UTD Ray and Beam Methods for Analysis of Large EM … · UTD Ray and Beam Methods for Analysis of...

UTD Ray and Beam Methods for Analysis of Large EM Wave Problems

Prabhakar H. Pathak ([email protected])

The Ohio State University, ElectroScience Laboratory 1320 Kinnear Road, Columbus Ohio, 43212, USA

2

COPYRIGHT

©The use of this work is restricted solely for academic purposes. The author of this work owns the copyright and no reproduction in any form is permitted without written permission by the author.

3

ABSTRACTIt is well known from the independent works of Keller, Deschamps, and Felsen, respectively, that an electromagnetic (EM) point current source positioned in complex space produces a beam wave field which is highly localized about its forward propagation axis. It is noted that the EM field of a complex source beam (CSB) constitutes an exact solution of Maxwell's equations, and it is simply obtained by an analytic continuation of the exact closed form expression of the EM field for a point current in real space, where the real source coordinates are replaced by complex values. In its paraxial region, a CSB automatically reduces to a Gaussian Beam (GB). By controlling the values of the complex source coordinates, one can produce either a CSB with a very broad ( not well focused ) beam or a very narrow ( highly focused ) beam; consequently, such a CSB field can be made to pass smoothly from the field of a real point source to a plane wave field. It is clear that CSBs can serve as highly useful basis functions to represent EM fields, and indeed they have been used in this fashion by some researchers in this area. Here, additional useful methods for developing convergent CSB expansions to represent the fields of EM sources, via appropriate EM equivalence theorems, will be illustrated. Applications of such CSB expansions to analyze a class of electrically large practical antenna and scattering problems will be presented to demonstrate their utility.

Index Terms: Ray Optics, Diffraction, GTD, UTD, PTD, Beams, Hybrid Methods

4

BIOGRAPHYPrabhakar Pathak received his Ph.D (NVTP) from the Ohio State Univ (OSU). Currently he is Professor (Emeritus) at OSU. He is also a Courtesy Professor and an Adàunct Professor at Univ. of South Florida. Prof. Pathak is regarded as a co-developer of the uniform geometrical theory of diffraction (UTD). His interests continue to be in the development of new UTD solutions in both frequency and time domains, as well as in the development of fast Beam and Hybrid methods, for solving large antenna/scattering problems of engineering interest. Prof. Pathak has been actively presenting short courses and invited talks at conferences and workshops both in the US and abroad. He has authored/coauthored over a hundred journal and conference papers, as well as contributed chapters to seven books. Prior to NVVP, he served two consecutive terms as an Associate Editor of IEEE Trans. AP-S. He was appointed as an IEEE (AP-S) Distinguished lecturer (DL) from 1991-1993. Prof. Pathak was also appointed as the chair of the IEEE AP-S DL program during 1995 – 2005. He served as a member of the IEEE AP-S AdCom in 2010. He received the 1996 Schelkunoff best paper award from IEEE-AP-S; the ISAP 2009 best paper award; the George Sinclair award (1996) from OSU ElectroScience Laboratory; and, IEEE Third Millennium Medal from AP- S in 2000. Prof. Pathak received the Distinguished Achievement Award from IEEE AP-S in 2013. He is an IEEE Life Fellow, and a member of URSI-commission B.

• Two Basic Asymptotic High Frequency (HF)Methodologies can be Categorized as follows:

1. RAY OPTICAL METHODSa) Geometrical Optics (GO)b) Geometrical Theory of Diffraction (GTD)

[GTD = GO + Diffraction]c) Uniform Version of the GTD

2. WAVE OPTICAL METHODSa) Physical Optics (PO)b) Physical Theory of Diffraction (PTD)

[PTD = PO + Diffraction Correction]c) Incremental Theory of Diffraction (ITD) and

Equivalent Current Method (ECM)

UTD UAT

5

ON THE LOCALIZATION OF THE WAVE PROPAGATION AT HF

• The HF localization principle can be demonstrated viaasymptotic evaluation of the radiation integral as depictedbelow:

a) Radiation integral for the scattered field in the spatialdomain.

QE, QR and QS are critical points corresp. to end, stationaryand confluence of two stationary points, respectively ofthe integrand. For other values of Q’ the integral isnegligible due to destructive interference in the asymptoticHF regime.

2

from from from

( ) ( ) ( )

( ) ( | ) ( )

( | ) ,

( ) ( ) ( ) ( )R E S

i s

see S

S

jkR

ee

s r d sd

Q Q Q Q Q Q

E P E P E P

E P P Q J Q dS

er r j I R r rk R

E P E P E P E P

6

b) Radiation integral for the scattered field in the spectraldomain.

QE, QR and QS transform into critical within the spectrum aspoles, saddle points, branch points, etc. Only those planewaves reaching P from the nbhd of QE, QR and QScontribute significantly; all others interfere destructively.

( ) ( , )

( ) ( )

( ) ( ) ( ) ( )

s j k rx y x y

j k rS

S

s r d sd

E P dk dk f k k e

f j C J Q e dS

E P E P E P E P

2 2 2 2 2 2 ;

x y z

z x y x y

k k x k y k z

k k k k j k k k z z

as sumed 7

8

RAY METHODS & SOME APPLICATIONS• Unlike most other computational electromagnetic (CEM) techniques, asymptotic high frequency(HF) ray methods offer a simple picture for describing EM antenna/scattering phenomena.

Examples of EM antenna radiation and coupling problems of interest and some typical UTD rays.

• Rays wave effects are highly LOCALIZED at HF.• Primary focus here will be on the uniform geometrical theory ofdiffraction (UTD) type ray solutions.

• The need for UTD arises because classical geometrical optics (GO)ray method fails to predict diffraction !

ISB: Incident Shadow Boundary, RSB: Reflection Shadow Boundary, SSB: Surface Shadow Boundary

,0,1

,0,1~

~

r

i

rjksr

RRir

ii

jks

Oi

U

U

UesfQRQEPE

Us

ePCPE

r

i

Lit side of RSB

Shadow side of RSB

Lit side of ISBShadow side of ISB * depends only on

surface and wavefront geometry at & near

rsf

RQ

0rE

0&0 ri EE

rE

iE

iE

is

rs

9

• Keller and coworkers (1958; 1962) introduced a new class of rays, i.e. diffracted rays, todescribe diffraction in his geometrical theory of diffraction (GTD).• Diffracted rays exist in addition to geometrical optics (GO) rays.• Diffracted rays are produced at structural and material discontinuities, as well as atgrazing incidence on a smooth convex surface.

