Research Article Design of Miniaturized Multiband Filters...

Research ArticleDesign of Miniaturized Multiband Filters UsingZero Order Resonators for WLAN Applications

Maryam Shafiee1 Mohammad Amin Chaychi Zadeh2 and Homayoon Oraizi2

1Department of Electrical Engineering Arizona State University Tempe AZ 85287 USA2Department of Electrical Engineering Iran University of Science and Technology Narmak Tehran 16846 13114 Iran

Correspondence should be addressed to Maryam Shafiee mshafieiasuedu

Received 23 November 2014 Revised 15 February 2015 Accepted 18 February 2015

Academic Editor Giancarlo Bartolucci

Copyright copy 2015 Maryam Shafiee et alThis is an open access article distributed under theCreative CommonsAttribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The objective of this paper is to design miniaturized narrow- and dual-band filters for WLAN application using zero orderresonators by the method of least squares The miniaturization of the narrow-band filter is up to 70 and that of the dual-bandfilter is up to 64 compared to the available models in the literature Two prototype models of the narrow-band and dual-bandfilters are fabricated and measured which verify the proposed structure for the filter and its design by the presented method usingan equivalent circuit model

1 Introduction

Mobility of components and equipment is a requirementin mobile communication systems Therefore considerableeffort has gone into the miniaturization of devices Further-more various techniques of multibanding have been devisedfor various functions Particularly different techniques havebeen proposed to make components dual-band which areconsidered conventional methods such as series connectionof two separate band pass filters step impedance resonatorsand defected ground structures All of these methods havesome limitations on the reduction of device dimensionsbecause they are based on half-wave resonators [1 2]

Recently metamaterials have found wide applicationssuch as miniaturization of microwave devices due to theirunique properties and nonlinear dispersion curves Zeroorder resonance (ZOR) metamaterials have the distinctproperty of infinite wavelength which makes the resonancefrequency independent of their dimensions Consequentlythe ZOR componentmay be fabricated as small as specified torealize the desired circuit elements for the required resonancefrequency [3] This unique characteristic of ZORs has beenused in microwave components such as filters [4ndash6]

In this paper we use the method of least mean squareerror (LMS) as a contribution to design and fabricate anarrow-band metamaterial filter and also a dual-band filter

for application in WLAN systems The main advantage andfeature of the proposed filters are their compactness andminiaturization which is up to 70 smaller than conven-tional components in the literature [7]

2 Design Procedure

The proposed zero order resonance (ZOR) cell is depictedin Figure 1 The two interdigital capacitors provide theleft-handed capacitance and right-handed inductance Themeander inductance andT-junction are inserted in the circuitto control and adjust the right-handed section of the deviceIts equivalent circuit is shown in Figure 2 The values ofcapacitors and inductors are extracted from the formulas inthe literature [8ndash11]

The error function is explicitly expressed as the functionof geometrical dimensions of various sections of the filter Inother words the lumped elements (119871 and119862) of the equivalentcircuit of the filter in Figure 2 are expressed in terms ofthe geometrical dimensions of the filter structure which aregiven in literature The overall transmission matrix of theequivalent circuit is first derived which is then converted tothe scattering parameters Thus the scattering parameters ofequivalent circuit might be obtained by the available relationsof the T-matrix in terms of its physical dimensions [12]

Hindawi Publishing CorporationInternational Journal of Microwave Science and TechnologyVolume 2015 Article ID 345326 7 pageshttpdxdoiorg1011552015345326

2 International Journal of Microwave Science and Technology

lf

wc

d1

wtj2 wtj1

d2

d3

wst

lstwm

lm

wf1wf2

Figure 1 The microstrip layout of the metamaterial unit cell 119908119898= 01mm 119897

119891= 455mm 119908

1198911= 03mm 119908

1198912= 04mm gap

1= 03mm

gap2= 02mm 119908

119904119905= 114mm 119897

119904119905= 13mm 119889

1= 1mm 119889

2= 045mm 119897

119898= 20mm 119908

1199051198951= 03mm and 119908

1199051198952= 01mm

Cf +N

2Csf Cf +

N

2Csf

Cf +N

2Csf

Cf +N

2Csf

Rf CiRf Ci

Rf Rf

Lsm Lsm

CsmCsm

Lm + Ll + Ltj1 Lm + Ll + Ltj1

Cst + Ctj

Rvia

Ltj2 + Lst + Lvia

2(Lsf + Lf)N 2(Lsf + Lf)N 2(Lsf + Lf)N 2(Lsf + Lf)N

Figure 2 The proposed equivalent circuit of metamaterial unit cell

We now refer to the specified frequency response of thebandpass filter as shown in Figure 3 It is composed of thelower (119891

119871to 119891119891) and upper (119891

119897to 119891119880) stop bands with

attenuation 119892119904 the lower (119891

119891to 1198910minus BW2) and upper (119891

0+

BW2 to 119891119897) transition bands with the attenuations 119892

119905119891=

(119892119904(119891119891minus (1198910minus BW2)))(119891

119894minus 119891119891) and 119892

119905119904= (119892119904(119891119891minus (1198910+

BW2)))(119891119894minus 119891119897) and the pass band with center frequency

1198910and bandwidth BW and insertion loss 119892

119901 Each frequency

interval is divided into119873119894discrete frequencies The accuracy

of the error function increases for larger number of discretefrequencies but theCPU time of computations also increasesTherefore a trade-off should be considered between accuracyand computing time We then construct an error function as

