Research Article A New Method of Designing Circularly...

Research ArticleA New Method of Designing Circularly Symmetric ShapedDual Reflector Antennas Using Distorted Conics

Mohammad Asif Zaman and Md Abdul Matin

Department of Electrical and Electronic Engineering Bangladesh University of Engineering and Technology Dhaka 1000 Bangladesh

Correspondence should be addressed to Mohammad Asif Zaman asifzaman13gmailcom

Received 30 July 2014 Accepted 1 December 2014 Published 17 December 2014

Academic Editor Ramon Gonzalo

Copyright copy 2014 M A Zaman and Md A Matin This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

A newmethod of designing circularly symmetric shaped dual reflector antennas using distorted conics is presentedThe surface ofthe shaped subreflector is expressed using a new set of equations employing differential geometry The proposed equations requireonly a small number of parameters to accurately describe practical shaped subreflector surfaces A geometrical optics (GO) basedmethod is used to synthesize the shaped main reflector surface corresponding to the shaped subreflector Using the proposedmethod a shaped Cassegrain dual reflector system is designedThe field scattered from the subreflector is calculated using uniformgeometrical theory of diffraction (UTD) Finally a numerical example is provided showing how a shaped subreflector producesmore uniform illumination over the main reflector aperture compared to an unshaped subreflector

1 Introduction

Reflector antennas are widely used in radars radio astron-omy satellite communication and tracking remote sensingdeep space communication microwave and millimetre wavecommunications and so forth [1ndash3]The rapid developmentsin these fields have created demands for development ofsophisticated reflector antenna configurationsThere is also acorresponding demand for analytical numerical and exper-imental methods of design and analysis techniques of suchantennas

The configuration of the reflectors depends heavily onthe application The dual reflector antennas are preferred inmany applications because they allow convenient positioningof the feed antenna near the vertex of the main reflectorand positioning of other bulky types of equipment behindthe main reflector [3] Also the feed waveguide length isreduced [4]They also have some significant electromagneticadvantage over single reflector systems [5] Although manydual reflector configurations exist the circularly symmetricdual reflector antennas remain one of the most popularchoices for numerous applications [1]

One of the most common circularly symmetric dualreflector antennas is the Cassegrain antenna The Cassegrainantenna is composed of a hyperboloidal subreflector anda paraboloidal main reflector A feed antenna (usually ahorn antenna) illuminates the subreflector which in turnilluminates the main reflector The main reflector producesthe radiated electric field that propagates into spaceThe radi-ation performance of the dual reflector antennas depends onthe radiation characteristics of the feed and the geometricalshapes of the main reflector and the subreflector Modernwireless communication and RADAR applications enforcestringent requirements on the far-field characteristics ofthe antenna For example satellite communications imposelimitations onmaximum beamwidth andmaximum sidelobelevels of the antenna to avoid interference with adjacentsatellites [2] The traditional Cassegrain antennas have fixedgeometries and offer limited flexibilities to antenna designersAs a result the maximum performance that can be extractedfrom these antennas is limited by geometrical constraints

For high performance applications the traditional hyper-boloidparaboloidal geometry must be changed Reflectorshaping is the method of changing the shape of the reflecting

Hindawi Publishing CorporationInternational Journal of Microwave Science and TechnologyVolume 2014 Article ID 849194 8 pageshttpdxdoiorg1011552014849194

2 International Journal of Microwave Science and Technology

surfaces to improve the performance of the antenna Shapedreflector antennas outperform conventional unshaped reflec-tor antennas Reflector shaping allows the designers addi-tional flexibility The antenna designers have independentcontrol over relative position of the reflectors diameter of thereflectors and the curvature of the reflectors when shapedreflectors are used instead of conventional reflectors Thismakes reflector shaping an essential tool for designing highperformance reflector antennas

Many methods of designing shaped reflectors are presentin literature One of the first major articles related to reflectorshaping was published by Galindo in 1964 [6] The method isbased on geometrical optics (GO)Galindorsquosmethod requiredsolution of multiple nonlinear differential equations whichsometimes may be computationally demanding A modifiedversion of this method was presented by Lee [7] Lee dividedthe reflector surfaces into small sections and assumed thesections to be locally planar This assumption converted thedifferential equations to algebraic equations which are mucheasier to solve However the reflector surfacemust be dividedinto a large number of sections to increase the accuracy of thismethod

Another popular method for designing shaped reflectorsinvolves expanding the shaped surfaces using a set of orthog-onal basis functions [8 9] Rahmat-Samii has publishedmul-tiple research papers on this area [8ndash10]The expansion coef-ficients determine the shape of the surface A small number ofterms of the expansion set are sufficient to accurately describea shaped surface So a few expansion coefficients must bedetermined to define the surface The differential equationbased methods determine the coordinates of the points onthe reflector surfaces whereas the surface expansion basedmethod only determines the expansion coefficients Due tothe decrease in number of unknowns the surface expansionmethod is computationally less demanding The surfaceexpansion method can be incorporated with geometricaltheory of diffraction (GTD) or its uniform version uniformtheory of diffraction (UTD) to produce an accurate designalgorithm [10]These design procedures are known as diffrac-tion synthesis [8ndash10]Thismethod has been successfully usedin many applications Recently a few new efficient methodsfor designing circularly symmetric shaped dual reflectordesign have been developed One of the first significant workson thismethodwas reported byKimandLee in 2009 [11]Thismethod divides the shaped reflector surfaces into electricallysmall sections Each section is assumed to be a conventionalunshaped dual reflector system Since well-established meth-ods for analyzing conventional dual reflector system existthe radiation characteristics of each section can easily beevaluatedThe shaped surface is defined by combining all thelocal conventional surfacesThemethod requires solutions ofseveral nonlinear algebraic equations So it is computationallyconvenient Another method based on the same principlewas proposed by Moreira and Bergmann in 2011 [12] Thismethod also divides the shaped surface into small localsections The local sections are represented by unshapedconics Each conic section is optimized to produce a desiredaperture distribution which is formulated by GO methodAs these methods have recently appeared in literature most

of the advantages and drawbacks of the method have notbeen investigated Reduction in computational complexity isan obvious advantage The proposed work concentrated onpresenting an alternative method rather than improving theexisting methods

