research article full-wave analysis of stable cross

Research ArticleFull-Wave Analysis of Stable Cross Fractal Frequency SelectiveSurfaces Using an Iterative Procedure Based on Wave Concept

V P Silva Neto12 M J Duarte1 and A G DrsquoAssunccedilatildeo2

1Federal Rural University of the Semiarid Region DEE RN 233 km 01 59780-000 Caraubas RN Brazil2Federal University of Rio Grande do Norte UFRN-CT-DCO PO Box 1655 59078-970 Natal RN Brazil

Correspondence should be addressed to V P Silva Neto valdemirnetoufersaedubr

Received 25 March 2015 Revised 25 June 2015 Accepted 1 July 2015

Academic Editor Xianming Qing

Copyright copy 2015 V P Silva Neto et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This work presents a full-wave analysis of stable frequency selective surfaces (FSSs) composed of periodic arrays of cross fractalpatch elementsThe shapes of these patch elements are defined conforming to a fractal concept where the generator fractal geometryis successively subdivided into parts which are smaller copies of the previous ones (defined as fractal levels) The main objective ofthis work is to investigate the performance of FSSs with cross fractal patch element geometries including their frequency responseand stability in relation to both the angle of incidence and polarization of the plane waveThe frequency response of FSS structuresis obtained using the wave concept iterative procedure (WCIP) This method is based on a wave concept formulation and theboundary conditions for the FSS structure Prototypes were manufactured and measured to verify the WCIP model accuracy Agood agreement between WCIP and measured results was observed for the proposed cross fractal FSSs In addition these FSSsexhibited good angular stability

1 Introduction

Frequency selective surfaces (FSSs) are being designed formany applications in modern communication systems Typi-cally they are used as spatial filters of electromagnetic wavesand are composed of two-dimensional periodic arrays ofmetallic patch or aperture elements behaving like band-stopor band-pass filters respectively [1ndash4] Therefore FSSs areused to reflect or transmit electromagnetic waves accordingto their array geometry resonant element type and shape andother structural parameters Therefore different FSS geome-tries may be used depending upon the desired frequencyresponse

Furthermore each FSS element resonates and dispensesenergy around its resonance frequency when there areincident electromagnetic waves Typically FSSs with aperturetype elements are used to provide band-pass filter responsewith incident wave transmission at resonant frequenciesSimilarly FSSs with metallic patch type elements are usedto provide band-stop filter response with incident wavereflection at resonant frequencies [5 6]

The frequency response of a periodic FSS geometrydepends on the element shape and type substrate parametersand array type and periodicity Therefore FSSs are used inmanymicrowave applications such as radomes subreflectorsantenna systems absorbers and ovens

Recently several papers were devoted to study improve-ments on the performance of FSS geometries such as polar-ization independence angular stability and array elementsize reduction [7ndash9] Particularly the use of fractal elementsin FSS designs has proved to be an effective method tominiaturize FSS elements and to obtain good angular stability(in relation to oblique incidence of the plane wave)

This work presents a full-wave analysis of periodic FSSstructures with cross fractal patch elements This elementshape is based on a combination of circular and crossdipole conductive patches Prototypes of band-stop filters aredesigned fabricated andmeasured for comparison purposes

Moreover this work aims to investigate the proposedFSS geometry properties regarding miniaturization angularstability and polarization independence An analysis of the

Hindawi Publishing CorporationInternational Journal of Antennas and PropagationVolume 2015 Article ID 401210 7 pageshttpdxdoiorg1011552015401210

2 International Journal of Antennas and Propagation

Tx

Ty

100mm300mm

30mm

(a)

Tx

Ty

300mm333mm

10mm

(b)

Figure 1 Cross fractal FSS elements (a) generator and (b) first fractal iteration

(a) (b)

Figure 2 Photographs of FSS prototypes fabricated with cross fractal patch element geometries (a) generator and (b) first fractal iteration

array periodicity was carried out to verify its influence on theFSS frequency response

Theoretical results were obtained using an integral anditerative method based on transversal wave concept (WCIP)The WCIP method solves electromagnetic problems usingthe concept of incident reflected and transmitted waves on acircuit interface Therefore this method is suitable to analyzeplanar microwave circuits [10] patch antennas and FSSs [11ndash16] The WICP method uses FFT to reduce the requiredcomputing time and provide both versatility and reliablerepresentation of the circuit structure