120576119904=

1198731

sum

119894isinstop(1003816100381610038161003816100381611990421

11989410038161003816100381610038161003816dB minus 119892119904)

2

120576119901=

1198732

sum

119894isinpass(1003816100381610038161003816100381611990421

11989410038161003816100381610038161003816dBminus 119892119901)2

120576119905119891=

1198733

sum

119894isinfirsttransition(1003816100381610038161003816100381611990421

11989410038161003816100381610038161003816dB minus 119892119905119891)

2

120576119905119904=

1198734

sum

119894isinsectransition(1003816100381610038161003816100381611990421

11989410038161003816100381610038161003816dBminus 119892119905119904)2

997904rArr 120576total

= 119908119901times 120576119901+ 119908119905times (120576119905119904+ 120576119905119891) + 119908119904times 120576119904

(1)

where 120576119904is the error due to the lower and upper stop bands

120576119901is the error related to the pass band and 120576

119905119891and 120576119905119904are the

errors from the lower and upper transition bands as depictedin the filter frequency response in Figure 3 The weighting

ff f0fL fl fU

gs

gp

f

BW

|S21| (dB)

Figure 3 Design specifications of filter frequency response

functions are 119908119904 119908119901 and 119908

119905in order to adjust the value of

each error term in the function of total errorThe error function can be constructed for multiband

filtersTheminimization of error function gives the optimumphysical dimensions of the filter The genetic algorithm (GA)as a global minimum seeking algorithm does not needinitial values for variable but it is quite slow Howeverthe conjugate gradient (CG) algorithm is a local minimumseeker algorithm which needs initial values but it is quitefast Therefore a MATLAB code is written which combinesGA and CG First GA is run to reach the vicinity of theabsolute minimum point but it is aborted prematurely andGC is activated to locate the minimum point quite fast Sincethe design procedure based on the transmission line circuitmodel suffers some approximation and does not account forfull wave performance of the filter the CSTMicrowave Studiocomputer simulator is used to adjust the filter performanceand obtain its optimum design [13]

International Journal of Microwave Science and Technology 3

Table 1 Line widths and lengths for the designed ZOR

Values before full wave optimization (mm) Values after full wave optimization (mm)119908119891= 047 119897

119904119905= 090 119908

1199051198952= 010 119908

1198911= 0305 119908

119904119905= 114 119908

1199051198952= 010

gap = 023 119908119898= 010 119897

119898= 200 119908

1198912= 040 119897

119904119905= 130 119897

119898= 200

119897119891= 495 119889

1= 114 119889

2= 067 gap1 = 030 119908

119898= 010 119889

2= 045

119908119904119905= 20 119908

1199051198951= 012 119889

3= 091 gap2 = 020 119889

1= 112 119889

3= 050

119897119891= 455 119908

1199051198951= 030

2 22 24 26 28 3

0

Frequency (GHz)

Circuit modelSimulation

minus25

minus20

minus15

minus10

minus5

S 21

andS 1

1(d

B)

Figure 4 The frequency response of the ZOR unit cell obtained bythe equivalent circuit and simulation software

3 Design of a ZOR Unit Cell

We design a ZOR unit cell according to the followingspecifications specified in Figure 3 namely centre frequency1198910= 24GHz BW = 10 maximum insertion loss 119892

119901=

05 dBmaximum stop band rejection 119892119904= 20 dB in the bands

1-2GHz and 3-4GHz 119891119891= 20GHz 119891

119897= 30GHz119873

1= 51

1198732= 212 and 119873

3= 1198734= 45 The minimization of error

function determines the dimensions of the ZOR filterThe frequency response of ZOR unit cell as obtained

by the circuit model and full wave simulation using CSTare shown in Figure 4 They are in a good agreement Inthe circuit model of the unit cell the interdigital sectionsof the filter are assumed identical whereas in the full wavefilter simulation they are assumed different by defining twoseparate variables for both gaps andwidths to achieve the bestpossible result Also another optimization is performed ondimensions in full wave simulation as reported in Table 1

The propagation constant of the structure could becomputed from transmission parameters

120573 = minus120593unwrapped

(11987821) + 120577 (2)

The unwrapped phase of 11987821

is obtained by connectingthe discontinuous sections of phase curves at +120587 and minus120587 asshown in Figure 5 For the determination of the referencewhere the electrical length (120579 = 120573119897) is zero it is necessaryto shift the unwrapped phase curve by 120577 to remove the phaseambiguity [3] Its value is obtained from the location of zerocrossing point at the phase response curve

0 2 4 61

2

3

4

5

6

Freq

uenc

y (G

Hz)

minus4 minus2

120573l

120573 = 0 at f = 24GHz

Figure 5 The dispersion diagram of the ZOR unit cell

1 15 2 25 3 35 4

0

10

20

Frequency (GHz)

minus40

minus30

minus20

minus10

120583eff

120576eff

120576eff120583eff

Figure 6 Constitutive parameters of the ZOR unit cell

For the accurate characterization of metamaterial cellits constitutive parameters are obtained by the procedure ofeffective media and are drawn in Figure 6 [14]

4 Design of a Narrow-Band Filter

In order to design a narrow-band filter with high rejection inthe stop band low insertion loss in the pass band and sharpdrop in the transition band two ZOR unit cells are connectedin series by an inductive coupling as shown in Figure 7 Thedesign procedure based on the LMS design method is similarto that of the unit cellThe scattering parameters are obtained