0rE

0&0 ri EE

rE

iE

iE

is

rs

GO

(GO + Diffraction)

RAY METHODS & SOME APPLICATIONS (cont.)

d

D

de

jksdDDSS

idS

jksdeEE

ide

esfQQTQEPE

esfDQEPPE

,~

~,

4

42

Examples of diffraction

0rE

0&0 ri EE

rE

iE

is

rs

des

dDs

dSE

dEE

des

DQ

10

RAY METHODS & SOME APPLICATIONS (cont.)• To find and , etc, in diffraction problems, one may:

(a) Solve appropriate, simpler canonical problems which model the LOCALgeometrical and electrical properties of the original surface in theneighborhood of diffraction points.

(b) An exact (or sometimes approximate) solution to a canonical problem isfirst expressed as an integral containing an exponent

(c) Canonical integral is then evaluated asymptotically, generally in closedform, as parameter becomes large (i.e. at HF).

(d) and are then typically found from (c) by inspection.(e) Canonical and generalized to arbitrary shapes by invoking

principle of locality of HF waves.

D

D T

dimensionsticcharacteri

2number wave

D

D TD

• Keller’s original GTD is not valid at and near ISB, RSB, SSB (i.e. in SB transitionregions).

• UTD developed to patch Keller’s original theory within the SB transition regions.

• GTD corrects GO, and GTD = GO + diffraction

• UTD corrects GTD, but usually UTD GTD outside SB transition regions.

D T

11

RAY METHODS & SOME APPLICATIONS (cont.)

• Additional Comments :

(a)Ufimtsev’s Physical Theory of Diffraction (PTD) (1950s) correctsPhysical Optics (PO). PO contains incomplete diffraction.

(b)PTD generally requires numerical integration on theradiating/scattering objects, hence, loses efficiency asfrequency increases.

(c) PTD does not describe creeping/surface wave diffraction onsmooth convex objects; hence, does not accurately predictpatterns in shadow zone of antennas on such complex objects.

(d)Conventional numerical CEM methods become rapidlyinefficient with increase in frequency.

(e) In contrast, UTD ray paths remain independent of frequency.(f) UTD offers an analytical (generally closed form) solution to

many complex problems that can not otherwise be solved in ananalytical fashion.

12

RAY METHODS & SOME APPLICATIONS (cont.)• In many practical applications of UTD, the following diffraction ray mechanisms dominate

[1] R. G. Kouyoumjian and P. H. Pathak,“A uniform geometrical theory ofdiffraction for an edge in a perfectlyconducting surface,” Proc. EEE, vol. 62,pp. 1448-1461, Nov. 1974.

Alternative ray solutions (UAT)[2] S. W. Lee and G. A. Deschamps, “AUniform Asymptotic theory of EMdiffraction by a curved wedge,”lEEETrans. Antennas Propagat., vol. AP-24,pp. 25-34, Jan. 1976.[3] Borovikov, V.A.and Kinber B.Ye,“Some problems in the asymptotic theoryof diffraction”, IEEE Proceeding, volume62, pp. 1416-1437, Nov. 1974.

[1] P.H Pathak, “An asymptotic analysis of the scatteringof plane waves by a smooth convex cylinder,” RadioScience, Vol 14 pp419-435, 1979[2] P.H Pathak et al, “A uniform GTD analysis of thediffraction of EM waves by a smooth convex surface,”IEEE Trans Ant and Propa. Vol 8 Sept 1980.

[1] P.H Pathak et al,”A uniform GTD solution for theradiation from sources on a convex surface,” IEEETrans Ant and Propa. Vol 29 July 1981.

[1] P.H Pathak and N. Wang,”Ray analysis of mutualcoupling between antennas on a convex surface,”IEEE Trans Ant and Propa. Vol 29 Nov 1981.

[1] K.C Hill and P.H Pathak, “A UTDsolution for EM diffraction by a corner ina plane angular sector,” IEEE Ant. Prop.Symp. June 1991.[2] K. C. Hill, “A UTD solution to the EMscattering by the vertex of a perfectlyconducting plane angular sector,” Ph.Ddissertation, The Ohio State University,1990.

(c) PEC Corner Diffraction

[1] G. Carluccio, “A UTD DiffractionCoefficient for a Corner Formed byTruncation of Edges in an OtherwiseSmooth Curved Surface,” IEEE Ant.Prop. Symp. June 2009.

(a) PEC Wedge Diffraction (b) PEC Convex Surface Diffraction

(Alt. Soln. by S.W. Lee in IEEE AP-S)

13

The Ohio State Univ. (OSU) ElectroScience Lab. (ESL) UTD based codes:

(a)OSU-ESL NEWAIR code(b)OSU-ESL BSC code

• Complex radiating and scattering objectsmodeled by simpler shapes consisting ofellipsoids, spheroids, cylinders, cone frustrums,flat plates, etc.

Some UTD code developments in USA during 1980’s – 1990’s

[1] R. G. Kouyoumjian and P. H. Pathak,“A uniform geometrical theory ofdiffraction for an edge in a perfectlyconducting surface,” Proc. EEE, vol. 62,pp. 1448-1461, Nov. 1974.

Alternative ray solutions (UAT)[2] S. W. Lee and G. A. Deschamps, “AUniform Asymptotic theory of EMdiffraction by a curved wedge,”lEEETrans. Antennas Propagat., vol. AP-24,pp. 25-34, Jan. 1976.[3] Borovikov, V.A.and Kinber B.Ye,“Some problems in the asymptotic theory of diffraction”, IEEE Proceeding, volume 62, pp. 1416-1437, Nov. 1974.