A design method that requires lesser number of parame-ters to define the shaped reflector surfaces without decreasingaccuracy of the obtained results is a challenging goal forreflector antenna designers In this paper the surface of theshaped reflector is defined using a novel equationThe shapedsubreflector surfaces are assumed to be distorted forms ofunshaped surfaces As most shaped subreflectors resem-ble their unshaped counterparts [11ndash13] the assumption islogical The shaped surfaces can therefore be representedby modified versions of the equations that represent theconventional unshaped surfaces This method of visualizingthe shaped surface as perturbeddistorted form of unshapedsurface has not been reported in literature yet As the generalformof the surface is generated from the conventional conicsonly a small number of parameters need be used to representthe shaped surface Once the subreflector surface is definedthe main reflector surface is defined using GO method andequal optical path length criterion

The paper is organized as follows Section 2 describes thegeometry of the shaped subreflector Surface equations anddifferential geometric analysis are presented in this sectionThe synthesis method of the main reflector is discussed inSection 3 Section 4 covers the numerical results Concludingremarks are made in Section 5

2 Geometry of the Shaped Subreflector

21 Surface Equations In a conventional Cassegrain geome-try the subreflector is hyperboloidal In shaped-Cassegraingeometry the subreflector is shaped to provide a desiredillumination over the aperture of themain reflector Howeverthe geometrical features of this shaped subreflector arevery similar to the geometrical features of the unshapedhyperboloid Due to these similarities the shaped surfacescan be considered a distorted form of the unshaped surfacesSo to define the geometry of the shaped subreflector theconventional Cassegrain geometry must be described first

The geometry of a Cassegrain dual reflector antenna isshown in Figure 1 The parameters describing the geometryof the Cassegrain system are

119889119901 diameter of the main reflector = 10m

119891119901 focal length of the main reflector = 5m

119889119904 diameter of the subreflector = 125m

2119888 distance between the foci = 4m

119890 = ca subreflector eccentricity = 14261

Δ119901 depth of the paraboloid = 125m

119897119901 distance from feed to paraboloid vertex = 1m

120595119900 120601119900 opening half angle of the main reflector andsubreflector = 5313∘ and 10037∘ respectively

International Journal of Microwave Science and Technology 3

ds dp

Primary focus

Hyperboloidal

Secondary focus Feed horn

Mainreflector

subreflector

(a + c)

Δp

Fs

Fp 120601o

120595o

fp

lp

2c

Figure 1 Geometry of a Cassegrain antenna

Some of these parameter values are selected from stan-dard values The other parameters are found using geometri-cal relations found in literature [3 14]

It is assumed that the feed antenna is located at the originand direction of feed radiation is towards the negative 119911 axisSo the hyperboloid must be located at the negative side ofthe 119911 axis One of the focuses of the hyperboloid must be atorigin so that it coincides with the feedThe equation of sucha hyperboloidal surface symmetric around the 119911 axis is givenby the following equation [15]

(119911 + 119888)2

1198862minus

1199092

+ 1199102

1198872= 1 (1)

where

1198872

= 1198882

minus 1198862 (2)

The parameters 119886 and 119888 are related to the position of thevertex and focus of the hyperboloid as shown in Figure 1 [15]The parameter 120588119904 is defined as the radius of surface pointprojected on the 119909119910 plane It is related to 119909 and 119910 by

1205882

119904= 1199092

+ 1199102 (3)

Equation (1) can be modified as

(119911 + 119888)2

1198862minus

1205882

119904

1198872= 1 (4)

119911 = minus119888 minus119886

119887radic1198872 + 1205882

119904 (5)

In (5) the negative square root is taken because thedefined coordinate system places the hyperboloid in thenegative 119911 axis

For a circularly symmetric shaped subreflector (4) andcorrespondingly (5) need bemodified A distortion function120575(sdot) is introduced in the equations to get the shaped surface

(119911 + 119888)2

1198862minus

1205882

119904

1198872120575 (120588119904) = 1 (6)

119911 = minus119888 minus119886

119887radic1198872 + 1205882

119904120575 (120588119904) (7)

The distortion function 120575(sdot) must be a function of 120588119904to maintain circular symmetry The shape of the surfacedepends on the expression of 120575(sdot) Through literature reviewit is found that shaped hyperboloidal subreflectors are usuallydifferent from unshaped hyperboloids near the edges [13]The shaped surface curves towards the vertex at the edges Anexponential functionwith arguments containing even powersof 120588119904 can give the desired shaped The following generalizedexpression of the distortion function is developed

120575 (120588119904) = exp(

119873

sum

119899=1

1205911198991205882119899120577119899

119904) (8)

The function contains 2119873 number of parameters denotedby 120591119899 and 120577119899 Here 120591119899 denotes the 119899th amplitude distor-tion parameter and 120577119899 denotes the 119899th exponent distortionparameter The parameters together are termed distortionparameters


Shaped 1Shaped 2

Shaped 3Unshaped

08

06

04

02

0

minus02

minus04

minus06

minus08

x(m

)

z (m)

minus36 minus35 minus34

Figure 2Multiple shaped subreflectors generated by using differentsets of distortion parameter values

For high values of 119873 the control over the curvature ofthe surface is more precise But it increases the number ofparameters required to define the surface It is found that forpractical subreflector surfaces 119873 = 2 is sufficient and only 4parameters are required to define the surface

