Modeling Varactor TunableMicrostrip Resonatorsfor Wireless Applications

By Boris Kapilevich and Roman LukjanetsSiberia State University of Telecommunications & Informatics

In a previous article [1], wedescribed single terminatedvaractor tunable microstrip

resonators. Configurations withseries and parallel varactor cir-cuits in the center of resonatorwere considered. The advantagesof those configurations are a sim-plicity and minimum number ofvaractors. Their major draw-backs are rather limited tuningrange and harmonic suppression.

Another configuration of tun-able resonator is considered here— Double-terminated VaractorTunable Microstrip Resonator(D-VTMR). This configurationdemonstrates some advantages as compared toits single terminated counterpart, due to addi-tional degrees of freedom. The general form ofresonance conditions, tuning facilities and high-er harmonics suppression of this configurationare investigated in this paper on the basis of theappropriate circuit model.

Resonance conditionsConsider the two basic configurations of D-

VTMR with different varactor circuits: in series,shown in Figure 1(a), and in parallel, shown inFigure 1(b). Using the result described in [1],the resonance conditions of both circuits can bewritten through transmission matrixes. Theanalysis of a resonator can be simplified byusing its symmetry property with respect toinput/output ports. Hence, a transmissionmatrix of left and right parts of a resonatorrelated to a symmetry plane are dependent oneach other:

(1)

So the only matrix in Equation (1) must bedetermined that simplifies an analysis. Finally,a resultant transmission matrix of the resonatorconsidered can be written as follows usingEquation (1).

(2)

The above expressions have been used to esti-mate resonance conditions |S11| = 0, which cor-respond to the following equalities:

AsBs – Z02CsDs = 0 (3)

for the series configuration, and

RA D B C A B

C D A D B Cs[ ] =× + × × ×

× × × + ×

22

AA BC D

AD BC Aleft right[ ]

[ ]

= = and

54 · APPLIED MICROWAVE & WIRELESS

� Figure 1. Configurations of double terminated varactor tunablemicrostrip resonator (D-VTMR) (a) in series; (b) in parallel.

56 · APPLIED MICROWAVE & WIRELESS

ApBp – Z02CpDp = 0 (4)

for the parallel configuration, where indices s and p cor-respond to series or parallel configurations, respectively,and Z0 is the system impedance.

Substituting the elements of matrixes into (3) and (4),the general resonance condition determining resonancelengths θ1res and θ2res/2 of the D-VTMR can be writtenin polynomial form:

P4T4 – P3T3 + P2T2 – P1T + P0 = 0 (5)

where

P0, P1, P2, P3 and P4 are polynomial coefficients,depending on the varactor’s parameters and impedanceof the coupling element, calculated as follows:

For a simplicity of consideration capacitive or induc-tive coupling elements are discussed here. Each of themprovides a specified value of θ1res and θ2res/2 through theparameters K1, K2, K3, R1, R2, ..., R9. General expres-sions determining resonance conditions of the D-VTMRconfigurations are written as:

(7)

Detailed specifications of R-parameters are given inthe Appendix for series and parallel configurations.

Calculating polynomial roots from Equation (5), wecan determine the resonance frequencies and estimatethe available tuning range of D-VTMR. As an illustra-tion, we assume that the absolute impedance of the cou-pling elements is 220 ohms at a design frequency of 1.8GHz. The impedances Z1 and Z1 are varied within 20ohms = Z1 = 80 ohms and 20 ohms = Z1 = 80 ohms.The junction capacitive of the idealized varactor isassumed to be 0.5 pF = Cv = 2.5 pF.