(a) PEC Wedge Diffraction

• In many practical applications of UTD, the following diffraction raymechanisms dominate

0

0

1

1 20

1 0 2 0

0 0

20 1

00

0 0 0 0

( ) ( )

lim ( ) ( ) ;

, , , ,

( ) ( )

d

Ed

d

d dd d jksd d d d

d

d dd id

E e d dP Q

e es eh

jksd dd i

E e d d d

E P E P es s

s P P

E P E Q Ds s

D D D

s eE P E Q Ds s

contains 4 terms which are a product of cot2

and

eDn

F kLa

2

UTD EDGE 2 TRANSITION

FUNCTION

jx j

x

F x j xe d e

14

where is theequivalent magnetic current in terms of the transmiting electric field in the slot aperture of area Sa; this replaces the aperture Sa which is now short circuited. Likewise, the radiation from a short thin monopole of height h and transmiting current

fed at the base Q’ on a convex surface can be found as

The UTD solution can predict complex surface dependent polarization effects resulting form surface ray torsion (see terms T1, T2, T3, T4, T5, T6).

( ) ( )S aM Q E Q n

[1] P.H Pathak et al,”A uniform GTD solution for theradiation from sources on a convex surface,” IEEE TransAnt and Propa. Vol 29 July 1981.

( )aE QSM

( )I l

15

The is obtained from uniform asymptotic solutions to problems of radiation by on conducting cylinders and spheres.

UTD Transition functions in A, B, C, D, H, S, n and N are the radiation Fock fcns.

,i m

p

0 at 0 at

S

L

PP

16

17

80

18

19

Geodesic surface ray

cylindercircular afor offunction a as )T(Q' a'

75 spheroid. prolate aon apart) 90 (phased antennaslot crossed Lindberg a of patternsRadiation

'

't

'b

'Q 'ˆ2

'ˆ1

't

'b

'Q

'n

t

b

n

curvature. of radii principal are R and R

Q'.at directionssurface principal denote 'ˆ and'ˆ

21

21

'at ;112

'2sin'

Torsion ; '''

1212

QRRRR

QT

TQQTQT gO

[1] P.H Pathak and N. Wang,”Ray analysis of mutualcoupling between antennas on a convex surface,”IEEE Trans Ant and Propa. Vol 29 Nov 1981.

(Alt. Soln. by S.W. Lee in IEEE AP-S)

contain the UTD transition functions corresponding to surface Fock fcns.

, , and ee he eh hh

20

21

CalculatedMeasured

Modeling of Boeing 737 aircraft

C. L. Yu, W. D. Burnside, and M. C. Gilreath, “Volumetric pattern analysis ofairborne antennas,” IEEE Trans. AP, Sep. 1978.

22

• J. J. Kim and W. D. Burnside, “Simulation and Analysis of Antennas Radiating in a ComplexEnvironment”, IEEE Trans. AP, April 1986.

23

-110

-100

-90

-80

-70

-60

0 4 8 12 16 20 24 28 32 36

Receive Antenna Distance from Nose (in.)

Cou

plin

g (d

B)

12 GHz (meas)12 GHz (calc)

wings not presentincludes double diffractionComputed by Dr. R. J. Marhefkaat OSU-ESL. This work was supported by Gary Roan at NRL

Comparison of Measured (NRL) and Calculated (NEC-BSC)Antenna Isolation with Receiver Moving above Center of Fuselage

24

Limitations of Existing UTD Codes• Existing UTD codes such as NEC-BSC and NEW-AIR have proven to

be successful over the past two decades.• However, these codes are based on the approximation of the

electrically large airborne platform in terms of canonical shapes, whichis a complicated task.

• Moreover, a canonical shape representation may lead toinaccuracies.

• Very limited capability to analyze material coatings

Canonical representation for NEC-BSC Canonical representation for NEW-AIR

25

New UTD code development

•Radiating object modeled with better fidelity viafacets based on CAD geometry data.

•UTD rays tracked in presence of facets. UTDray parameters obtained by mapping facets backto the original geometry (via bi-quadraticsurfaces or splines, etc).

•Does not require an expert user.

•Will eventually incorporate thin material coatingon metallic (or PEC) platform.

26

Quarter wavelength monopole on a KC-130 at 500 MHz(18.3λ0 height, 69.2λ0 wingspan, 49.7λ0 length at 500 MHz).

New code (Applied EM)

27

Quarter wavelength monopole on a KC-130 at 500 MHz(18.3λ0 height, 69.2λ0 wingspan, 49.7λ0 length at 500 MHz).

Mesh Geometry

Frequency (GHz)

# of facets for UTD

CPU time for UTD

# of facets for MLFMM

CPU time for MLFMM

KC-135 0.500 6,496 3.57 sec 420,466 1 h 28 min

Preliminary Validation

Near Field Far Field

28

29

30

Preliminary Validation

31

32

New UTD analytical development

•UTD + slope for PEC planar faced wedges with thinmaterial coating in form useful and accurate forengineering applications.

•UTD + slope for curved PEC wedges with thin materialcoating.

•UTD for PEC corners in curved edges and surfaces withthin material coating.

•UTD for PEC edge excited surface (or creeping) rays(and its reciprocal problem) in a form useful forengineering applications.

• Material coatings are generally replaced by approximate boundary conditions (e.g. Impedance Boundary Condition [IBC])• Solutions to canonical problems with “approx.” boundary conditions formulated exactly via Wiener-Hopf (W-H) or

Maliuzhinets (MZ) methods for surfaces made up of planar structures. Ray solutions extracted analytically from them viaasymptotic procedures.

• W-H method:

– J. L. Volakis and T. B. A. Senior, “Diffraction by a Thin Dielectric Half-Plane”, IEEE Trans. AP, Dec.1987

– R. G. Rojas, “Wiener-Hopf Analysis of the EM Diffraction by an Impedance Discontinuity in a Planar Surface andby an Impedance Half-Plane”, IEEE Trans. AP, Jan. 1988.

– R. G. Rojas and P. H. Pathak, “Diffraction of EM Waves by a Dielectric/Ferrite Half-Plane and RelatedConfigurations”, IEEE Trans. AP, June 1989.

– J. L. Volakis and T. B. A. Senior, "Application of a Class of Generalized Boundary Conditions to Scattering by aMetal-Backed Dielectric Half Plane”, Proc. IEEE, May 1989.