Two-dimensional and three-dimensional representationsof shaped surfaces obtained using (7) are shown in Figures2 and 3 The figures are generated using arbitrary valuesof the distortion parameters However it is necessary toestablish that practical shaped subreflector surfaces can beaccurately represented by these equations Practical data ofa shaped hyperboloidal and subreflector surfaces are takenfrom literature [13] The dimensions are normalized withrespect to subreflector diameter in [13] By adjusting thedistortion parameters in (8) and using them in (7) a surfaceclosely resembling the shaped subreflector surface of [13] isgenerated The results are shown in Figure 4

It is clear that the unshaped surface is very differentfrom the shaped surface With 1205911 = minus33363 1205771 = 057061205912 = 21697 and 1205772 = 01951 the sum of squares of theerror between the defined shaped hyperboloidal surface andthe practical data of [13] is only 32 times 10

minus5 So the derivedequations represent the practical shaped surface very closely

22 Differential Geometric Analysis The design proceduresof reflector antennas are interrelated with the numericaltechniques that are used to analyse those antennas GO andUTD are two common numerical techniques employed forthese purposes To apply these methods it is necessary thatthe reflecting surfaces be expressed in differential geometricform Also the normal vector at each point of the surfacemust also be defined to compute scattered field using GO orUTD method [16 17]

The shaped hyperboloidal surface defined by (7) canbe represented using parameters 120588119904 and 120601119904 in differentialgeometrical form as [16 18]

r = 120588119904 cos120601119904x + 120588119904 sin120601119904y minus [119888 +119886

119887radic1198872 + 1205882

119904120575 (120588119904)] z (9)

Here (120588119904 120601119904) are the polar coordinates of the projectionof a surface point on the 119909119910 planeThe unit normal vector nsis defined by the following equation [19]

ns =(120597r120597120588119904) times (120597r120597120601119904)

1003816100381610038161003816(120597r120597120588119904) times (120597r120597120601119904)1003816100381610038161003816

(10)

The differential geometrical parameters along with thecoordinate system are shown in Figure 5 The incident rayvector (si) and the reflected ray vector (sr) are also shownin Figure 5 As the unit normal vector is required forlater calculations (10) needs to be evaluated The partialderivatives can be calculated from (9) using (8)

120597r120597120588119904

= cos120601119904x + sin120601119904y

minus[[

[

119886

2119887

120588119904120575 (120588119904) sum119873

119899=11205911198991205882119899120577119899 + 2

radic1198872 + 1205882119904120575 (120588119904)

]]

]

z

120597r120597120601119904

= minus120588119904 sin120601119904x + 120588119904 cos120601119904y

(11)

Substituting these values in (10)

ns =Λ (120588119904) cos120601119904

radicΛ2 (120588119904) + Δ2119867

(120588119904)

x +Λ (120588119904) sin120601119904

radicΛ2 (120588119904) + Δ2119867

(120588119904)

y

+Δ119867 (120588119904)

radicΛ2 (120588119904) + Δ2119867

(120588119904)

z

(12)

where

Δ119867 (120588119904) = 2119887radic1198872 + 1205882119904120575 (120588119904)

Λ (120588119904) = 119886120575 (120588119904) 120576 (120588119904)

120576 (120588119904) = 120588119904

119873

sum

119899=1

1205911198991205882119899120577119899 + 2

(13)

For a given set of distortion parameter values the shapedsubreflector surface and the normal vectors can be evaluatedusing these equations

3 Synthesizing the Main Reflector

In a dual reflector system the shape of the main reflector iscompletely dependent on the shape of the subreflector Oncethe shape of the subreflector is defined the main reflectormust be shaped so that the path lengths of the rays are


minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus050

051

minus1minus05

005

1

(a)

minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus05

005

1

minus1minus05

005

1

(b)

Figure 3 Three-dimensional representation of (a) conventional hyperboloid and (b) shaped hyperboloid

0

005

01

015

02

025

03

035

04

045

05

minus14 minus135 minus13 minus125 minus12

zds

xds

Shaped hyperboloid dataUnshaped hyperboloid

Defined shaped hyperboloid

Figure 4 Representation of practical hyperboloidal shaped subre-flector using the proposed equations

constant at an observation plane perpendicular to the mainreflector axis Using GO methods to calculate the incidentand reflected ray vectors the required position of a point onthemain reflector surface for a given point on the subreflectorsurface can be formulated

The geometry of dual reflector system with a shapedhyperboloidal subreflector is shown in Figure 6 An obser-vation plane is defined parallel to the 119909119910 plane In the two-dimensional diagram of Figure 6 the observation plane isshown as line parallel to 119909 axis going through the point(0 0 119911ref) For a ray emitted from the feed the sum of thedistances 1198891 1198892 and 1198893 must be constant (1198961) irrespectiveof the position of reflection on the subreflector surface So

1198891 + 1198892 + 1198893 = 1198961 (14)

The distance 1198961 can be calculated by considering an axialray from the feed (along the 119911 axis) From Figure 6 for theaxial ray the distance from feed to subreflector is (119886 + 119888)then the distance from the subreflector to main reflector is(119886 + 119888 + 119897119901) and finally the distance from main reflector toobservation point is (119897119901 minus 119911ref) when considering 119911ref to be anegative quantity So 1198961 is calculated as

1198961 = (119886 + 119888) + (119886 + 119888 + 119897119901) + (119897119901 minus 119911ref)

= 2 (119886 + 119888 + 119897119901) minus 119911ref(15)

The parameter 119897119901 can be related to the geometricalparameters of the unshaped subreflector 119888 and the unshapedmain reflector 119891119901 as seen in Figure 6

119897119901 = 119891119901 minus 2119888 (16)

Using (16) in (15)

1198961 = 2 (119886 minus 119888 + 119891119901) minus 119911ref (17)

The distance 1198891 is related to the coordinates of thesubreflector surface point (119909119904 119910119904 119911119904) as

1198891 = radic1199092119904

+ 1199102119904

+ 1199112119904 (18)