Since θ1 and θ2/2 are the independent polynomialparameters, families of curves for different relative var-actor position ξ were calculated, where

(8)

The total electric lengths θ1 + θ2/2 for series and par-

ξ θ

θ θ=+

1

12

2

K R t R t

K R t R t

K R tR

t RR

1 6 1 9 1

2 7 1 8 1

3 12

14

1 32

22

2

= ( ) + ( )= ( ) + ( )= ( ) + + +

cos sin

sin cos

cos sin( )

PR

R K R K K K K K

P K R K R R K R KK K R K

P R K R K R K R K

R K

022

2 1 2 3 12

1 3 32

1 22

52

1 2 22

12

2 3

1 3 5 2

2 2 3 5 2 22

12

2 1

52

22

42

4 4 10 4 4 84 8

8 24 20 4 24

20 4

= − + + − +

= + − + + +− −

= − + + −

+ +

PP R R K R R K

P R R3 5

22 1 2

25 2

4 22

52

32 16 32 16

16 16

= − + −

= +

� Figure 2. Behaviour of the calculated electric length θ2/2for PDC of D-VTMR.

� Figure 3. Behaviour of the calculated electric length θ2/2for SIC of D-VTMR

Tt

t t= = =+

cos ; ;2 21

1 22

1 2

2 2 2

−θ θ θ θand

58 · APPLIED MICROWAVE & WIRELESS

allel configurations were chosen to be 2.1 and 0.87 ,respectively, in order to provide an existence of lowestresonance within frequency band 1.5 to 2.5 GHz. In anycase, total electric length θ1 + θ2/2 is the constant valueregardless of ξ.

The eight different configurations of D-VTMR wereinvestigated:

1. Series configuration with Increasing impedance andCapacitive coupling element (SIC);

2. Series configuration with Decreasing imped-ance and Capacitive coupling element (SDC);

3. Series configuration with Increasing imped-ance and Inductive coupling element (SII);

4. Series configuration with Decreasing imped-ance and Inductive coupling element (SDI);

5. Parallel configuration with Increasing imped-ance and Capacitive coupling element (PIC);

6. Parallel configuration with Decreasing imped-ance and Capacitive coupling element (PDC);

7. Parallel configuration with Increasing imped-ance and Inductive coupling element (PII);

8. Parallel configuration with Decreasing imped-ance and Inductive coupling element (PDI).

It should be pointed out that the variety of res-onator configurations depends on combining atype of coupling element (capacitive or inductive),a type of varactor circuit (parallel or series) andthe behaviour of the transmission line impedance(increasing or decreasing) from the ends of a res-onator toward the center of a resonator.

Figure 2 demonstrates the behavior of the cal-culated electric length θ2/2 as a function of theabsolute admittance of varactor Yv(mSm) for dif-ferent relative position ξ corresponding to thePDC configuration of D-VTMR. Figure 3 demon-strates the similar characteristics as a function ofthe absolute impedance of varactor Zv(ohm) fordifferent relative position ξ corresponding to theSIC configuration of D-VTMR. Sometimes thenegative electric length can be obtained duringanalysis of resonance conditions. In this case, it isnecessary to add a transmission line section withelectric length π.

Comparisons of the calculated characteristicsof different D-VTMR configurations are shown inTables 1 and 2 and discussed in more detail in thefollowing sections.

Analysis and discussion The two key criteria — maximum relative fre-

quency tuning range and frequency separatingbetween 1st and 2nd modes — were used to findout the better configurations of D-VTMR. Typicalvalues of transmission line impedances ratio and

relative positions of varactor were used in the analysis.

a) Parallel configuration:When the relative position of a varactor ξ is varied

from 0 to 0.2, both resonances move along the frequen-cy axis almost synchronously. Maximum relative fre-quency tuning range corresponds to ξ ≈ 0.15. It is neces-sary to note that the parallel configurations with induc-tive coupling elements give wider tuning range com-pared to PIC and PDC. The frequency separation

� Table 1. Comparison of calculated characteristics of differentparallel D-VTMR configurations.

� Table 2. Comparison of calculated characteristics of differentseries D-VTMR configurations.

60 · APPLIED MICROWAVE & WIRELESS

between 1st and 2nd modes is approximately the same.Preference should be given to configurations with capac-itive coupling elements that are simpler to be used inpractice.

b) series configuration:When the relative position of a varactor ξ is fixed near

0.6 to 0.7, the maximum relative frequency tuning rangeof all possible configurations with different ratios oftransmission line impedances and coupling elements isachieved. However, only SIC and SII configurations pro-vide the maximum frequency separation between 1st

and 2nd modes. SDC and SDI configurations demon-strate an unfavorable effect of modes converging.