– V. G. Daniele and G. Lombardi, “Wiener-Hopf Solution for Impedance Wedges at Skew Incidence”, IEEE Trans.AP, Sep. 2006 .

• MZ method:

– G. D. Maliuzhinets, “Excitation, Reflection and Emission of Surface Waves from a Wedge with Given FaceImpedance”, Sov. Phys.-Dokl., 1958.

– R. G. Rojas, “Electromagnetic Diffraction of an Obliquely Incident Plane Wave Field by a Wedge with ImpedanceFaces”, IEEE AP, July 1988.

– R. Tiberio and G. Pelosi and G. Manara and P. H. Pathak, “High-Frequency Scattering from a Wedge withImpedance Faces Illuminating by a Line Source, Part I: Diffraction”, IEEE Trans. AP, Feb. 1989, see also IEEETrans. AP, July 1993.

– M. A. Lyalinov and N.Y. Zhu, “Diffraction of a Skew Incident Plane Electromagnetic Wave by an ImpedanceWedge”, Wave Motion, 2006.

• Approx. skew incidence solution (MZ) for imp. wedges based on modifying the HP solution:– H. Syed and J. L. Volakis, “Skew incidence diffraction by an impedance wedge with arbitrary face impedances”,

Electromagnetics, Vol. 15, No.3, 1995.

Previous Work on Thin Material Coated Metallic Wedge Structures

33

An Approximate UTD Ray Solutionfor Skew Incidence Diffraction

by Material Coated Wedges of Arbitrary AngleT Lertwiriyaprapa, P. H. Pathak and J. L. Volakis

ElectroScience LaboratoryDepartment of Electrical and Computer Engineering

The Ohio State University

URSI, Chicago 2008

• Present solution based on spectral synthesis.• Solution useful and accurate for engineering applications.

3232

0

30

6090

120

150

180

-40

-20

0dB

0

30

6090

120

150

180

-40

-20

0dB

0

30

6090

120

150

180

210

240 300

330

-20

-10

0dB

Grounded Material Junction

UTD-MZUTDMZ

0

30

6090

120

150

180

210

240 300

330

-20

-10

0dB

Grounded Material Junction

UTD-MZUTDMZ

y

x

0

0

0

ˆ 0

ds

is

Semi infinite Material Slab

Diffracted Field

ro , ro

z

rn , rn

Scatt. Field

Total Field

E-pol H-pol

• MZ is R. G. Rojas, “Electromagnetic Diffraction ofan Obliquely Incident Plane Wave Field by a Wedgewith Impedance Faces”, IEEE AP, July 1988.

PW illumination

Numerical Results(3-D Junction Planar Material Slabs on a PEC Ground Plane)

•Comparison of UTD-MZ, UTD and MZ at r=5, =45, o=65, =/20,ro=2, ro=4, rn=5, and rn=1.

35

0

30

6090

120

150

180

210

240270

300

330

-40

-20

0dB

Coated PEC Wedge

PEC

UTD-MZMZ

0

30

6090

120

150

180

210

240270

300

330

-40

-20

0dB

Coated PEC Wedge

PEC

UTD-MZMZ

0

30

6090

120

150

180

210

240270

300

330

-40

-20

0dB

Coated PEC Wedge

UTD-MZMZ

0

30

6090

120

150

180

210

240270

300

330

-40

-20

0dB

Coated PEC Wedge

UTD-MZMZ

36

Scatt. Field

Total Field

r


E-pol H-pol

PW o-face illumination

Numerical Results(3-D Material Coated PEC Right-Angled Wedge)

•Comparison of UTD-MZ and MZ at r=10, =45, o=65, =/20, rn=2, and rn=5.

3734

Total Field


0 20 40 60 80 100 120 140 160 180-40

-20

0

20

HH

-pol

[dB

]

in Degrees

in Degrees0 20 40 60 80 100 120 140 160 180

-15

-14

-13

-12

-11

-10

HE

-pol

[dB

]

UTD-MZMZ

0 20 40 60 80 100 120 140 160 180-16

-14

-12

-10

EH

-pol

[dB

]

in Degrees

0 20 40 60 80 100 120 140 160 180-30

-20

-10

0

10

in Degrees

EE

-pol

[dB

]

UTD-MZMZ

0

0

0

ˆ 0

ds

is

Numerical Results(3-D Junction Planar Material Slabs on a PEC Ground Plane)

•Comparison of UTD-MZ and MZ at r=10, =45, o=65, =/20, ro=4, ro=2,rn=2, and rn=5.

3835

Total Field


r

0 50 100 150 200 250 300 350-100

-50

0

50

HH

-pol

[dB

]

in Degrees

0 50 100 150 200 250 300 350-80

-60

-40

-20

0

in Degrees

HE

-pol

[dB

]

UTD-MZMZ

0 50 100 150 200 250 300 350-80

-60

-40

-20

0

EH

-pol

[dB

]

in Degrees

0 50 100 150 200 250 300 350-80

-60

-40

-20

0

20

in Degrees

EE

-pol

[dB

]

UTD-MZMZ

Numerical Results(3-D Material Coated PEC Half Plane)

•Comparison of UTD-MZ and MZ at r=10, =45, o=65, =/20, ro=4, ro=2,rn=2, and rn=5.

3936

Total Fieldr


0 50 100 150 200 250-150

-100

-50

0

EH

-pol

[dB

]

in Degrees

0 50 100 150 200 250-80

-60

-40

-20

0

20

EE

-pol

[dB

]

UTD-MZMZ

0 50 100 150 200 250-60

-40

-20

0

20

HH

-pol

[dB

]

in Degrees

0 50 100 150 200 250-120

-100

-80

-60

-40

-20

0

HE

-pol

[dB

]

UTD-MZMZ

Numerical Results(3-D Material Coated PEC Right-Angled Wedge)

• Comparison of UTD-MZ and MZ at r=10, =45, o=65, =/20, ro=4, ro=2,rn=2, and rn=5.

0

30

6090

120

150

180

210

240270

300

330

-40

-20

0

Coated PEC Wedge

TETM

40

Scatt. Field Total Field

0

30

6090

120

150

180

210

240270

300

330

-40

-20

0

Coated PEC Wedge

TETM

Numerical Results(3-D Material Coated PEC Wedge, WA = 54o)

• r=5, =117, =66, r=5,=/20,rn=2.4, and rn=8.