The reflected ray from the main reflector is parallel to thez axisThe distance 1198893 can easily be related to the coordinatesof the main reflector surface point (119909119898 119910119898 119911119898) as

1198893 = 119911119898 minus 119911ref (19)

To calculate distance 1198892 it is necessary to calculate thereflected ray vector The unit vector in the direction of thereflected ray is found from GOmethod and is given by [16]

sr = si minus 2 (ns sdot si) ns (20)


120588s 120588s

F (focus)Feed antenna position

Hyperboloidal subreflector surface

120579

O Oz x

x y

ns

r = si

sr

(a + c)

2c

120594 120601s

Figure 5 Differential geometric representation of the shaped hyperboloidal subreflector

2c

O

(a + c)

(xs ys zs)

(0 0 zref)

lp

fp

d2

d3

d1

(xm ym zm)

Main reflector

Subreflector

Feed antenna

ns

r = si

sr

srd

srm

x

z

Figure 6 Synthesis of main reflector surface for a given shaped subreflector

where

si = r =r|r|

(21)

The vector r is defined by (9) and ns is defined by (12)Once the unit vector along the reflected ray is formulated thereflected ray vector srd defined from the reflection point onthe subreflector surface can be defined as

srd = sr1198892 + 119909119904x + 119910119904y + 119911119904z (22)

In (22) the subreflector surface point vector appears asadditive termThis term is used to shift the vector from originto the subreflector surface point (119909119904 119910119904 119911119904)

Equation (22) can be reorganized as

srd = (sr sdot x) 1198892 + 119909119904 x + (sr sdot y) 1198892 + 119910119904 y

+ (sr sdot z) 1198892 + 119911119904 z(23)


The components of this vector indicate the coordinates ofthe main reflector [18 19] So

119909119898 = (sr sdot x) 1198892 + 119909119904

119910119898 = (sr sdot y) 1198892 + 119910119904

119911119898 = (sr sdot z) 1198892 + 119911119904

(24)

Substituting the value of 119911119898 from (24) to (19)

1198893 = (sr sdot z) 1198892 + 119911119904 minus 119911ref (25)

Now using (18) and (25) in (14) and solving for 1198892

1198892 =

2 (119886 minus 119888 + 119891119901) minus radic1199092119904

+ 1199102119904

+ 1199112119904

minus 119911119904

1 + (sr sdot z) (26)

Equation (26) can be used to calculate 1198892 for a givensubreflector surface point and it can be replaced in (24) tocalculate corresponding main reflector surface point Thusthe derived equations can synthesize the main reflectorsurface for an arbitrary subreflector surface

To verify the method the main reflector for an unshapedhyperboloid is synthesized It is found that the synthesizedmain reflector is an exact paraboloid which is expectedThusthe method is verified

4 Numerical Example

For numerical analysis the feed is assumed to be a conicalcorrugated horn antenna The radiated field from the horn iscalculated using standard equations [1] The fields scatteredfrom the subreflector are calculated using UTDmethod Theobservation points are taken on the main reflector surfaceFirst the scattered field from an unshaped hyperboloid iscalculatedThe results are shown in Figure 7The rapid fall ofthe reflected field indicates the reflection shadow boundary(RSB) of the subreflector [16] It can be observed fromFigure 7 that the scattered field tapers gradually as the obser-vation angle increases This creates a tapered illuminationof the main reflector Using the proposed method a shapedhyperboloidal subreflector is defined which produces a moreuniform field distribution An optimization algorithm can beused to find the optimum set of distortion parameter valuesthat will create a subreflector surface that will create a desiredillumination The scattered field for a shaped subreflectordefined by the parameters 1205911 = minus33928 1205912 = 25015 1205771 =

17212 and 1205772 = 07184 is shown in Figure 8 The valuesof the distortion parameters are calculated using differentialevolution (DE) optimization algorithm [20ndash22] The fitnessfunction in the optimization process is defined in terms ofuniformity of the scattered field As a result the algorithmconverges to a set of distortion parameter values that resultin a uniform aperture distribution It can be observed thatthe scattered field is more uniform when the subreflector isshaped

0 20 40 60 80 100

Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

Angle 120579 (deg)

Reflected fieldDiffracted field

Total scattered field

Figure 7 Scattered field from an unshaped hyperboloidal subreflec-tor

Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus450 20 40 60 80 100

Angle 120579 (deg)



Figure 8 Scattered field from a shaped hyperboloidal subreflector

5 Conclusion

A novel method of expressing the shaped reflector surfaceshas been presented in this paper The method expresses theshaped subreflector surface as distorted form of the unshapedconventional conic surface This allows us to mathematicallyexpress the shaped subreflector surface with only a fewparameters The shaped main reflector surface is synthesizedemploying GO method and using the derived equation ofthe shaped subreflector A numerical example is provided toshow how a shaped reflector system defined by the proposedmethod can create a uniform field distribution over themain reflector aperture As only a few parameters completely


describe the shaped reflector surfaces the computational bur-den of design and optimization process of shaped reflectorscan be reduced using the proposed method

Conflict of Interests

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

References

[1] J L Volakis EdAntennaEngineeringHandbookMcGraw-Hill4th edition 2007

[2] C A Balanis Ed Modern Antenna Handbook John Wiley ampSons New York NY USA 2008

[3] J W M Baars The Paraboloidal Reflector Antenna in RadioAstronomy and CommunicationsTheory and Practice Springer2007

[4] W L Stutzman and G A Thiele Antenna Theory and DesignJohn-Wiley amp Sons 1981

[5] S Silver EdMicrowave AntennaTheory and Design McGraw-Hill New York NY USA 1st edition 1949

[6] V Galindo ldquoDesign of dual-reflector antennas with arbitraryphase and amplitude distributionsrdquo IEEE Transactions onAntennas and Propagation vol 12 pp 403ndash408 1964

[7] C S Lee ldquoA simple method of dual-reflector geometrical opticssynthesisrdquoMicrowave and Optical Technology Letters vol 1 no10 pp 367ndash371 1988