Table 2 Dimensions of narrow-band filter designed by the proposed method

(a) Dimensions of first unit cell


119904119905= 100 119908

1199051198952= 010 119908

119891= 043 119897

119904119905= 100 119908

1199051198952= 010

gap = 0588 119908119898= 0218 119897

119898= 265 gap = 056 119908

119898= 025 119897

119898= 270

119897119891= 4226 119889

1= 120 119889

2= 050 119897

119891= 344 119889

1= 12 119889

2= 080

119908119904119905= 117 119908

1199051198951= 020 119889

3= 000 119908

119904119905= 133 119908

1199051198951= 020 119889

3= 000

(b) Dimensions of second unit cell


119904119905= 105 119908

1199051198952= 010 119908

119891= 052 119897

119904119905= 100 119908

1199051198952= 010

gap = 069 119908119898= 0218 119897

119898= 265 gap = 07 119908

119898= 025 119897

119898= 270

119897119891= 359 119889

1= 110 119889

2= 070 119897

119891= 347 119889

1= 092 119889

2= 070

119908119904119905= 133 119908

1199051198951= 020 119889

3= 000 119908

119904119905= 175 119908

1199051198951= 020 119889

3= 000

CRLH unit cell 1

CRLH unit cell 2Input Output

Lt

Figure 7 The equivalent circuit of the two metamaterial ZOR cellsin the narrow-band and dual-band filters

0

minus80

minus60

minus40

minus20

2 22 24 26 28 3

Frequency (GHz)


S 21

andS 1

1(d

B)

S21

S11

Figure 8 The frequency response of the narrow-band filter as 11987821

obtained by simulations and circuit model

from the transmission matrix which are then used in theerror function The specifications of the narrow-band filterare 1198910= 244GHz BW = 100MHz 119892

119901= 05 dB 119892

119904= 30 dB

1198731= 51119873

2= 212 and119873

3= 1198734= 45

The physical dimensions of the narrow-band filterobtained by the LMS design procedure and CST are given inTable 2

The frequency responses of filter as the amplitudes of11987811

and 11987821

obtained by the circuit model and computersimulation using CST Microwave Studio are illustrated inFigure 8 The full wave result shows a bandwidth of 80MHzfrom 236GHz to 248GHz The dispersion diagram of theproposed filter is depicted in Figure 9 The appearance of

0 2 4 62

22

24

26

28

3

Freq

uenc

y (G

Hz)

minus8 minus6 minus4 minus2

120573l (rad)

120573l = 0 at f = 24GHz

Figure 9 The dispersion response of the narrow-band filter

ripples in the response curve is due to the CST simulationsoftware which is based on the time-domain analysis usinga truncated Gaussian excitation signal Consequently theresulting bounded frequency band of excitation and theresonance nature of filter structure generate the ripples in thesimulated responseThemagnitude of ripplesmay be reducedby increasing the accuracy of simulations but the time ofcomputation greatly increases

The frequency response of the single-band filter is shownin Figure 10The graph covers a frequency range up to 10GHzto show that spurious response suppression is very good andis about 65 dB

The main advantage of the designed filter is its compact-ness and miniaturization It is almost 70 smaller than thosereported in the literature [7]

5 Design of a Dual-Band Filter

At first a ZOR unit cell filter is designed for dual-bandapplications The lower pass band is due to its left-handedresonance and the upper pass band is due to the right-handed


1 2 3 4 5 6 7 8 9 10

0

Frequency (GHz)

minus80

minus60

minus40

minus20

S 21

(dB)

Figure 10 The spurious response of narrow-band filter as 11987821in the stop band up to 10GHz

2 3 4 5 6

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

(dB)

(a)

2 25 3 35 4 45 5 55 6

0

5

Frequency (GHz)


minus30

minus25

minus20

minus15

minus10

minus5

S 11

(dB)

(b)

Figure 11 The frequency response of the dual-band filter as obtained by the circuit model and simulation (a) 11987821 (b) 119878

11

resonance Both upper and lower bands can be designed inde-pendently The frequency response of the designed filter hasachieved its specification In order to improve the isolationbetween the two bands and also higher quality factor (119876)at each band two ZOR unit cells are connected again usinginductive coupling Its design specifications are as follows

first band center frequency 1198910= 244GHz band-

width BW = 80MHz and maximum insertion lossIL = 05 dBsecond band center frequency 119891

0= 52GHz

bandwidth BW = 150MHz andmaximum insertionloss IL = 05 dBisolation between the two pass bands = 30 dB

For the construction of error function for the dual-bandfilter the frequency intervals of each pass-band each stopband and each transition band are divided into 51 212 and45 discrete frequenciesThe physical dimensions of each unitcell of the dual-band filter obtained for the optimum designare given in Table 3

The frequency responses by circuit model and full wavecomputer simulation using CST Microwave Studio are illus-trated in Figure 11 An isolation of 37 dB is observed betweentwo pass bands in full wave simulation The dispersioncharacteristic is extracted and shown in Figure 12

0 2 4 6 82

3

4

5

6

Freq

uenc

y (G

Hz)


120573l

f = 24GHz 120573l = minus28

f = 52GHz 120573l = +6

Figure 12 The dispersion curve of the dual-band filter

The compactness of this filter is 64 percent better than theavailable designs reported in the literature [2 15]

The difference between circuit model and full wavesimulation results in Figures 8 and 11 is due to the fact that weused two different variables for gaps andwidths of interdigitalstructures to provide more degree of freedom in full waveoptimization