WCIP results for the frequency behavior of the designedFSS were obtained and compared to simulated results byAnsoft HFSS and experimental results for validation pur-pose Measured results were obtained using a vector networkanalyzer (Agilent Technologies model N5230A) and twohorn antennas A good agreement between simulated andmeasured results was verified

2 FSS Design

The FSS elements used in this work are presented in Figure 1The generator is based on the combination of a circular

patch element with diameter equal to 10mm and a crossdipole element with length equal to 30mm and width3mm as shown in Figure 1(a) Figure 1(b) presents the fractalgeometry for the first fractal iteration according to the crossfractal formation rule

Figure 2 depicts photographs of the fabricated FSS pro-totypes The design of FSS structures with cross fractalgeometries was developed taking into account the resonantproperties of the cross fractal elements For example theautosimilarity provided by the cross fractal element natureincreases the electrical length traversed by the surface currenton the patch resulting in lower resonant frequencies orreduced sizes (if the frequency design requirement is main-tained) Therefore the proposed cross fractal patch geome-tries can be used to decrease the FSS resonant frequency toprovide multiband behavior or to reduce the patch elementdimensions and to develop FSS structures with angularstability and independence of polarization

3 WCIP Theoretical Formulation

The wave concept iterative procedure (WCIP) is an integraland iterative method that consists in separating the FSS

International Journal of Antennas and Propagation 3

rarr

A0 rarr

A1

rarr

B1 rarrn1

rarrn2

rarr

A2rarr

B2

S

Γ2

Γ1

Medium 1

Medium 2

Figure 3 WCIP formulation at the circuit interface

structure in interfaces with upper and lower homogeneousmedia as shown in Figure 3 The boundary conditions at thecircuit surface are represented by a scattering parameter 119878defined in the spatial domain The propagation conditionsin the homogeneous media are described by the reflectioncoefficient operator Γ defined in the modal domain

Basically theWCIPmethod is described by two equationsthat relate the incident ( 119860

119894) and reflected ( 119861

119894) waves at the

circuit interface These equations are given in

119861

119894=

119878

119860

119894+

119860

0

119860

119894=

Γ

119861

119894

(1)

where 1198600is a local electromagnetic wave source defined

throughout the space domain that focuses on the FSSinterface from one of the means The relationship betweenthe incident and the reflected waves and the independentvariables which are the tangential electric field 997888119864

119879119894and the

tangential magnetic field 119867119879119894 are given by

119860

119894=

1

2radic119885

0119894

(

119864

119894+119885

0119894(

119867

119879119894times 119899))

119861

119894=

1

2radic119885

0119894

(

119864

119894minus119885

0119894(

119867

119879119894times 119899))

(2)

where 119894 indicates medium 1 or 2 at the interface and1198850119894is the

characteristic impedance of medium 119894 given by

119885

0119894= radic

120583

0

120576

0120576

119903119894

(3)

1205830 and 1205760 are the free space permeability and permittivityrespectively and 120576

119903119894is the relative permittivity of medium 119894

In the WCIP method (1) are established in each pointat the discontinuity surface which is divided in pixels Foreach pixel a different scattering operator is associated andthis operator will consider the boundary conditions on thisparticular point of the circuit surface Different boundaryconditions can be considered such as those related to dielec-tric metal source or load pixel For the FSS studied in this

work we can identify two different cases for the scatteringoperator At metal pixels we use the operator shown in (4)and at dielectric pixels we use the operator shown in (5)

119878metal = (minus1 00 minus1) (4)

119878dielectric = (

120578

2minus 1120578

2+ 1

2120578120578

2+ 1

2120578120578

2+ 1minus

120578

2minus 1120578

2+ 1

) (5)

where 120578 = radic11988501119885

02and 119885

01and 119885

02are the characteristic

impedance of media 1 and 2 respectively The source ismodeled as a distributed source defined in themodal domainwithout changing the scattering parameter

The reflection coefficient operator Γ assigns the propaga-tion boundary conditions on the upper and lower media andtakes into account the characteristics of the wave propagationin the medium around the circuit interface This operatordefines the relationships between the incident and reflectedwaves in the media around the circuit interfaces In additionit is defined in the modal domain because the wave isdecomposed into TE

119898119899and TM

119898119899modes which are taken

into account considering the reflection coefficients given by(6) and (7) where 119898 = 1 2 3 and 119899 = 1 2 3 respectively

Γ

TE119898119899119894=

1 minus 1198850119894119884TE119898119899119894

1 + 1198850119894119884TE119898119899119894 (6)

