lumped tuningelements. Detailed theoretical analysis on the tuning range and

of these resonators/filters are presented. A systematic de-sign methodology is also proposed. To validate the theory andthe design procedure, various tunable resonators were fabricatedwith up to 214 at 1.1 GHz. Additionally, a two-pole tunablefilter with tuning range of 0.5–1.1 GHz at a constant FBW of4 0.1% and measured insertion loss of 1.67 dB at 1.1 GHz, anda three-pole tunable filter with tuning range of 0.58–1.22 GHzat a constant FBW of 4 0.2% and measured insertion loss of2.05 dB at 1.22 GHz have been demonstrated.

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Akash Anand (S’12) received the B.S. degreein electrical engineering from the University ofCalifornia at Davis, Davis, CA, USA, in 2009, andis currently working toward the Ph.D. degree atUniversity of California at Davis.His research interest includes RF/microwave tun-

able filters, RF and analog integrated circuit (IC) de-sign, and reconfigurable RF systems.

Joshua Small (S’11–M’12) received the Bachelor’sdegree in electrical engineering from Morgan StateUniversity, Baltimore, MD, USA, in 2005, andthe Ph.D. degree from Purdue University, WestLafayette, IN, USA, in 2012.He is currently a Postdoctoral Research Engineer

with the Department of Electrical and ComputerEngineering, University of California at Davis,Davis, CA, USA. His research interests includesinvestigating novel high- microstructures forreconfigurable radio front ends.


Dimitrios Peroulis (S’99–M’04) received the Ph.D.degree in electrical engineering from The Universityof Michigan at Ann Arbor, Ann Arbor, MI, USA, in2003.Since August 2003, he has been with Purdue Uni-

versity, West Lafayette, IN, USA, where he is cur-rently a Professor leading a group of graduate stu-dents on a variety of research projects in the areas ofRF-MEMS, sensing, and power harvesting applica-tions, as well as RF identification (RFID) sensors forcondition monitoring of sensitive equipment. He has

been a Principle Investigator (PI) or a co-PI in numerous projects funded bygovernment agencies and industry in these areas. He has been a key contrib-utor on developing very high quality RF MEMS tunable filtersin mobile form factors. Furthermore, he has been investigating failure modesof RF MEMS and MEMS sensors through the Defense Advanced ResearchProjects Agency (DARPA) MEMS/Nanoelectromechanical systems (NEMS)Science and Technology (S&T) Fundamentals Program, Phases I and II) andthe Center for the Prediction of Reliability, Integrity and Survivability of Mi-crosystems (PRISM) funded by the National Nuclear Security Administration.He has coauthored over 200 journal and conference papers.Dr. Peroulis was the recipient of the National Science Foundation (NSF) CA-

REER Award (2008) and the IEEE Ultrasonics, Ferroelectrics, and FrequencyControl Society (Ferroelectrics section) Outstanding Paper Award (2012). Hisstudents have been the recipients of numerous Student Paper Awards and otherstudent research-based scholarships. He is a Purdue University Faculty Scholarand has also been the recipient of ten teaching awards including the 2010 HKNC. Holmes MacDonald Outstanding Teaching Award and the 2010 Charles B.Murphy award, which is Purdue University’s highest undergraduate teachinghonor.

Dr. Xiaoguang “Leo” Liu (S’07–M’10) receivedthe B.S. degree from Zhejiang University, Zhejiang,China, in 2004, and the Ph.D. degree from PurdueUniversity, West Lafayette, IN, USA, in 2010.He is currently an Assistant Professor with the

Department of Electrical and Computer Engineering,University of California at Davis, Davis, CA, USA.He has authored or coauthored over 30 papersin peer-reviewed journals and conferences. Hisresearch interests include RF-MEMS and otherreconfigurable RF/microwave components, soft-

ware-defined radios, and terahertz components and systems.Dr. Liu was the recipient (as a student) of the IEEE Antenna and Propagation

Society Graduate Research Fellowship in 2009. He was selected as a Universityof Californiat at Davis Hellman Fellow for the 2013–2014 academic year.

Documents

Theory and Design of Octave Tunable Filters With Lumped Tuning Elements