To ensure maximum power transfer and reduction ofreturn loss a single open stub matching network is designedand added in parallel to the input of both filters to matchsource and load impedances


Table 3 Dimensions of dual-band filter designed by the proposed method



119904119905= 200 119908

1199051198952= 010 119908

11989112= 027 amp 020 119897

119904119905= 110 119908

1199051198952= 010

gap = 048 119908119898= 024 119897

119898= 200 gap = 045 119908

119898= 010 119897

119898= 200

11989711989112

= 127 amp 126 1198891= 035 119889

2= 037 119897

11989112= 130 amp 170 119889

1= 040 119889

2= 040

119908119904119905= 117 119908

1199051198951= 020 119889

3= 403 119908

119904119905= 112 119908

1199051198951= 020 119889

3= 495



119904119905= 190 119908

1199051198952= 010 119908

119891= 025 119897

119904119905= 120 119908

1199051198952= 010

gap = 047 119908119898= 024 119897

119898= 200 gap = 040 119908

119898= 010 119897

119898= 200

11989711989112

= 140 amp 130 1198891= 050 119889

2= 050 119897

11989112= 110 amp 130 119889

1= 050 119889

2= 050

119908119904119905= 110 119908

1199051198951= 020 119889

3= 430 119908

119904119905= 114 119908

1199051198951= 020 119889

3= 495

(a) (b)

Figure 13 Photographs of the fabricated prototype models (a) Narrow-band filter (b) dual-band filter

6 Measurement Result

The prototypes of the fabricated narrow-band and dual-bandband pass filters are depicted in Figure 13 Substrate Rogers4003 is used with dielectric constant 355 thickness 20miland tan 120575 = 00027 For the miniaturization of filter profilesthe two unit cells of filters are located together back to backby a pin so that the ground plane is placed between themThe dimensions of the pin are optimized in the full wavesimulationThe frequency responses of the narrow-band anddual-band filters as obtained by full wave simulation andmeasurement are illustrated in Figures 14 and 15 for 119878

21

versus frequencyThe agreement among simulation and measurement is

fairly good The difference between the simulation resultsand measurement data is due to the mediocre fabricationtechniques and low precision of the equipment in ourlaboratory

7 Conclusion

The zero order resonators derived from the theory of meta-materials are used as the building blocks to design narrow-band and dual-band filters A circuit model is derived for

2 22 24 26 28 3

0

Frequency (GHz)

SimulationMeasurement

minus80

minus60

minus40

minus20

S 21

andS 1

1(d

B)

S21 S11

Figure 14 Results for simulation and measurement of the narrow-band filter (the dimensions of substrate are 14 times 32mm2)

the filters by themethod of least squaresThe filter designs arefor single- and dual-band application inWLAN systemsTheproposed designs have achieved 70 and 64 miniaturiza-tion for the narrow-band and dual-band filters compared tothe available models in the literature The measurement andcomputer simulation data of fabricated prototypes agree quite


2 25 3 35 4 45 5 55 6

0

Frequency (GHz)


minus40

minus30

minus20

minus10

S 21

andS 1

1(d

B) S21

S11

Figure 15 Results for simulation and measurement of 11987821

for thedual-band filter (the dimensions of substrate are 2 times 14mm2)

well with the results of the proposed equivalent circuits of thefilter Consequently the proposed filter structures and designmethod are verified for application in microwave systems

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] J-T Kuo and H-S Cheng ldquoDesign of quasi-elliptic functionfilters with a dual-passband responserdquo IEEE Microwave andWireless Components Letters vol 14 no 10 pp 472ndash474 2004

[2] S Sun and L Zhu ldquoCoupling dispersion of parallel-coupledmicrostrip lines for dual-band filters with controllable frac-tional pass bandwidthsrdquo in Proceedings of the IEEE MTT-SInternational Microwave Symposium p 4 IEEE June 2005

[3] C Caloz and T Itoh Electromagnetic Metamaterials Transmis-sion Line Theory and Microwave Applications John Wiley ampSons 2005

[4] S Kahng G Jang B Lee J Ju and S Lee ldquoCompact UHF band-pass filter with the subwavelength metamaterial ZORs andtransmission zeros for enhanced channel selectivityrdquo in Pro-ceedings of the 41st European Microwave Conference (EuMC rsquo11)pp 567ndash570 IEEE October 2011

[5] G Jang and S Kahng ldquoDesign of a dual-band metamaterialband-pass filter using zeroth order resonancerdquo Progress InElectromagnetics Research C vol 12 pp 149ndash162 2010

[6] C H Tseng and T Itoh ldquoDual-band bandpass and bandstopfilters using composite rightleft-handed metamaterial trans-mission linesrdquo in Proceedings of the IEEE MTT-S InternationalMicrowave Symposium Digest pp 931ndash934 IEEE June 2006

[7] X Chen J Li and Y Shen ldquoWLAN bandpass filter with widestopband using spurlines and microstrip stubsrdquo in Proceedingsof the 9th International Symposium on Antennas Propagationand EMTheory (ISAPE rsquo10) pp 1188ndash1191 December 2010

[8] T C Edwards Foundations for Microstrip Circuit Design WileyChichester UK 1981

[9] G D Alley ldquoInterdigital capacitors and their application tolumped-element microwave integrated circuitsrdquo IEEE Transac-tions on Microwave Theory and Techniques vol 18 no 12 pp1028ndash1033 1970