Slope diffraction is included

z-directed currentmoment excitation

ANTENNAS ON CONVEX COATED STRUCTURESKittisak Phaebua and Prabhakar Pathak

• A Uniform Geometrical Theory of Diffraction (UTD) Ray Solution is developedto predict the radiation by antennas on smooth convex metallic surfaces withthin material coating.

• Metallic surface is assumed to be a perfect electric conductor (PEC).

• Thin coating ; ; = surface radii of curvatureAlso,

• For sufficiently thin material coating, one can approximate the actual boundary on the external surface by a surface impedance Zs

gkd k 2k

g

)( , rr

ˆ ˆ sn n E Z n H121 ( )1 (1 )

2o

r rr r

rs jZ kd kdZ

sZ

n

Arbitrarily oriented electric or magnetic point current at Q’ on external boundary

d (coating thickness)

PL

Q’

Q

PsPEC

Direct Ray

Surface diffracted Ray

Geodesic surface ray

Thin uniform material coating

1kd

41

MOTIVATION•UTD Ray Analysis can be applied to analyze radiation by conformal antennas

and antenna arrays in the presence of a smooth PEC convex surface with thinmaterial coating.

Single printed patchPrinted patch array

Printed cross dipole element

dPEC

Single slotSlot array

dPEC

Single monopole

42

SOME PREVIOUS RELATED WORK[1] P. Munk and P. H. Pathak, "A UTD Analysis of the Radiation and Mutual Coupling Associated with

Antennas on a Smooth Perfectly Conducting Arbitrary Convex Surface with a Uniform Material Coating,"Antennas and Propagation Society International Symposium, vol. 1, pp. 696 - 699, Jul. 1996.- UTD ray solution not in form convenient for applications. Also, not all UTD transition functions computed.

[2] N. A. Logan and K. S. Lee, "A Mathematical Model for Diffraction by Convex Surface," In Electromagneticwaves. R. ranger, Ed, Univ. Wisconsin Press, 1962.- No specific ray solution for radiation available.

[3] Wait, J. R., Electromagnetic Waves in Stratified Media, A Pergamon Press Book, McMillan Co., New York,1962.- Propagation of waves around the earth, spherical surface analyzed. No UTD ray solution presented- Similar to work by V. A. Fock, Electromagnetic Diffraction and Propagation Problems, New York, PergamonPress, 1965 (Original work in Russian was published in 1940s)

[4] L. W. Pearson, “A scheme for automatic computation of Fock-type integrals,” IEEE Trans. AntennasPropagat.,vol. AP-35, pp. 1111–1118, Oct. 1987.- Solution presented for only the scattering into shadow region of a coated circular cylinder.

[5] C. Tokgöz, P. H. Pathak and R. J. Marhefka," An Asymptotic Solution for the Surface Magnetic Field Withinthe Paraxial Region of a Circular Cylinder With an Impedance Boundary Condition", IEEE Trans. AntennasPropagat., vol. 53, no. 4, April 2005.- Mostly restricted to surface fields on cylinders due to point magnetic currents on the same surface.

[6] P. H. Pathak, N. Wang, W. D. Burnside and R. G. Kouyoumjian, “A uniform GTD solution for the radiationfrom sources on a convex surface”, IEEE Trans. Antennas Propagat., vol. AP-29, no. 4, pp. 609-622, July 1981.- UTD analysis restricted to smooth convex PEC surfaces.

[7] P. H. Pathak, R. J. Burkholder, Y. Kim and J. Lee, "A Hybrid Numerical-Ray Based Analysis of Large ConvexConformal Antenna Array on Large Platforms," Presented at ACES conference in Finland, April, 2010.- Hybrid numerical UTD solution restricted to complex antennas on locally smooth convex PEC surfaces.

43

ANALYTICAL FORMULATION• A UTD solution for radiation by an arbitrarily oriented or on an arbitrary

smooth convex surface with a uniform surface impedance boundary condition (IBC)is developed from canonical solutions.

(Prabhakar Pathak & Kittisak Phaebua)

• Canonical problems to be solved pertain to (or ) with arbitrary orientation on circular cylinders and spheres with IBC.

• Generalization of canonical solutions to arbitrary convex surface performedheuristically based on the principal of locality of HF wave phenomena

ed p md p

( )edp Q ( )mdp Q

(a) Canonical circular cylinder problem geometry (b) Canonical spherical problem geometry

44

10 20 30 40

30

210

60

240

90270

120

300

150

330

180

0

Ray SolutionCST-Microwave Studio

NUMERICAL RESULTS (CYL)Radius of cylinder 4Thickness of dielectric coating 0.02Length of cylinder 50

2 .1 (T eflo n )r 1r

a t 8 0 oE a t 6 0 oE

a t 9 0 oE

Frequency of operation = 10 GHz

10 20 30 40

30

210

60

240

90270

120

300

150

330

180

0


10 20 30 40

30

210

60

240

90270

120

300

150

330

180

0


ˆNormal electric current source, ( . )J J n

45

NUMERICAL RESULTS (CYL)

a t 9 0 oE

ˆTangential magnetic current source, M ( . )t M b

Radius of cylinder 4Thickness of dielectric coating 0.02Length of cylinder 50

2 .1 (T eflo n )r 1r


10 20 30 40

30

210

60

240

90270

120

300

150

330

180

0


46

NUMERICAL RESULTS (SPH)

nE E

ˆNormal electric current source, ( . )rJ J n ˆTangential magnetic current source, M ( . )t M b

Radius of sphere 4Thickness of dielectric coating 0.02 2 .1 (T eflon ) ; r 1r


10 20 30 40 50

30

210

60

240

90270

120

300

150

330

180

0


10 20 30 40

30

210

60

240

90270

120

300

150

330

180

0


nE E

47

48

A UTD Diffraction Coefficient for a Corner Formed by Truncation of Edges

in an Otherwise Smooth Curved Surface

Giorgio Carluccio(1), Matteo Albani(1), and Prabhakar H. Pathak(2)