Based on the results considered, some recommenda-tions and limitations concerning the choice of the D-VTMR configurations may be formulated.

1. All parallel configurations give better relative fre-quency separating between 1st and 2nd modes.

2. SDC and SDI configurations are not recommendedbecause of existing effect of modes converging.

3. SIC and SII configurations give better tuning facilities.

It should be pointed out that we have neglected lossesin transmission lines and varactor in the above describedresults. This approach allows the designer to gather gen-eral information concerning the resonators studied andchoose a configuration for further experiments.However, the suggested circuit model can also be usedwhen losses in transmission lines and varactors are notequal zero. In this case, the varactor must be describedby the real equivalent circuit [2]. The latter has beenapplied in estimating real world characteristics of D-VTMR and comparison with experimental results below.

Experimental resultsThis section describes the design and performance of

experimental D-VTMRs with parallel and series config-urations, comparison their calculated and measuredcharacteristics.

The designed resonator frequency was 1.8 GHz. Asubstrate of a dielectric constant εr = 2.6 and thickness1.5 mm has been used for fabrication. A GaAs D-610Cpackaged varactor diode V with a capacitor ratio of 3:1from 0 V to 30 V having Cj(0) = 1.65 pF and Cj(b) = 0.55pF was used in the experiments. The unloaded qualityfactor of resonators was 200.

The first experimental D-VTMR was realized as theSIC configuration providing the maximum absolute fre-quency tuning range. The relative position of varactor ξwas chosen to be 0.7 as the optimal. The impedances Z1and Z2 were chosen to be 20 and 80 ohms, respectively.The layout of D-VTMR is shown in Figure 4.

In Figure 5, the dotted line illustrates the theoreticalresults of |S21| and the solid one corresponds to mea-sured characteristics of |S21|. Capacitive coupling ele-ments with C = 0.6 pF were used. In this case, the 3-dBbandwidths of |S21| were 38 MHz and 22 MHz for Cj(b)= 0.55 pF and Cj(0) = 1.65 pF, respectively. Insertionlosses of the experimental D-VTMR with series configu-ration were varied near 2.8 dB at a whole tuning range,compared with 2.5 dB predicted by theoretical model.

The second experimental D-VTMR was realized asthe PDC configuration providing the maximum frequen-cy separating between 1st and 2nd modes. Relative posi-tion of varactor ξ was chosen to be 0.1 as the optimalone. The impedances Z1 and Z2 were chosen to be 70 and

� Figure 5. Characteristics of S21 for the theoretical modeland experimental performance of the D-VTMR series con-figuration.

� Figure 4. Experimental D-VTMR with series configurationand simple capacitive coupling element.

62 · APPLIED MICROWAVE & WIRELESS

30 ohms, accordingly. The layout of the D-VTMR isshown in Figure 6.

In Figure 7, the dotted line illustrates the theoreticalresults of |S21| and the solid one corresponds to themeasured characteristics of |S21|. L-type of capacitivecoupling element having C1 = 3.6 pF, C = 1.2 pF and C2= 3.6 pF was implemented to achieve the minimumbandwidth change with frequency tuning. In this case,the 3 dB bandwidths of S21 were 36 MHz and 26 MHzfor Cj(b) = 0.55 pF and Cj(0) = 1.65 pF, respectively. Thesecond mode was near 4 GHz. A good agreement of res-onance frequencies for both calculated and measuredcharacteristics was achieved with resonator tuning.Insertion losses of the experimental D-VTMR with par-allel configuration were varied near 2.0 dB at a whole

tuning range compared with 1.6 dB predicted by the the-oretical model. An active device can be implemented fora loss compensation. One of the possible realizations ofsuch an approach has been reported in [3].

ConclusionAn analysis of double terminated varactor tunable

microstrip resonators has been presented in this paperusing a proper circuit model. The resonance conditionshave been obtained and applied to find out a better con-figuration of D-VTMR. The two experimental D-VTMRswith series and parallel varactor circuits have been fab-ricated. A good agreement has been demonstratedbetween the calculated and experimental results.