[10] R Siragusa H V Nguyen P Lemaıtre-Auger S Tedjini and CCaloz ldquoModeling and synthesis of the interdigitalstub compos-ite rightleft-handed artificial transmission linerdquo InternationalJournal of RF and Microwave Computer-Aided Engineering vol19 no 5 pp 549ndash560 2009

[11] B C Wadell Transmission Line Design Handbook ArtechHouse 1991

[12] D M PozarMicrowave Engineering John Wiley amp Sons 2009[13] H Oraizi and N Azadi-Tinat ldquoOptimum design of novel UWB

multilayer microstrip hairpin filters with harmonic suppressionand impedance matchingrdquo International Journal of Antennasand Propagation vol 2012 Article ID 762790 7 pages 2012

[14] S-GMao S-L Chen and C-W Huang ldquoEffective electromag-netic parameters of novel distributed left-handed microstriplinesrdquo IEEE Transactions on Microwave Theory and Techniquesvol 53 no 4 pp 1515ndash1521 2005

[15] J T Kuo and H S Cheng ldquoDesign of quasi-elliptic functionfilters with a dual-passband responserdquo IEEE Microwave andWireless Components Letters vol 14 no 10 pp 472ndash474 2004

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lf

wc

d1

wtj2 wtj1

d2

d3

wst

lstwm

lm

wf1wf2

Figure 1 The microstrip layout of the metamaterial unit cell 119908119898= 01mm 119897

119891= 455mm 119908

1198911= 03mm 119908

1198912= 04mm gap

1= 03mm

gap2= 02mm 119908

119904119905= 114mm 119897

119904119905= 13mm 119889

1= 1mm 119889

2= 045mm 119897

119898= 20mm 119908

1199051198951= 03mm and 119908

1199051198952= 01mm

Cf +N

2Csf Cf +

N

2Csf

Cf +N

2Csf

Cf +N

2Csf

Rf CiRf Ci

Rf Rf

Lsm Lsm

CsmCsm

Lm + Ll + Ltj1 Lm + Ll + Ltj1

Cst + Ctj

Rvia

Ltj2 + Lst + Lvia

2(Lsf + Lf)N 2(Lsf + Lf)N 2(Lsf + Lf)N 2(Lsf + Lf)N

Figure 2 The proposed equivalent circuit of metamaterial unit cell

We now refer to the specified frequency response of thebandpass filter as shown in Figure 3 It is composed of thelower (119891

119871to 119891119891) and upper (119891

119897to 119891119880) stop bands with

attenuation 119892119904 the lower (119891

119891to 1198910minus BW2) and upper (119891

0+

BW2 to 119891119897) transition bands with the attenuations 119892

119905119891=

(119892119904(119891119891minus (1198910minus BW2)))(119891

119894minus 119891119891) and 119892

119905119904= (119892119904(119891119891minus (1198910+

BW2)))(119891119894minus 119891119897) and the pass band with center frequency

1198910and bandwidth BW and insertion loss 119892

119901 Each frequency

interval is divided into119873119894discrete frequencies The accuracy

of the error function increases for larger number of discretefrequencies but theCPU time of computations also increasesTherefore a trade-off should be considered between accuracyand computing time We then construct an error function as

120576119904=

1198731

sum

119894isinstop(1003816100381610038161003816100381611990421

11989410038161003816100381610038161003816dB minus 119892119904)

2

120576119901=

1198732

sum

119894isinpass(1003816100381610038161003816100381611990421

11989410038161003816100381610038161003816dBminus 119892119901)2

120576119905119891=

1198733

sum

119894isinfirsttransition(1003816100381610038161003816100381611990421

11989410038161003816100381610038161003816dB minus 119892119905119891)

2

120576119905119904=

1198734

sum

119894isinsectransition(1003816100381610038161003816100381611990421

11989410038161003816100381610038161003816dBminus 119892119905119904)2

997904rArr 120576total

= 119908119901times 120576119901+ 119908119905times (120576119905119904+ 120576119905119891) + 119908119904times 120576119904

(1)

where 120576119904is the error due to the lower and upper stop bands

120576119901is the error related to the pass band and 120576

119905119891and 120576119905119904are the

errors from the lower and upper transition bands as depictedin the filter frequency response in Figure 3 The weighting

ff f0fL fl fU

gs

gp

f

BW

|S21| (dB)

Figure 3 Design specifications of filter frequency response

functions are 119908119904 119908119901 and 119908

119905in order to adjust the value of

each error term in the function of total errorThe error function can be constructed for multiband

filtersTheminimization of error function gives the optimumphysical dimensions of the filter The genetic algorithm (GA)as a global minimum seeking algorithm does not needinitial values for variable but it is quite slow Howeverthe conjugate gradient (CG) algorithm is a local minimumseeker algorithm which needs initial values but it is quitefast Therefore a MATLAB code is written which combinesGA and CG First GA is run to reach the vicinity of theabsolute minimum point but it is aborted prematurely andGC is activated to locate the minimum point quite fast Sincethe design procedure based on the transmission line circuitmodel suffers some approximation and does not account forfull wave performance of the filter the CSTMicrowave Studiocomputer simulator is used to adjust the filter performanceand obtain its optimum design [13]




119904119905= 090 119908

1199051198952= 010 119908

1198911= 0305 119908

119904119905= 114 119908

1199051198952= 010

gap = 023 119908119898= 010 119897

119898= 200 119908

1198912= 040 119897

119904119905= 130 119897

119898= 200

119897119891= 495 119889

1= 114 119889

2= 067 gap1 = 030 119908

119898= 010 119889

2= 045

119908119904119905= 20 119908

1199051198951= 012 119889

3= 091 gap2 = 020 119889

1= 112 119889

3= 050

119897119891= 455 119908

1199051198951= 030

2 22 24 26 28 3

0

Frequency (GHz)


minus25

minus20

minus15

minus10

minus5

S 21

andS 1

1(d

B)