[8] Y Rahmat-Samii and JMumford ldquoReflector diffraction synthe-sis using global coefficients optimization techniquesrdquo in Pro-ceedings of the Antennas and Propagation Society InternationalSymposium vol 3 pp 1166ndash1169 IEEE San Jose Calif USAJune 1989

[9] D-W Duan and Y Rahmat-Samii ldquoGeneralized diffractionsynthesis technique for high performance reflector antennasrdquoIEEE Transactions on Antennas and Propagation vol 43 no 1pp 27ndash40 1995

[10] Y Rahmat-Samii and V Galindo-Israel ldquoShaped reflectorantenna analysis using the jacob-bessel seriesrdquo IEEE Transac-tions on Antennas and Propagation vol 28 no 4 pp 425ndash4351980

[11] Y Kim and T-H Lee ldquoShaped circularly symmetric dual reflec-tor antennas by combining local conventional dual reflectorsystemsrdquo IEEE Transactions on Antennas and Propagation vol57 no 1 pp 47ndash56 2009

[12] F J S Moreira and J R Bergmann ldquoShaping axis-symmetricdual-reflector antennas by combining conic sectionsrdquo IEEETransactions on Antennas and Propagation vol 59 no 3 pp1042ndash1046 2011

[13] M S Narasimhan P Ramanujam and K Raghavan ldquoGTDanalysis of the radiation patterns of a shaped subreflectorrdquo IEEETransactions on Antennas and Propagation vol 29 no 5 pp792ndash795 1981

[14] C Granet ldquoDesigning axially symmetric cassegrain or grego-rian dual-reflector antennas from combinations of prescribedgeometric parametersrdquo IEEE Antennas and Propagation Maga-zine vol 40 no 2 pp 76ndash81 1998

[15] A D Polyanin and A V Manzhirov Handbook of Mathematicsfor Engineers and Scientists Chapman amp HallCRC Taylor ampFrancis 2007

[16] D A McNamara C W I Pistorius and J A G MalherbeIntroduction to the Uniform Geometrical Theory of DiffractionArtech House London UK 1990

[17] K Lim H Ryu and J Choi ldquoUTD analysis of a shapedsubreflector in a dual offset-reflector antenna systemrdquo IEEETransactions on Antennas and Propagation vol 46 no 10 pp1555ndash1559 1998

[18] B OrsquoNiell Elementary Differential Geometry Academic PressElsevier 2nd edition 2006

[19] D V Widder Advanced Calculus Prentice-Hall New DelhiIndia 2nd edition 2004

[20] R Storn and K Price ldquoDifferential evolutionmdasha simple andefficient heuristic for global optimization over continuousspacesrdquo Journal of Global Optimization vol 11 no 4 pp 341ndash359 1997

[21] K V Price R M Storn and J A Lampinen Differential Evo-lution A Practical Approach to Global Optimization Springer2005

[22] A Qing and C K Lee Differential Evolution in Electromagnet-ics Springer Berlin Germany 2010

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surfaces to improve the performance of the antenna Shapedreflector antennas outperform conventional unshaped reflec-tor antennas Reflector shaping allows the designers addi-tional flexibility The antenna designers have independentcontrol over relative position of the reflectors diameter of thereflectors and the curvature of the reflectors when shapedreflectors are used instead of conventional reflectors Thismakes reflector shaping an essential tool for designing highperformance reflector antennas

Many methods of designing shaped reflectors are presentin literature One of the first major articles related to reflectorshaping was published by Galindo in 1964 [6] The method isbased on geometrical optics (GO)Galindorsquosmethod requiredsolution of multiple nonlinear differential equations whichsometimes may be computationally demanding A modifiedversion of this method was presented by Lee [7] Lee dividedthe reflector surfaces into small sections and assumed thesections to be locally planar This assumption converted thedifferential equations to algebraic equations which are mucheasier to solve However the reflector surfacemust be dividedinto a large number of sections to increase the accuracy of thismethod

Another popular method for designing shaped reflectorsinvolves expanding the shaped surfaces using a set of orthog-onal basis functions [8 9] Rahmat-Samii has publishedmul-tiple research papers on this area [8ndash10]The expansion coef-ficients determine the shape of the surface A small number ofterms of the expansion set are sufficient to accurately describea shaped surface So a few expansion coefficients must bedetermined to define the surface The differential equationbased methods determine the coordinates of the points onthe reflector surfaces whereas the surface expansion basedmethod only determines the expansion coefficients Due tothe decrease in number of unknowns the surface expansionmethod is computationally less demanding The surfaceexpansion method can be incorporated with geometricaltheory of diffraction (GTD) or its uniform version uniformtheory of diffraction (UTD) to produce an accurate designalgorithm [10]These design procedures are known as diffrac-tion synthesis [8ndash10]Thismethod has been successfully usedin many applications Recently a few new efficient methodsfor designing circularly symmetric shaped dual reflectordesign have been developed One of the first significant workson thismethodwas reported byKimandLee in 2009 [11]Thismethod divides the shaped reflector surfaces into electricallysmall sections Each section is assumed to be a conventionalunshaped dual reflector system Since well-established meth-ods for analyzing conventional dual reflector system existthe radiation characteristics of each section can easily beevaluatedThe shaped surface is defined by combining all thelocal conventional surfacesThemethod requires solutions ofseveral nonlinear algebraic equations So it is computationallyconvenient Another method based on the same principlewas proposed by Moreira and Bergmann in 2011 [12] Thismethod also divides the shaped surface into small localsections The local sections are represented by unshapedconics Each conic section is optimized to produce a desiredaperture distribution which is formulated by GO methodAs these methods have recently appeared in literature most

of the advantages and drawbacks of the method have notbeen investigated Reduction in computational complexity isan obvious advantage The proposed work concentrated onpresenting an alternative method rather than improving theexisting methods