(1) Department of Information Engineering, University of SienaVia Roma 56, 53100 Siena, Italy, http://www.dii.unisi.it

(2) ElectroScience Laboratory, The Ohio State University1320 Kinnear Road, 43212 Columbus – OH, USA,

http://electroscience.osu.edu

IEEE International Symposium on Antennas and Propagation and USNC/URSI NationalRadio Science Meeting

June 01-05, 2009

49

UTD Vertex Diffraction CoefficientShadow Boundary Cones (SBCs) and Shadow Boundary Planes (SBPs):

50 -180 -150 -120 -90 -60 -30 05

10

15

20

25

30

[dB

]

Tot UTDTot Uniform Asym PO

-180 -150 -120 -90 -60 -30 0-10

-5

0

5

10

15

20

25

30

[dB

]

GOD-UTD-ABD-UTD-DAV-UTD-ATot

-4 -2 0 2-4-202

-3

-2

-1

0

1

2

3

4

5

BC

x

A

D

z

y

III Example: Vertex Double Transition

Scan Center on the Vertex A

FieldE

3 , 45 , 180 0r

We consider a smooth convex parabolic surface illuminated by an electric point source

FieldE

51

Remarks• A UTD diffraction coefficient for a corner formed by truncation of edges in asmooth curved surface was presented.

• A PO diffraction coefficient is derived by asymptotical evaluation of the POintegral, to understand how the surface curvature affects the diffracted fieldtransitional behavior.

• The UTD diffraction coefficient was obtained by heuristically modifying theUTD diffraction coefficient for a corner in a flat surface, on the basis of theprevious PO result.

• Numerical examples show how the proposed diffracted coefficient smoothlycompensates for the abrupt discontinuity occurring when the GO field or thesingly diffracted at edges abruptly vanish.

• Valid for astigmatic ray tube illumination.

• Can be extended to include thin material coating.

52

53

i to

'Pie

Edge excited surface rays

• Presently UTD solution has been obtained for ISB and SSB farapart.

• Work is in progress to obtain an asymptotic solution useful forengineering applications when ISB and SSB regions overlap.

54

Comments• Keller’s original GTD is not valid at and near ISB, RSB, SSB (i.e. in

SB transition regions).

• UTD developed to patch Keller’s original theory within the SBtransition regions.

• GTD corrects GO, and GTD = GO + diffraction

• UTD corrects GTD, but usually UTD GTD outside SB transitionregions.

• UTD ray paths remain independent of frequency.

• UTD offers an analytical (generally closed form) solution to manycomplex problems that can not otherwise be solved in an analyticalfashion.

• UTD in some cases must be augmented by PO/PTD or ECM

• PO/PTD can give rise to spurious contributions from theshadow boundary line in a smooth body.

• PO/PTD does not incorporate creeping wave effects.

•PO can correct for UTD transport singularities at and nearthe confluence of edge induced GO ray shadow boundariesand GO/diffracted ray caustics (e.g. forward radiation fromparabolic reflector antennas).

RSB + diffracted caustic

offset paraboloidal reflector

reflected ray caustic

feed

55

SPECTRAL THEORY OF DIFFRACTION (STD)

• UTD/GTD requires a RAY OPTICAL incident field

• If the incident field is NON-RAY OPTICAL, then it must berepresented by:a) continuous set of PLANE WAVES (e.g. Plane Wave

Spectrum)b) discrete set of RAY OPTICAL fields

• Each constituent RAY OPTICAL field in the SPECTRALDECOMPOSITION of a NON-RAY OPTICAL incident wavecan be treated by UTD.

• The total UTD solution is then a summation of the UTDresponse to each constituent RAY OPTICAL incident field

R. Tiberio, G. Manara, G. Pelosi, R. KouyoumJian: ‘’HF EM scattering of plane waves fromdouble wedges,” IEEE Trans AP-37, pp. 1172-1180, Sept. 1989

Y. Rahmat-Samii, R. Mittra ‘’A spectral domain interpretation of HF phenomena,” IEEETrans AP-25 pp. 676-687, Sept. 1977

56

57

58

ECM for Interior Waveguide Discontinuities

Previous related work

59

ADDITIONAL COMMENTS ON ASYMPTOTIC HF METHODS

• Asymptotic HF methods are powerful for analyzing a wide varietyof electrically large EM problems.

• Conventional CEM numerical solution methods based on selfconsistent wave formulations become rapidly inefficient, or evenintractable, for solving large ‘’EM problems’’.

• UTD is more developed, especially for handling smooth convexboundary diffraction.

• HF wave optical methods (PO, PTD, ITD, SBR) have not directlyincorporated creeping waves.

• Ray optical methods require ray tracing. More efficient but lessrobust. Ray paths independent of frequency.

• Wave optical methods require numerical integration on the largeobject. Less efficient but more robust. Does not scale withfrequency.

60

61

62

Beam Methods & Some Applications• Beams provide useful basis functions for representing EM fields.• Ray methods fail at caustics (focii) of ray systems. Caustics are formed byintersection or envelopes associated with the same class of rays. Beamsremain valid in regions of real ray caustics

• Beams can be used to treat large reflector systems and radome problemsefficiently.• Beams can be used to improve the speed of conventional Moment Method(MoM) solution of governing EM integral equation (IE) for the radiating object.• Beams can also be used for Near Field Far Field transformations requiredin near field measurements.