119901=


1-2GHz and 3-4GHz 119891119891= 20GHz 119891

119897= 30GHz119873

1= 51

1198732= 212 and 119873






(11987821) + 120577 (2)



0 2 4 61

2

3

4

5

6

Freq

uenc

y (G

Hz)

minus4 minus2

120573l

120573 = 0 at f = 24GHz


1 15 2 25 3 35 4

0

10

20

Frequency (GHz)

minus40

minus30

minus20

minus10

120583eff

120576eff

120576eff120583eff









119904119905= 100 119908

1199051198952= 010 119908

119891= 043 119897

119904119905= 100 119908

1199051198952= 010

gap = 0588 119908119898= 0218 119897

119898= 265 gap = 056 119908

119898= 025 119897

119898= 270

119897119891= 4226 119889

1= 120 119889

2= 050 119897

119891= 344 119889

1= 12 119889

2= 080

119908119904119905= 117 119908

1199051198951= 020 119889

3= 000 119908

119904119905= 133 119908

1199051198951= 020 119889

3= 000



119904119905= 105 119908

1199051198952= 010 119908

119891= 052 119897

119904119905= 100 119908

1199051198952= 010

gap = 069 119908119898= 0218 119897

119898= 265 gap = 07 119908

119898= 025 119897

119898= 270

119897119891= 359 119889

1= 110 119889

2= 070 119897

119891= 347 119889

1= 092 119889

2= 070

119908119904119905= 133 119908

1199051198951= 020 119889

3= 000 119908

119904119905= 175 119908

1199051198951= 020 119889

3= 000

CRLH unit cell 1


Lt


0

minus80

minus60

minus40

minus20

2 22 24 26 28 3

Frequency (GHz)


S 21

andS 1

1(d

B)

S21

S11




119901= 05 dB 119892

119904= 30 dB

1198731= 51119873

2= 212 and119873

3= 1198734= 45



and 11987821


0 2 4 62

22

24

26

28

3

Freq

uenc

y (G

Hz)


120573l (rad)

120573l = 0 at f = 24GHz








1 2 3 4 5 6 7 8 9 10

0

Frequency (GHz)

minus80

minus60

minus40

minus20

S 21

(dB)


2 3 4 5 6

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

(dB)

(a)

2 25 3 35 4 45 5 55 6

0

5

Frequency (GHz)


minus30

minus25

minus20

minus15

minus10

minus5

S 11

(dB)

(b)


11




0= 52GHz




0 2 4 6 82

3

4

5

6

Freq

uenc

y (G

Hz)


120573l

f = 24GHz 120573l = minus28

f = 52GHz 120573l = +6









119904119905= 200 119908

1199051198952= 010 119908

11989112= 027 amp 020 119897

119904119905= 110 119908

1199051198952= 010

gap = 048 119908119898= 024 119897

119898= 200 gap = 045 119908

119898= 010 119897

119898= 200

11989711989112

= 127 amp 126 1198891= 035 119889

2= 037 119897

11989112= 130 amp 170 119889

1= 040 119889

2= 040

119908119904119905= 117 119908

1199051198951= 020 119889

3= 403 119908

119904119905= 112 119908

1199051198951= 020 119889

3= 495



119904119905= 190 119908

1199051198952= 010 119908

119891= 025 119897

119904119905= 120 119908

1199051198952= 010

gap = 047 119908119898= 024 119897

119898= 200 gap = 040 119908

119898= 010 119897

119898= 200

11989711989112

= 140 amp 130 1198891= 050 119889

2= 050 119897

11989112= 110 amp 130 119889

1= 050 119889

2= 050

119908119904119905= 110 119908

1199051198951= 020 119889

3= 430 119908

119904119905= 114 119908

1199051198951= 020 119889

3= 495

(a) (b)




21



7 Conclusion


2 22 24 26 28 3

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

andS 1

1(d

B)

S21 S11




2 25 3 35 4 45 5 55 6

0

Frequency (GHz)


minus40

minus30

minus20

minus10

S 21

andS 1

1(d

B) S21

S11






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












119904119905= 090 119908

1199051198952= 010 119908

1198911= 0305 119908

119904119905= 114 119908

1199051198952= 010

gap = 023 119908119898= 010 119897

119898= 200 119908

1198912= 040 119897

119904119905= 130 119897

119898= 200

119897119891= 495 119889

1= 114 119889

2= 067 gap1 = 030 119908

119898= 010 119889

2= 045

119908119904119905= 20 119908

1199051198951= 012 119889

3= 091 gap2 = 020 119889

1= 112 119889

3= 050

119897119891= 455 119908

1199051198951= 030

2 22 24 26 28 3

0

Frequency (GHz)


minus25

minus20

minus15

minus10

minus5

S 21

andS 1

1(d

B)




119901=


1-2GHz and 3-4GHz 119891119891= 20GHz 119891

119897= 30GHz119873

1= 51

1198732= 212 and 119873






(11987821) + 120577 (2)