A design method that requires lesser number of parame-ters to define the shaped reflector surfaces without decreasingaccuracy of the obtained results is a challenging goal forreflector antenna designers In this paper the surface of theshaped reflector is defined using a novel equationThe shapedsubreflector surfaces are assumed to be distorted forms ofunshaped surfaces As most shaped subreflectors resem-ble their unshaped counterparts [11ndash13] the assumption islogical The shaped surfaces can therefore be representedby modified versions of the equations that represent theconventional unshaped surfaces This method of visualizingthe shaped surface as perturbeddistorted form of unshapedsurface has not been reported in literature yet As the generalformof the surface is generated from the conventional conicsonly a small number of parameters need be used to representthe shaped surface Once the subreflector surface is definedthe main reflector surface is defined using GO method andequal optical path length criterion

The paper is organized as follows Section 2 describes thegeometry of the shaped subreflector Surface equations anddifferential geometric analysis are presented in this sectionThe synthesis method of the main reflector is discussed inSection 3 Section 4 covers the numerical results Concludingremarks are made in Section 5

2 Geometry of the Shaped Subreflector

21 Surface Equations In a conventional Cassegrain geome-try the subreflector is hyperboloidal In shaped-Cassegraingeometry the subreflector is shaped to provide a desiredillumination over the aperture of themain reflector Howeverthe geometrical features of this shaped subreflector arevery similar to the geometrical features of the unshapedhyperboloid Due to these similarities the shaped surfacescan be considered a distorted form of the unshaped surfacesSo to define the geometry of the shaped subreflector theconventional Cassegrain geometry must be described first

The geometry of a Cassegrain dual reflector antenna isshown in Figure 1 The parameters describing the geometryof the Cassegrain system are

119889119901 diameter of the main reflector = 10m

119891119901 focal length of the main reflector = 5m

119889119904 diameter of the subreflector = 125m

2119888 distance between the foci = 4m

119890 = ca subreflector eccentricity = 14261

Δ119901 depth of the paraboloid = 125m

119897119901 distance from feed to paraboloid vertex = 1m

120595119900 120601119900 opening half angle of the main reflector andsubreflector = 5313∘ and 10037∘ respectively


ds dp

Primary focus

Hyperboloidal


Mainreflector

subreflector

(a + c)

Δp

Fs

Fp 120601o

120595o

fp

lp

2c




(119911 + 119888)2

1198862minus

1199092

+ 1199102

1198872= 1 (1)

where

1198872

= 1198882

minus 1198862 (2)


1205882

119904= 1199092

+ 1199102 (3)


(119911 + 119888)2

1198862minus

1205882

119904

1198872= 1 (4)

119911 = minus119888 minus119886

119887radic1198872 + 1205882

119904 (5)



(119911 + 119888)2

1198862minus

1205882

119904

1198872120575 (120588119904) = 1 (6)

119911 = minus119888 minus119886

119887radic1198872 + 1205882

119904120575 (120588119904) (7)


120575 (120588119904) = exp(

119873

sum

119899=1

1205911198991205882119899120577119899

119904) (8)



Shaped 1Shaped 2

Shaped 3Unshaped

08

06

04

02

0

minus02

minus04

minus06

minus08

x(m

)

z (m)









r = 120588119904 cos120601119904x + 120588119904 sin120601119904y minus [119888 +119886

119887radic1198872 + 1205882

119904120575 (120588119904)] z (9)


ns =(120597r120597120588119904) times (120597r120597120601119904)

1003816100381610038161003816(120597r120597120588119904) times (120597r120597120601119904)1003816100381610038161003816

(10)


120597r120597120588119904

= cos120601119904x + sin120601119904y

minus[[

[

119886

2119887

120588119904120575 (120588119904) sum119873

119899=11205911198991205882119899120577119899 + 2

radic1198872 + 1205882119904120575 (120588119904)

]]

]

z

120597r120597120601119904

= minus120588119904 sin120601119904x + 120588119904 cos120601119904y

(11)


ns =Λ (120588119904) cos120601119904

radicΛ2 (120588119904) + Δ2119867

(120588119904)

x +Λ (120588119904) sin120601119904

radicΛ2 (120588119904) + Δ2119867

(120588119904)

y

+Δ119867 (120588119904)

radicΛ2 (120588119904) + Δ2119867

(120588119904)

z

(12)

where

Δ119867 (120588119904) = 2119887radic1198872 + 1205882119904120575 (120588119904)

Λ (120588119904) = 119886120575 (120588119904) 120576 (120588119904)

120576 (120588119904) = 120588119904

119873

sum

119899=1

1205911198991205882119899120577119899 + 2

(13)





minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus050

051

minus1minus05

005

1

(a)

minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus05

005

1

minus1minus05

005

1

(b)


0

005

01

015

02

025

03

035

04

045

05


zds

xds






1198891 + 1198892 + 1198893 = 1198961 (14)


1198961 = (119886 + 119888) + (119886 + 119888 + 119897119901) + (119897119901 minus 119911ref)

= 2 (119886 + 119888 + 119897119901) minus 119911ref(15)


119897119901 = 119891119901 minus 2119888 (16)

Using (16) in (15)

1198961 = 2 (119886 minus 119888 + 119891119901) minus 119911ref (17)


1198891 = radic1199092119904

+ 1199102119904

+ 1199112119904 (18)


1198893 = 119911119898 minus 119911ref (19)




120588s 120588s



120579

O Oz x

x y

ns

r = si

sr

(a + c)

2c

120594 120601s


2c

O

(a + c)

(xs ys zs)

(0 0 zref)

lp

fp

d2

d3

d1

(xm ym zm)

Main reflector

Subreflector

Feed antenna

ns

r = si

sr

srd

srm

x

z


where

si = r =r|r|

(21)


srd = sr1198892 + 119909119904x + 119910119904y + 119911119904z (22)




+ (sr sdot z) 1198892 + 119911119904 z(23)