Γ

TM119898119899119894=

1 minus 1198850119894119884TM119898119899119894

1 + 1198850119894119884TM119898119899119894 (7)

1198850119894 is the characteristic impedance of medium 119894 and119884TE119898119899119894

and119884

TM119898119899119894

are respectively the TE119898119899

and TM119898119899

modes equivalentadmittances for medium 119894 The link between the spatialand modal domains is obtained using the Modal FourierTransform (FMT) Initially the waves are described in thespectral domain by the fast Fourier transform (FFT) Thenthey are decomposed into waves of TE

119898119899and TM

119898119899modes

passing them to the modal domain [14] This procedure isrepeated until convergence is achieved

4 Results and Discussion

The first step of this work is search the best value for thearray periodicity of the FSS geometry for a given frequencyband In this case a parametric study was performed onthe periodicity of FSS Figures 4(a) and 4(b) show thetransmission coefficient as function of frequency for differentvalues of the FSS array periodicity We should search forthe best value of array periodicity to get a good frequencyresponse for the FSS geometry with the generator elementsshown in Figure 2(a) in the WiMax frequency range

Taking into account the results shown in Figure 4 wechose those corresponding to an array periodicity 119879 equal


1 2 3 4 5 6 7 8 9 10 11 12minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

T = 30mmT = 31mmT = 32mmT = 33mm

T = 34mmT = 35mmT = 36mm

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

T = 30mmT = 31mmT = 32mmT = 33mm

T = 34mmT = 35mmT = 36mm

(b)

Figure 4 Simulation results for the transmission coefficient for different FSS array periodicity values (a) FSS1(generator) and (b) FSS

2(first

fractal iteration)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

FSS1(HFSS)FSS2(HFSS)

FSS1(WCIP)FSS2(WCIP)

Figure 5 Simulated results for the transmission coefficient for FSSwith cross fractal patch elements shown in Figure 2

to 32mm (FSS1) In this particular case the FSS presents a

good frequency response in theWiMax frequency range witha resonant frequency equal to 32 GHz a bandwidth (BW)equal to 09GHz and a transmission coefficient (119878

21) equal

to minus2972 dBFigure 5 shows the results for the transmission coefficient

for the FSS geometries with cross fractal elements shown inFigure 2 We confirmed an inversely proportional relation-ship between the FSS resonant frequencies and the fractallevel of the proposed FSS patch elementsThe accuracy of theWCIP method was verified by comparing simulated WCIPand HFSS results

Figure 6 presents simulated (WCIP and HFSS) and mea-sured results for the transmission coefficient of the FSSwith generator elements (FSS

1) The WCIP results for the

resonant frequency and bandwidth (for a minus10 dB reference)are 326GHz and 102GHz respectivelyThemeasured resultsare in good agreement with the WCIP simulated ones

presenting a relative error of 17 at the resonant frequencyThemeasured bandwidth (BW) result is 103GHz presentinga relative error of 01 with respect to the WCIP result

Figure 7 depicts the frequency response of the FSS withcross fractal patch elements (FSS

2) The curve shows that

the FSS has a first resonant frequency at 273GHz withan insertion loss equal to minus2568 dB and a bandwidth of650MHz This FSS geometry presents a second resonanceat 977GHz with 119878

21= minus2490 dB and a bandwidth BW =

910MHz We verified that the FSS with cross fractal patchelements (FSS

2) exhibited a compression factor of 1534

enabling a FSS patch element size reduction Also we verifieda dual band behavior in the considered frequency range

The angular stability and independence of polarizationfor the proposed FSS structures were investigated by varyingthe plane wave incidence angle from 0∘ (normal incidence) to60∘ with a 20∘ step as shown in Figures 8 and 9

Figure 8 showsmeasured results of the transmission coef-ficient frequency behavior for the proposed FSS structures(a) FSS

1and (b) FSS

2and TE polarization Figure 9 shows

measured results of the transmission coefficient frequencybehavior for the proposed FSS structures (a) FSS

1and (b)

FSS2and TM polarization

Table 1 exhibits measured values for the proposed FSSsfirst resonant frequency 119891

119903 and bandwidth BW in order

to compare their angular stability and polarization indepen-dence performances

According to measured results the FSS with cross fractalelements has a good angular stability with respect to the planewave incidence angle In addition the analyzed FSS

1struc-

ture with generator elements presented high independencein relation to the polarization plane wave incident at the firstresonance and a tuning possibility at the second resonanceband However the analyzed FSS