0 2 4 61

2

3

4

5

6

Freq

uenc

y (G

Hz)

minus4 minus2

120573l

120573 = 0 at f = 24GHz


1 15 2 25 3 35 4

0

10

20

Frequency (GHz)

minus40

minus30

minus20

minus10

120583eff

120576eff

120576eff120583eff









119904119905= 100 119908

1199051198952= 010 119908

119891= 043 119897

119904119905= 100 119908

1199051198952= 010

gap = 0588 119908119898= 0218 119897

119898= 265 gap = 056 119908

119898= 025 119897

119898= 270

119897119891= 4226 119889

1= 120 119889

2= 050 119897

119891= 344 119889

1= 12 119889

2= 080

119908119904119905= 117 119908

1199051198951= 020 119889

3= 000 119908

119904119905= 133 119908

1199051198951= 020 119889

3= 000



119904119905= 105 119908

1199051198952= 010 119908

119891= 052 119897

119904119905= 100 119908

1199051198952= 010

gap = 069 119908119898= 0218 119897

119898= 265 gap = 07 119908

119898= 025 119897

119898= 270

119897119891= 359 119889

1= 110 119889

2= 070 119897

119891= 347 119889

1= 092 119889

2= 070

119908119904119905= 133 119908

1199051198951= 020 119889

3= 000 119908

119904119905= 175 119908

1199051198951= 020 119889

3= 000

CRLH unit cell 1


Lt


0

minus80

minus60

minus40

minus20

2 22 24 26 28 3

Frequency (GHz)


S 21

andS 1

1(d

B)

S21

S11




119901= 05 dB 119892

119904= 30 dB

1198731= 51119873

2= 212 and119873

3= 1198734= 45



and 11987821


0 2 4 62

22

24

26

28

3

Freq

uenc

y (G

Hz)


120573l (rad)

120573l = 0 at f = 24GHz








1 2 3 4 5 6 7 8 9 10

0

Frequency (GHz)

minus80

minus60

minus40

minus20

S 21

(dB)


2 3 4 5 6

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

(dB)

(a)

2 25 3 35 4 45 5 55 6

0

5

Frequency (GHz)


minus30

minus25

minus20

minus15

minus10

minus5

S 11

(dB)

(b)


11




0= 52GHz




0 2 4 6 82

3

4

5

6

Freq

uenc

y (G

Hz)


120573l

f = 24GHz 120573l = minus28

f = 52GHz 120573l = +6









119904119905= 200 119908

1199051198952= 010 119908

11989112= 027 amp 020 119897

119904119905= 110 119908

1199051198952= 010

gap = 048 119908119898= 024 119897

119898= 200 gap = 045 119908

119898= 010 119897

119898= 200

11989711989112

= 127 amp 126 1198891= 035 119889

2= 037 119897

11989112= 130 amp 170 119889

1= 040 119889

2= 040

119908119904119905= 117 119908

1199051198951= 020 119889

3= 403 119908

119904119905= 112 119908

1199051198951= 020 119889

3= 495



119904119905= 190 119908

1199051198952= 010 119908

119891= 025 119897

119904119905= 120 119908

1199051198952= 010

gap = 047 119908119898= 024 119897

119898= 200 gap = 040 119908

119898= 010 119897

119898= 200

11989711989112

= 140 amp 130 1198891= 050 119889

2= 050 119897

11989112= 110 amp 130 119889

1= 050 119889

2= 050

119908119904119905= 110 119908

1199051198951= 020 119889

3= 430 119908

119904119905= 114 119908

1199051198951= 020 119889

3= 495

(a) (b)




21



7 Conclusion


2 22 24 26 28 3

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

andS 1

1(d

B)

S21 S11




2 25 3 35 4 45 5 55 6

0

Frequency (GHz)


minus40

minus30

minus20

minus10

S 21

andS 1

1(d

B) S21

S11






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119904119905= 100 119908

1199051198952= 010 119908

119891= 043 119897

119904119905= 100 119908

1199051198952= 010

gap = 0588 119908119898= 0218 119897

119898= 265 gap = 056 119908

119898= 025 119897

119898= 270

119897119891= 4226 119889

1= 120 119889

2= 050 119897

119891= 344 119889

1= 12 119889

2= 080

119908119904119905= 117 119908

1199051198951= 020 119889

3= 000 119908

119904119905= 133 119908

1199051198951= 020 119889

3= 000



119904119905= 105 119908

1199051198952= 010 119908

119891= 052 119897

119904119905= 100 119908

1199051198952= 010

gap = 069 119908119898= 0218 119897

119898= 265 gap = 07 119908

119898= 025 119897

119898= 270

119897119891= 359 119889

1= 110 119889

2= 070 119897

119891= 347 119889

1= 092 119889

2= 070

119908119904119905= 133 119908

1199051198951= 020 119889

3= 000 119908

119904119905= 175 119908

1199051198951= 020 119889

3= 000

CRLH unit cell 1


Lt


0

minus80

minus60

minus40

minus20

2 22 24 26 28 3

Frequency (GHz)


S 21

andS 1

1(d

B)

S21

S11




119901= 05 dB 119892

119904= 30 dB

1198731= 51119873

2= 212 and119873

3= 1198734= 45



and 11987821


0 2 4 62

22

24

26

28

3

Freq

uenc

y (G

Hz)


120573l (rad)

120573l = 0 at f = 24GHz








1 2 3 4 5 6 7 8 9 10

0

Frequency (GHz)

minus80

minus60

minus40

minus20

S 21

(dB)


2 3 4 5 6

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

(dB)

(a)

2 25 3 35 4 45 5 55 6

0

5

Frequency (GHz)


minus30

minus25

minus20

minus15

minus10

minus5

S 11

(dB)