119909119898 = (sr sdot x) 1198892 + 119909119904

119910119898 = (sr sdot y) 1198892 + 119910119904

119911119898 = (sr sdot z) 1198892 + 119911119904

(24)




1198892 =


+ 1199102119904

+ 1199112119904

minus 119911119904

1 + (sr sdot z) (26)



4 Numerical Example



0 20 40 60 80 100

Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

Angle 120579 (deg)




Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus450 20 40 60 80 100

Angle 120579 (deg)




5 Conclusion






References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










ds dp

Primary focus

Hyperboloidal


Mainreflector

subreflector

(a + c)

Δp

Fs

Fp 120601o

120595o

fp

lp

2c




(119911 + 119888)2

1198862minus

1199092

+ 1199102

1198872= 1 (1)

where

1198872

= 1198882

minus 1198862 (2)


1205882

119904= 1199092

+ 1199102 (3)


(119911 + 119888)2

1198862minus

1205882

119904

1198872= 1 (4)

119911 = minus119888 minus119886

119887radic1198872 + 1205882

119904 (5)



(119911 + 119888)2

1198862minus

1205882

119904

1198872120575 (120588119904) = 1 (6)

119911 = minus119888 minus119886

119887radic1198872 + 1205882

119904120575 (120588119904) (7)


120575 (120588119904) = exp(

119873

sum

119899=1

1205911198991205882119899120577119899

119904) (8)



Shaped 1Shaped 2

Shaped 3Unshaped

08

06

04

02

0

minus02

minus04

minus06

minus08

x(m

)

z (m)









r = 120588119904 cos120601119904x + 120588119904 sin120601119904y minus [119888 +119886

119887radic1198872 + 1205882

119904120575 (120588119904)] z (9)


ns =(120597r120597120588119904) times (120597r120597120601119904)

1003816100381610038161003816(120597r120597120588119904) times (120597r120597120601119904)1003816100381610038161003816

(10)


120597r120597120588119904

= cos120601119904x + sin120601119904y

minus[[

[

119886

2119887

120588119904120575 (120588119904) sum119873

119899=11205911198991205882119899120577119899 + 2

radic1198872 + 1205882119904120575 (120588119904)

]]

]

z

120597r120597120601119904

= minus120588119904 sin120601119904x + 120588119904 cos120601119904y

(11)


ns =Λ (120588119904) cos120601119904

radicΛ2 (120588119904) + Δ2119867

(120588119904)

x +Λ (120588119904) sin120601119904

radicΛ2 (120588119904) + Δ2119867

(120588119904)

y

+Δ119867 (120588119904)

radicΛ2 (120588119904) + Δ2119867

(120588119904)

z

(12)

where

Δ119867 (120588119904) = 2119887radic1198872 + 1205882119904120575 (120588119904)

Λ (120588119904) = 119886120575 (120588119904) 120576 (120588119904)

120576 (120588119904) = 120588119904

119873

sum

119899=1

1205911198991205882119899120577119899 + 2

(13)





minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus050

051

minus1minus05

005

1

(a)

minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus05

005

1

minus1minus05

005

1

(b)


0

005

01

015

02

025

03

035

04

045

05


zds

xds






1198891 + 1198892 + 1198893 = 1198961 (14)


1198961 = (119886 + 119888) + (119886 + 119888 + 119897119901) + (119897119901 minus 119911ref)

= 2 (119886 + 119888 + 119897119901) minus 119911ref(15)


119897119901 = 119891119901 minus 2119888 (16)

Using (16) in (15)

1198961 = 2 (119886 minus 119888 + 119891119901) minus 119911ref (17)


1198891 = radic1199092119904

+ 1199102119904

+ 1199112119904 (18)


1198893 = 119911119898 minus 119911ref (19)




120588s 120588s



120579

O Oz x

x y

ns

r = si

sr

(a + c)

2c

120594 120601s


2c

O

(a + c)

(xs ys zs)

(0 0 zref)

lp

fp

d2

d3

d1

(xm ym zm)

Main reflector

Subreflector

Feed antenna

ns

r = si

sr

srd

srm

x

z


where

si = r =r|r|

(21)


srd = sr1198892 + 119909119904x + 119910119904y + 119911119904z (22)




+ (sr sdot z) 1198892 + 119911119904 z(23)



119909119898 = (sr sdot x) 1198892 + 119909119904

119910119898 = (sr sdot y) 1198892 + 119910119904

119911119898 = (sr sdot z) 1198892 + 119911119904

(24)




1198892 =


+ 1199102119904

+ 1199112119904

minus 119911119904

1 + (sr sdot z) (26)



4 Numerical Example



0 20 40 60 80 100

Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

Angle 120579 (deg)




Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus450 20 40 60 80 100

Angle 120579 (deg)




5 Conclusion






References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Shaped 1Shaped 2

Shaped 3Unshaped

08

06

04

02

0

minus02

minus04

minus06

minus08

x(m

)

z (m)









r = 120588119904 cos120601119904x + 120588119904 sin120601119904y minus [119888 +119886

119887radic1198872 + 1205882

119904120575 (120588119904)] z (9)


ns =(120597r120597120588119904) times (120597r120597120601119904)

1003816100381610038161003816(120597r120597120588119904) times (120597r120597120601119904)1003816100381610038161003816

(10)


120597r120597120588119904

= cos120601119904x + sin120601119904y

minus[[

[

119886

2119887

120588119904120575 (120588119904) sum119873

119899=11205911198991205882119899120577119899 + 2

radic1198872 + 1205882119904120575 (120588119904)

]]

]

z

120597r120597120601119904

= minus120588119904 sin120601119904x + 120588119904 cos120601119904y

(11)


ns =Λ (120588119904) cos120601119904

radicΛ2 (120588119904) + Δ2119867

(120588119904)

x +Λ (120588119904) sin120601119904

radicΛ2 (120588119904) + Δ2119867

(120588119904)

y

+Δ119867 (120588119904)

radicΛ2 (120588119904) + Δ2119867

(120588119904)

z

(12)

where

Δ119867 (120588119904) = 2119887radic1198872 + 1205882119904120575 (120588119904)