62

• UTD for real source excitation of wedge developedvia first order Pauli-Clemmow method (PCM) [1-4] forasymptotic solution of canonical wedge diffractionintegral along a steepest descent path (SDP)•First order PCM not strictly valid (for poles crossingthe SDP away from saddle point); hence analyticcontinuation of UTD for complex source locationwithout further study is questionable!•First order Van der Waerden method (VWM) [1-4] isvalid even where PCM fails.

y

x

EQ~Oz

)2(

nW

A

,,r

'~,'~,'~ r

Complex Source Beam (CSB) Diffraction by a Wedge

• However, one can show that the first order VWM method, upon using a key rearrangement(and combination) of terms, yields:

PCM) (Extended EPCMΔPCMVWM Key Step• Next, for the special wedge case of interest, it is shown analytically (and verified numerically)that .0

)negligible is (since UTD UTD EUTD

for wedge EUTD EPCM VWM for wedge UTD PCM :Note

For a wedge

Therefore, analytic continuation of UTD for a wedge is OK for

complex waves

1. T. B. A. Senior and J. L. Volakis, “Approximate Boundary Conditions in Electromagnetics,” The Institute of Electrical Engineerings, London, 1995.2. L. B. Felsen and N. Marcuvitz, “Radiation and Scattering of Waves,” Englewood Cliffs, NJ: Prentice-Hall, 1973.3. C. Gennarelli and L. Palumbo, “A uniform asymptotic expansion of typical diffraction integrals with many coalescing simple pole singularities and a

first-order saddle point,” IEEE Trans. Antennas and Propagat., vol. AP-32, pp. 1122-1124, Oct. 1984.4. R. G. Rojas, “Comparison between two asymptotic methods,” IEEE Trans. Antenn- as and Propagation, vol. 35, no. 12, pp 1492-1493, Dec 1987.

61

Numerical Result 1

bbb ˆ

y

x

eE zQ ~,0,0~ Oz

'~,'~,'~ r

)2(

nW

A

bb

,,rP

r 'r

0 50 RSB(135.86) ISB(224.14) 3000

1

2

3

4

5

6

7x 10-3

,degM

ag

Einc Eref EUTDdiff EUTD

total EEUTDdiff EEUTD

total

0 50 100 150 ISB(224.14) 300

0.5

1

1.5

2

, deg

0 50 RSB(135.86) 200 250 300

0.5

1

1.5

2

, deg

V2a V2

b V2=V2a*V2

b

V2a V2

b V2=V2a*V2

b

222.5 223 223.5 224 224.5 225 225.5 226-8

0

8x 10

-3

, Deg

Im[pi ]

134 134.5 135 135.5 136 136.5 137 137.5-8

0

8x 10

-3

, Deg

Im[pr ]

• Additional term in EUTD solution• Trajectories of

0Im rp

0Im ip

nss

nV

p

pp

a

2

~cot'~~

2~cos

~~

4,2

~~1~4,2

kFV pb

rip,Im

30WA

10b

6.60b 4.217b

2r 45zp ri ˆˆ ,

5'r 140' 50'

Complex Source Beam (CSB) Diffraction by a Wedge(cont.)

62

0 50 RSB(112.37) 150 200 ISB(247.67) 3000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x 10-3

, degM

ag

Einc Eref EUTDdiff EUTD

total EEUTDdiff EEUTD

total

Numerical Result 2

246.5 247 247.5 248 248.5 249-0.015

0

0.015

, Deg

Im[pi ]

111 111.5 112 112.5 113 113.5-0.015

0

0.015

, Deg

Im[pr ]

0 50 100 150 200 ISB(247.67) 300

0.5

1

1.5

2

, deg

0 50 RSB(112.37)150 200 250 300

0.5

11.5

22.5

, deg

V2a V2

b V2=V2a*V2

b

V2a V2

a V2=V2a*V2

b

bbb ˆ

y

x

eE zQ ~,0,0~Oz

'~,'~,'~ r

)2(

nW

A

bb

,,rP

r 'r

• Additional term in EUTD solution• Trajectories of

0Im rp

0Im ip

nss

nV

p

pp

a

2

~cot'~~

2~cos

~~

4,2

~~1~4,2

kFV pb

rip,Im

45WA

10b

57.78b 57.258b

12r 40

zp ri ˆˆ ,

20'r 100' 70'

Complex Source Beam (CSB) Diffraction by a Wedge(cont.)

66 SB1(5.3) 40 60 80 100 120 140 SB2(174.7)0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

, deg

Mag

HzPCM Hz

EPCM

Surface Wave 58.35jZS

0 20 SB1(54.7) 80 100 SB2(125.3) 160 1800

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

, deg

Mag

HzPCM Hz

EPCM

Leaky Wave 79.1794.327 jZS

2sin2 2,14

2,1

ppj

pp e

M

y

x

jXRZ s Shadow boundaries are determined when

0Im 2,1 pp

Other Complex Waves

67

Keller Cone of Edge Diffraction

68

Sequence of CSP beams

Feed Antenna

Reflected & Diffracted Fields of each CSP beam

Feed Radiation Pattern

Fast analysis of the reflector antennas.Radome

AntennaCSP beams

Antenna radiation in the presence of radomes.

• A GB -UTD (PO based) method was previouslyreported in [1].

• With the CSP method, feed pattern is expandedinto a set of CSBs.

• Each CSP beam field is scattered from thereflector by using complex extension of UTD.

• A 2-D case for a single beam illumination wasreported in [2].

• This new fully 3-D CSP-UTD approach (UTD forbeams) is expected to be more accurate then [1].

• The field of the antenna is first expanded intoa set of CSP beams.

• Each beam is next tracked through theradome.

• The transmitted beams are summed up atthe observer location.

• Complex ray tracing can be employed forbeam tracking through the radome [3,4].

[1] H. T. Chou, P. H. Pathak and R. J. Burkholder, “Novel Gaussian Beam Method for the Rapid Analysis of Large Reflector Antennas”, IEEE Trans. AntennasPropagat., 2001[2] G.A.Suedan and E.V. Jull, ”Beam diffraction by planar and parabolic reflectors,” IEEE Trans. Antennas Propagat., 1991.[3] X. J. Gao and L. B. Felsen, “Complex ray analysis of beam transmission through two-dimensional radomes”, IEEE Trans. Antennas Propagat., 1985[4] J. J. Maciel and L. B. Felsen, “Gaussian beam analysis of propagation from an extended plane aperture distribution through dielectric layers, part 1 - planelayer,” IEEE Trans. Antennas Propagat., 1990.

Large Antenna Applications

69

'~r

b

p • The CSB-UTD solution is valid for analyzing CSBexcited focus-fed parabolic reflector antennas since the caustics are now in complex space for the CSB excitation case.

• The PO analysis for a CSB excited parabolic reflector(a) loses its accuracy in the region of the main beam

when a CSB axis hits near the edge. can beimproved by adding the additional edge diffractionterm based on Physical Theory of Diffraction (PTD).