2structure with first fractal

iteration elements presented high independence in relationto the polarization planewave incident at the first and secondresonance bands


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements

Figure 6 Simulated and measured results of the transmission coefficient for the FSS with cross fractal patch elements (FSS1)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)

Figure 8 Measured transmission coefficient concerning the plane wave incidence angle and TE polarization for the proposed FSS structures(a) FSS

1and (b) FSS

2


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)

Figure 9Measured transmission coefficient concerning the plane wave incidence angle and TMpolarization for the proposed FSS structures(a) FSS

1and (b) FSS

2

Table 1 Measured results for FSS transmission coefficient frequency behaviors concerning the plane wave incidence angle and TE and TMpolarization

120579

FSS1 FSS2TE TM TE TM

119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz)

0∘ 326 102 318 102 273 0650 270 064920∘ 328 125 324 122 269 0900 271 095340∘ 319 136 317 142 270 0960 270 095360∘ 324 209 317 211 273 114 272 115

5 Conclusions

In this work we analyzed and designed FSSs with crossfractal frequency selective surfaces The FSS structures wereanalyzed by theoretical and experimental investigation Par-ticularly the proposed FSS geometries were analyzed throughthe efficient and accurate WCIP method Thereafter HFSSandmeasured results were obtained for comparison purposeIt was shown that the proposed FSS geometries presentedexcellent polarization stability angular stability and patchelementminiaturization In addition the FSS designmethod-ology described in this paper was validated by excellentagreement observed between theoretical (WCIP) simulation(HFSS) and measured results

Conflict of Interests

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

Acknowledgments

This work was supported by CNPq under covenant5739392008-0 (INCT-CSF) and under contract 5526592011-8 CAPES Federal University of Rio Grande do Norte(UFRN) and Federal Rural University of the SemiaridRegion (UFERSA)

References

[1] J I A Trindade P H F Silva A L P S Campos andA G DrsquoAssuncao ldquoAnalysis of stop-band frequency selectivesurfaces with Durerrsquos pentagon pre-fractals patch elementsrdquoIEEE Transactions on Magnetics vol 47 no 5 pp 1518ndash15212011

[2] B Sanz-Izquierdo and E A Parker ldquoDual polarized recon-figurable frequency selective surfacesrdquo IEEE Transactions onAntennas and Propagation vol 62 no 2 pp 764ndash771 2014

[3] J Romeu and Y Rahmat-Samii ldquoFractal FSS a novel dual-bandfrequency selective surfacerdquo IEEETransactions onAntennas andPropagation vol 48 no 7 pp 1097ndash1105 2000

[4] H-Y Yang S-X Gong P-F Zhang F-T Zha and J Ling ldquoAnovel miniaturized frequency selective surface with excellentcenter frequency stabilityrdquo Microwave and Optical TechnologyLetters vol 51 no 10 pp 2513ndash2516 2009

[5] T KWu Ed Frequency Selective Surface andGrid ArrayWileyNew York NY USA 1995

[6] B A Munk Frequency-Selective Surfaces Theory and DesignWiley New York NY USA 2000

[7] A G DrsquoAssuncao Jr G Fontgalland A Gomes Neto and HBaudrand ldquoFrequency selective surface filters with polarizedband passband reject performancesrdquo Microwave and OpticalTechnology Letters vol 56 no 2 pp 483ndash487 2014

[8] P H F Silva C L Nobrega M R Silva and A G drsquoAssuncaoldquoOptimal design of frequency selective surfaces with fractalmotifsrdquo IET Microwaves Antennas amp Propagation vol 8 no 9pp 627ndash631 2014


[9] M R Silva A G DrsquoAssuncao P H Fonseca da Silva andC L Nobrega ldquoStable and compact multiband frequencyselective surfaces with Peano pre-fractal configurationsrdquo IETMicrowaves Antennas amp Propagation vol 7 pp 543ndash551 2013

[10] V P Silva Neto Characterization of microwave printed circuitsusing WCIP method [MS thesis] Federal University of RioGrande do Norte Natal Brazil 2013 (Portuguese)

[11] P B C Medeiros V P Silva Neto and A G DrsquoAssuncaoldquoA compact and stable design of FSS with radial slit circularelements using an iterative methodrdquo Microwave and OpticalTechnology Letters vol 57 no 3 pp 729ndash733 2015

[12] M Titaouine A G Neto H Baudrand and F Djahli ldquoAnalysisof frequency selective surface on isotropicanisotropic layersusing WCIP methodrdquo ETRI Journal vol 29 no 1 pp 36ndash442007