Λ (120588119904) = 119886120575 (120588119904) 120576 (120588119904)

120576 (120588119904) = 120588119904

119873

sum

119899=1

1205911198991205882119899120577119899 + 2

(13)





minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus050

051

minus1minus05

005

1

(a)

minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus05

005

1

minus1minus05

005

1

(b)


0

005

01

015

02

025

03

035

04

045

05


zds

xds






1198891 + 1198892 + 1198893 = 1198961 (14)


1198961 = (119886 + 119888) + (119886 + 119888 + 119897119901) + (119897119901 minus 119911ref)

= 2 (119886 + 119888 + 119897119901) minus 119911ref(15)


119897119901 = 119891119901 minus 2119888 (16)

Using (16) in (15)

1198961 = 2 (119886 minus 119888 + 119891119901) minus 119911ref (17)


1198891 = radic1199092119904

+ 1199102119904

+ 1199112119904 (18)


1198893 = 119911119898 minus 119911ref (19)




120588s 120588s



120579

O Oz x

x y

ns

r = si

sr

(a + c)

2c

120594 120601s


2c

O

(a + c)

(xs ys zs)

(0 0 zref)

lp

fp

d2

d3

d1

(xm ym zm)

Main reflector

Subreflector

Feed antenna

ns

r = si

sr

srd

srm

x

z


where

si = r =r|r|

(21)


srd = sr1198892 + 119909119904x + 119910119904y + 119911119904z (22)




+ (sr sdot z) 1198892 + 119911119904 z(23)



119909119898 = (sr sdot x) 1198892 + 119909119904

119910119898 = (sr sdot y) 1198892 + 119910119904

119911119898 = (sr sdot z) 1198892 + 119911119904

(24)




1198892 =


+ 1199102119904

+ 1199112119904

minus 119911119904

1 + (sr sdot z) (26)



4 Numerical Example



0 20 40 60 80 100

Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

Angle 120579 (deg)




Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus450 20 40 60 80 100

Angle 120579 (deg)




5 Conclusion






References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus050

051

minus1minus05

005

1

(a)

minus33

minus335

minus34

minus345

minus35

minus355

minus36

z(m

)

y (m)x (m)

minus1

minus05

005

1

minus1minus05

005

1

(b)


0

005

01

015

02

025

03

035

04

045

05


zds

xds






1198891 + 1198892 + 1198893 = 1198961 (14)


1198961 = (119886 + 119888) + (119886 + 119888 + 119897119901) + (119897119901 minus 119911ref)

= 2 (119886 + 119888 + 119897119901) minus 119911ref(15)


119897119901 = 119891119901 minus 2119888 (16)

Using (16) in (15)

1198961 = 2 (119886 minus 119888 + 119891119901) minus 119911ref (17)


1198891 = radic1199092119904

+ 1199102119904

+ 1199112119904 (18)


1198893 = 119911119898 minus 119911ref (19)




120588s 120588s



120579

O Oz x

x y

ns

r = si

sr

(a + c)

2c

120594 120601s


2c

O

(a + c)

(xs ys zs)

(0 0 zref)

lp

fp

d2

d3

d1

(xm ym zm)

Main reflector

Subreflector

Feed antenna

ns

r = si

sr

srd

srm

x

z


where

si = r =r|r|

(21)


srd = sr1198892 + 119909119904x + 119910119904y + 119911119904z (22)




+ (sr sdot z) 1198892 + 119911119904 z(23)



119909119898 = (sr sdot x) 1198892 + 119909119904

119910119898 = (sr sdot y) 1198892 + 119910119904

119911119898 = (sr sdot z) 1198892 + 119911119904

(24)




1198892 =


+ 1199102119904

+ 1199112119904

minus 119911119904

1 + (sr sdot z) (26)



4 Numerical Example



0 20 40 60 80 100

Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

Angle 120579 (deg)




Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus450 20 40 60 80 100

Angle 120579 (deg)




5 Conclusion






References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










120588s 120588s



120579

O Oz x

x y

ns

r = si

sr

(a + c)

2c

120594 120601s


2c

O

(a + c)

(xs ys zs)

(0 0 zref)

lp

fp

d2

d3

d1

(xm ym zm)

Main reflector

Subreflector

Feed antenna

ns

r = si

sr

srd

srm

x

z


where

si = r =r|r|

(21)


srd = sr1198892 + 119909119904x + 119910119904y + 119911119904z (22)




+ (sr sdot z) 1198892 + 119911119904 z(23)



119909119898 = (sr sdot x) 1198892 + 119909119904

119910119898 = (sr sdot y) 1198892 + 119910119904

119911119898 = (sr sdot z) 1198892 + 119911119904

(24)




1198892 =


+ 1199102119904

+ 1199112119904

minus 119911119904

1 + (sr sdot z) (26)



4 Numerical Example



0 20 40 60 80 100

Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

Angle 120579 (deg)




Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus450 20 40 60 80 100

Angle 120579 (deg)




5 Conclusion






References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119909119898 = (sr sdot x) 1198892 + 119909119904

119910119898 = (sr sdot y) 1198892 + 119910119904

119911119898 = (sr sdot z) 1198892 + 119911119904

(24)




1198892 =


+ 1199102119904

+ 1199112119904

minus 119911119904

1 + (sr sdot z) (26)



4 Numerical Example



0 20 40 60 80 100

Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus45

Angle 120579 (deg)




Relat

ive p

ower

(dB)

0

minus5

minus10

minus15

minus20

minus25

minus30

minus35

minus40

minus450 20 40 60 80 100

Angle 120579 (deg)




5 Conclusion






References

























RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













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 A New Method of Designing Circularly...

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