On the basis of the analysis carried out, some recom-mendations have been formulated for better choice ofresonator’s configuration. Finally, it is necessary tonotice existing a natural limitation concerning D-VTMRs; namely, there is no universal D-VTMR configu-ration that satisfies both above mentioned criteriasimultaneously. The designer will need to make a prop-er choice of D-VTMRs, taking into account specified sys-tem requirements. �

AcknowledgementThis work was supported by INTAS Project no. 96-

0851. The authors would like to thank the project coor-dinators, Dr. B. Jarry (IRCOM, France) and Dr. M.Guglielmy (ESA ESTEC, The Netherlands), for currentdiscussions and supporting the topics considered. Theauthors would like also to thank Vladimir Smirnov forhis useful advice and valuable assistance in the processof the experiments.

References1. B.Yu. Kapilevich and R.L. Lukjanets, “Modeling

Varactor Tunable Microstrip Resonators for WirelessApplications,” Applied Microwave & Wireless, Vol. 10,No. 7 (September 1998): 32-44.

2. K.C. Gupta, Garg Ramesh and Chadha Rakesh,Computer Aided Design of Microwave Circuits, NewYork: Artech House, 1981.

3. B.Yu. Kapilevich and R.L. Lukjanets, “TunableActive Bandpass Filter with Fixed Bandwidth,” FourthInternational Wireless and TelecommunicationsSymposium (IWTS-2000), Malaysia, May 15-19, 2000:36-40.

Author informationBoris Kapilevich is a professor of microwave and RF

and head of the Applied Electromagnetics Departmentat Siberia State University of Telecommunication &Informatics, Novosibirsk, Russia. He received an MSdegree from Tomsk State University in 1965, a Ph.D. in1969 from Novosibirsk State Technical University and aDr.Sc. in 1986 from Moscow Power Energy University.

� Figure 6. Experimental D-VTMR with parallel configura-tion and capacitive coupling elements.

� Figure 7. Characteristics of S21 for the theoretical modeland experimental performance of D-VTMR parallel config-uration.

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Appendix

� Table 3. Series and parallel configurations.

The general expressions for R1, R2... R9 parametersare written as:

R1 = a1 × b1 – a3 × b3 – z02 × c1 × d1 + z0

2 × c3 × d3R2 = a2 × b2 – a4 × b4 – z0

2 × c2 × d2 + z02 × c4 × d4;

R3 = a3 × b3 + a4 × b4 – z02 × c3 × d3 – z0

2 × c4 × d4;R4 = a1 × b3 + a3 × b1 – z0

2 × c1 × d3 – z02 × c3 × d1;

R5 = a2 × b4 + a4 × b2 – z02 × c2 × d4 – z0

2 × c4 × d2;

R6 = a1 × b2 + a2 × b1 – z02 × c1 × d2 – z0

2 × c2 × d1;R7 = a4 × b3 + a3 × b4 – z0

2 × c3 × d4 – z02 × c4 × d3;

R8 = a4 × b1 + a1 × b4 – z02 × c1 × d4 – z0

2 × c4 × d1;R9 = a2 × b3 + a3 × b2 – z0

2 × c2 × d3 – z02 × c3 × d2.

Using Table 3, R1, R2... R9 parameters can be calcu-lated for series or parallel configurations with capaci-tive and inductive coupling elements.

He has more than 30 years of experience in analysis,design and implementation of microwave componentsand devices for the communications industry and is asenior member of the IEEE. His interests are inmicrowave and RF components for mobile and spacecommunications. He may be reached by telephone at+007 3832 660943; by fax at +007 3832 222581; or by

e-mail at boris@neic.nsk.su.Roman Lukjanets is a Ph.D. student at Siberia State

University of Telecommunication & Informatics. Hereceived his MS degree from the university in 1996. Hisresearch interests are focused in microwave and RFcomponents for mobile communications, includingmicrowave active filters.