[13] A Serres G K F Serres G Fontgalland R C S Freire andH Baudrand ldquoAnalysis of multilayer amplifier structure by anefficient iterative techniquerdquo IEEE Transactions on Magneticsvol 50 no 2 pp 185ndash188 2014

[14] R S NrsquoGongo and H Baudrand ldquoApplication of wave conceptiterative procedure in planar circuitrdquo Recent Research Develop-ments in Microwave Theory and Technique vol 1 pp 187ndash1971999

[15] G A Cavalcante A G DrsquoAssuncao Jr and A G DrsquoAssuncaoldquoAn iterative full-wave method for designing bandstop fre-quency selective surfaces on textile substratesrdquo Microwave andOptical Technology Letters vol 56 no 2 pp 383ndash388 2014

[16] M Titaouine A Gomes Neto H Baudrand and F DjahlildquoWCIP method applied to active frequency selective surfacesrdquoJournal of Microwaves and Optoelectronics vol 6 no 1 pp 1ndash162007

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Propagation




Navigation and Observation



DistributedSensor Networks



Tx

Ty

100mm300mm

30mm

(a)

Tx

Ty

300mm333mm

10mm

(b)

Figure 1 Cross fractal FSS elements (a) generator and (b) first fractal iteration

(a) (b)

Figure 2 Photographs of FSS prototypes fabricated with cross fractal patch element geometries (a) generator and (b) first fractal iteration

array periodicity was carried out to verify its influence on theFSS frequency response

Theoretical results were obtained using an integral anditerative method based on transversal wave concept (WCIP)The WCIP method solves electromagnetic problems usingthe concept of incident reflected and transmitted waves on acircuit interface Therefore this method is suitable to analyzeplanar microwave circuits [10] patch antennas and FSSs [11ndash16] The WICP method uses FFT to reduce the requiredcomputing time and provide both versatility and reliablerepresentation of the circuit structure

WCIP results for the frequency behavior of the designedFSS were obtained and compared to simulated results byAnsoft HFSS and experimental results for validation pur-pose Measured results were obtained using a vector networkanalyzer (Agilent Technologies model N5230A) and twohorn antennas A good agreement between simulated andmeasured results was verified

2 FSS Design

The FSS elements used in this work are presented in Figure 1The generator is based on the combination of a circular

patch element with diameter equal to 10mm and a crossdipole element with length equal to 30mm and width3mm as shown in Figure 1(a) Figure 1(b) presents the fractalgeometry for the first fractal iteration according to the crossfractal formation rule

Figure 2 depicts photographs of the fabricated FSS pro-totypes The design of FSS structures with cross fractalgeometries was developed taking into account the resonantproperties of the cross fractal elements For example theautosimilarity provided by the cross fractal element natureincreases the electrical length traversed by the surface currenton the patch resulting in lower resonant frequencies orreduced sizes (if the frequency design requirement is main-tained) Therefore the proposed cross fractal patch geome-tries can be used to decrease the FSS resonant frequency toprovide multiband behavior or to reduce the patch elementdimensions and to develop FSS structures with angularstability and independence of polarization

3 WCIP Theoretical Formulation

The wave concept iterative procedure (WCIP) is an integraland iterative method that consists in separating the FSS


rarr

A0 rarr

A1

rarr

B1 rarrn1

rarrn2

rarr

A2rarr

B2

S

Γ2

Γ1

Medium 1

Medium 2







119861

119894=

119878

119860

119894+

119860

0

119860

119894=

Γ

119861

119894

(1)



119879119894and the


119860

119894=

1

2radic119885

0119894

(

119864

119894+119885

0119894(

119867

119879119894times 119899))

119861

119894=

1

2radic119885

0119894

(

119864

119894minus119885

0119894(

119867

119879119894times 119899))

(2)



119885

0119894= radic

120583

0

120576

0120576

119903119894

(3)







120578

2minus 1120578

2+ 1

2120578120578

2+ 1

2120578120578

2+ 1minus

120578

2minus 1120578

2+ 1

) (5)

where 120578 = radic11988501119885

02and 119885

01and 119885




119898119899and TM



Γ

TE119898119899119894=

1 minus 1198850119894119884TE119898119899119894

1 + 1198850119894119884TE119898119899119894 (6)