(b)


11




0= 52GHz




0 2 4 6 82

3

4

5

6

Freq

uenc

y (G

Hz)


120573l

f = 24GHz 120573l = minus28

f = 52GHz 120573l = +6









119904119905= 200 119908

1199051198952= 010 119908

11989112= 027 amp 020 119897

119904119905= 110 119908

1199051198952= 010

gap = 048 119908119898= 024 119897

119898= 200 gap = 045 119908

119898= 010 119897

119898= 200

11989711989112

= 127 amp 126 1198891= 035 119889

2= 037 119897

11989112= 130 amp 170 119889

1= 040 119889

2= 040

119908119904119905= 117 119908

1199051198951= 020 119889

3= 403 119908

119904119905= 112 119908

1199051198951= 020 119889

3= 495



119904119905= 190 119908

1199051198952= 010 119908

119891= 025 119897

119904119905= 120 119908

1199051198952= 010

gap = 047 119908119898= 024 119897

119898= 200 gap = 040 119908

119898= 010 119897

119898= 200

11989711989112

= 140 amp 130 1198891= 050 119889

2= 050 119897

11989112= 110 amp 130 119889

1= 050 119889

2= 050

119908119904119905= 110 119908

1199051198951= 020 119889

3= 430 119908

119904119905= 114 119908

1199051198951= 020 119889

3= 495

(a) (b)




21



7 Conclusion


2 22 24 26 28 3

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

andS 1

1(d

B)

S21 S11




2 25 3 35 4 45 5 55 6

0

Frequency (GHz)


minus40

minus30

minus20

minus10

S 21

andS 1

1(d

B) S21

S11






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1 2 3 4 5 6 7 8 9 10

0

Frequency (GHz)

minus80

minus60

minus40

minus20

S 21

(dB)


2 3 4 5 6

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

(dB)

(a)

2 25 3 35 4 45 5 55 6

0

5

Frequency (GHz)


minus30

minus25

minus20

minus15

minus10

minus5

S 11

(dB)

(b)


11




0= 52GHz




0 2 4 6 82

3

4

5

6

Freq

uenc

y (G

Hz)


120573l

f = 24GHz 120573l = minus28

f = 52GHz 120573l = +6









119904119905= 200 119908

1199051198952= 010 119908

11989112= 027 amp 020 119897

119904119905= 110 119908

1199051198952= 010

gap = 048 119908119898= 024 119897

119898= 200 gap = 045 119908

119898= 010 119897

119898= 200

11989711989112

= 127 amp 126 1198891= 035 119889

2= 037 119897

11989112= 130 amp 170 119889

1= 040 119889

2= 040

119908119904119905= 117 119908

1199051198951= 020 119889

3= 403 119908

119904119905= 112 119908

1199051198951= 020 119889

3= 495



119904119905= 190 119908

1199051198952= 010 119908

119891= 025 119897

119904119905= 120 119908

1199051198952= 010

gap = 047 119908119898= 024 119897

119898= 200 gap = 040 119908

119898= 010 119897

119898= 200

11989711989112

= 140 amp 130 1198891= 050 119889

2= 050 119897

11989112= 110 amp 130 119889

1= 050 119889

2= 050

119908119904119905= 110 119908

1199051198951= 020 119889

3= 430 119908

119904119905= 114 119908

1199051198951= 020 119889

3= 495

(a) (b)




21



7 Conclusion


2 22 24 26 28 3

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

andS 1

1(d

B)

S21 S11




2 25 3 35 4 45 5 55 6

0

Frequency (GHz)


minus40

minus30

minus20

minus10

S 21

andS 1

1(d

B) S21

S11






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













119904119905= 200 119908

1199051198952= 010 119908

11989112= 027 amp 020 119897

119904119905= 110 119908

1199051198952= 010

gap = 048 119908119898= 024 119897

119898= 200 gap = 045 119908

119898= 010 119897

119898= 200

11989711989112

= 127 amp 126 1198891= 035 119889

2= 037 119897

11989112= 130 amp 170 119889

1= 040 119889

2= 040

119908119904119905= 117 119908

1199051198951= 020 119889

3= 403 119908

119904119905= 112 119908

1199051198951= 020 119889

3= 495



119904119905= 190 119908

1199051198952= 010 119908

119891= 025 119897

119904119905= 120 119908

1199051198952= 010

gap = 047 119908119898= 024 119897

119898= 200 gap = 040 119908

119898= 010 119897

119898= 200

11989711989112

= 140 amp 130 1198891= 050 119889

2= 050 119897

11989112= 110 amp 130 119889

1= 050 119889

2= 050

119908119904119905= 110 119908

1199051198951= 020 119889

3= 430 119908

119904119905= 114 119908

1199051198951= 020 119889

3= 495

(a) (b)




21



7 Conclusion


2 22 24 26 28 3

0

Frequency (GHz)


minus80

minus60

minus40

minus20

S 21

andS 1

1(d

B)

S21 S11




2 25 3 35 4 45 5 55 6

0

Frequency (GHz)


minus40

minus30

minus20

minus10

S 21

andS 1

1(d

B) S21

S11






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










2 25 3 35 4 45 5 55 6

0

Frequency (GHz)


minus40

minus30

minus20

minus10

S 21

andS 1

1(d

B) S21

S11






References


















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Research Article Design of Miniaturized Multiband Filters...

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Transcript of Research Article Design of Miniaturized Multiband Filters...