Γ

TM119898119899119894=

1 minus 1198850119894119884TM119898119899119894

1 + 1198850119894119884TM119898119899119894 (7)


and119884

TM119898119899119894


and TM119898119899


119898119899and TM

119898119899modes






1 2 3 4 5 6 7 8 9 10 11 12minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

T = 30mmT = 31mmT = 32mmT = 33mm

T = 34mmT = 35mmT = 36mm

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

T = 30mmT = 31mmT = 32mmT = 33mm

T = 34mmT = 35mmT = 36mm

(b)


2(first

fractal iteration)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)






21) equal
















1and (b) FSS



1and (b)






1struc-





1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


120579


119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891


0∘ 326 102 318 102 273 0650 270 064920∘ 328 125 324 122 269 0900 271 095340∘ 319 136 317 142 270 0960 270 095360∘ 324 209 317 211 273 114 272 115

5 Conclusions




Acknowledgments


References




















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










rarr

A0 rarr

A1

rarr

B1 rarrn1

rarrn2

rarr

A2rarr

B2

S

Γ2

Γ1

Medium 1

Medium 2







119861

119894=

119878

119860

119894+

119860

0

119860

119894=

Γ

119861

119894

(1)



119879119894and the


119860

119894=

1

2radic119885

0119894

(

119864

119894+119885

0119894(

119867

119879119894times 119899))

119861

119894=

1

2radic119885

0119894

(

119864

119894minus119885

0119894(

119867

119879119894times 119899))

(2)



119885

0119894= radic

120583

0

120576

0120576

119903119894

(3)







120578

2minus 1120578

2+ 1

2120578120578

2+ 1

2120578120578

2+ 1minus

120578

2minus 1120578

2+ 1

) (5)

where 120578 = radic11988501119885

02and 119885

01and 119885




119898119899and TM



Γ

TE119898119899119894=

1 minus 1198850119894119884TE119898119899119894

1 + 1198850119894119884TE119898119899119894 (6)

Γ

TM119898119899119894=

1 minus 1198850119894119884TM119898119899119894

1 + 1198850119894119884TM119898119899119894 (7)


and119884

TM119898119899119894


and TM119898119899


119898119899and TM

119898119899modes






1 2 3 4 5 6 7 8 9 10 11 12minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

T = 30mmT = 31mmT = 32mmT = 33mm

T = 34mmT = 35mmT = 36mm

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

T = 30mmT = 31mmT = 32mmT = 33mm

T = 34mmT = 35mmT = 36mm

(b)


2(first

fractal iteration)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)






21) equal
















1and (b) FSS



1and (b)






1struc-





1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


120579


119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891


0∘ 326 102 318 102 273 0650 270 064920∘ 328 125 324 122 269 0900 271 095340∘ 319 136 317 142 270 0960 270 095360∘ 324 209 317 211 273 114 272 115

5 Conclusions




Acknowledgments


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 11 12minus40

minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

T = 30mmT = 31mmT = 32mmT = 33mm

T = 34mmT = 35mmT = 36mm

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

T = 30mmT = 31mmT = 32mmT = 33mm

T = 34mmT = 35mmT = 36mm

(b)


2(first

fractal iteration)

1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)






21) equal
















1and (b) FSS



1and (b)






1struc-





1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


120579


119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891


0∘ 326 102 318 102 273 0650 270 064920∘ 328 125 324 122 269 0900 271 095340∘ 319 136 317 142 270 0960 270 095360∘ 324 209 317 211 273 114 272 115

5 Conclusions




Acknowledgments


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 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

HFSSWCIP

Measurements


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


1 2 3 4 5 6 7 8 9 10 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


120579


119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891


0∘ 326 102 318 102 273 0650 270 064920∘ 328 125 324 122 269 0900 271 095340∘ 319 136 317 142 270 0960 270 095360∘ 324 209 317 211 273 114 272 115

5 Conclusions




Acknowledgments


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 11 12minus35

minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(a)

1 2 3 4 5 6 7 8 9 10 11 12minus30

minus25

minus20

minus15

minus10

minus5

0

Frequency (GHz)

Tran

smiss

ion

coeffi

cien

t (dB

)

0∘

20∘

40∘

60∘

(b)


1and (b) FSS

2


120579


119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891

119903(GHz) BW (GHz) 119891


0∘ 326 102 318 102 273 0650 270 064920∘ 328 125 324 122 269 0900 271 095340∘ 319 136 317 142 270 0960 270 095360∘ 324 209 317 211 273 114 272 115

5 Conclusions




Acknowledgments


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 full-wave analysis of stable cross

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