(b) becomes more accurate when a CSB axis hits theactual reflector surface away from the edge.

• The present CSB-UTD & CSB-EUTD solution for aCSB excited PEC curved wedge is obtained by analyticallycontinuing the UTD solution for a PEC curved wedgeexcited by a real point source (or even real astigmatic ray)to deal with a CSB (or even more generally a complexastigmatic beam, i.e. CAB) illumination of a curved edgein a curved surface.

• The CSB reflected from doubly curved surface become anastigmatic Gaussian beam in paraxial region.

mfyxz

4

22

CSB-UTD Diffraction by a Curved Wedge

70 -50 0 50 100 150 200 250

0.2

0.4

0.6

0.8

1

1.2

, deg

MA

G

EPO

ECSB-EUTD

ECSB-UTD

Numerical Result : Finite parabolic reflector illuminated by a CSB

-50 0 50 100 150 200 250

0.2

0.4

0.6

0.8

1

1.2

1.4

, deg

MA

G

EPO

ECSB-EUTD

ECSB-UTD Mfyxz

4

22

b

5000r

S S

)270,100,40(),50,10(,ˆˆ,10,35,5,60 SQbpbfhd BMM

Symmetry Plane

Transverse Plane

CSB-UTD Diffraction by a Curved Wedge (cont.)

71


-50 0 50 100 150 200 250

0.1

0.2

0.3

0.4

0.5

, deg

MA

G

EPO

ECSB-EUTD

ECSB-UTD

-50 0 50 100 150 200 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

, deg

MA

G

EPO

ECSB-EUTD

ECSB-UTD Mfyxz

4

22

y

z

b

5000r

QB

SfM

y

xOO

QB

S

dM

h

)270,100,40(),60,20(,ˆˆ,10,35,5,60 SQbpbfhd BMM

Symmetry Plane

Transverse Plane


72


-50 0 50 100 150 200 250

0.05

0.1

0.15

0.2

0.25

, deg

MA

G

EPO

ECSB-EUTD

ECSB-UTD

-50 0 50 100 150 200 250

0.05

0.1

0.15

0.2

, deg

MA

G

EPO

ECSB-EUTD

ECSB-UTDMfyxz

4

22

b

5000r

S

S

)90,0,40(),10,17(ˆˆ,10,30,50 SQxpbfd BMM

Symmetry Plane

Transverse Plane


73


-50 0 50 100 150 200 250

0.5

1

1.5

2

, deg

MA

G

EPO

ECSB-EUTD

ECSB-UTD

-50 0 50 100 150 200 250

5

10

15

20

, deg

MA

G

EPO

ECSB-EUTD

ECSB-UTDMfyxz

4

22

y

z

b

200rQB

SfM

y

xOO

QB

S

dM

)0,10,40(),35,20(ˆˆ,15,30,50 SQbpbfd BMM

Symmetry Plane

Transverse Plane


74

Feed Radiation

GB Expansion

n = launching points in feed plane

m = number of GBs from each n

CSBs/GBs Illuminating a reflector

75

< 200 GBs

NUM-PO

Time < 5 min/iter

Time = 5 or 6 hrs/iter

Approx. 30 iter’s*

Offset Shaped Reflector for CONUS Contour Beam Using GBs

85D

* Rahmat-Samii’s paper

Normalized co-polarized contours based on the GB approach CONUS coverage by a shaped concave reflector with a feed pattern at 12 GHz with l =18.51. Approximately 200GBs were used.

Normalized co-polarized gain contours based on the numerical PO integration approach for the same shaped reflector case as above. ?

76

Conclusions

• CSB expansion methods for EM radiation arepresented employing three different variants of thesurface equivalence theorem.

• The analytical properties (validity region, truncation,etc.) of the approach are investigated.

• It is shown that accurate and efficient fieldrepresentations can be obtained by convenientlytruncating the beam expansion.

• It is demonstrated that the expansion idea is applicableto a class of EM radiation/scattering problems.

77

Band Width/Complexity Materials

Elec

tric

al S

ize

UTD

PO/GO

MOM

FEM

MLFMM

Hybrid Methods are required for in-situ

analysis

Source: http://www.feko.info/feko-product-info/technical

Computational Methods

78

Hybrid Method & Some Applications• In many applications, large antennas (arrays) and largeantenna platforms contain both large and small featuresin terms of wavelength.

•For electrically large parts on radiating object, UTD raymethod is useful but not valid for electrically smallportions.

•For highly inhomogeneous and electrically small region(e.g. complex antenna elements/arrays) the FE-BI ornumerical methods are useful, and UTD is not applicablehere.

•A hybrid combination of FE-BI (or other suitablenumerical methods) and UTD could handle the entireproblem not otherwise tractable by each single approachby itself.

79

Simple slot phased array in a convex PEC surface

Material Treatment

Tapered Absorber

Antenna Array Elements

Radome Cover Flush with Platform

PEC Platform

PEC Platform

Complex phased antenna array slightly recessed in a convex platform and covered by a radome

Conformal Array Configurations

80

Local Array Parttreated by full wave numerical methods.

Present Collective UTD Solution

converts numerical array solution into rays launched from

array aperture.

External Platform PartCollective UTD rays launched

from aperture efficiently excite external platform which

is analyzed by UTD.

Proposed Hybrid Numerical-UTD Approach

81

Actual problem

A local array modeling for FEM

PEC

Local UTD RayeqsM

Structure outside the local aperture region is ignored

A local array modeling for FE-BI

ABC

Local Array Part Treated by FEM, FE-BI

82

• Collective UTD rays launched by arrayaperture distribution obtained via numericalsolution to local array part.

• Collective UTD rays launched from a few flashpoints in the array aperture.

Observation point

Collective UTD RaysPanuwat Janpugdee and Prabhakar H. Pathak

83

• Collective UTD rays efficiently launched from arrayaperture interact with external platform.

• Rays from platform could interact back with the arrayand so on. These effects can be included if desired.

• Collective UTD surface rays provide direct couplingbetween two arrays located on the same platform.

Convex Surface

Observation point

External Platform Interaction Part

84

Less efficient for large arrays – need to trace a large number of rays (with all the constructive & destructive interferenceeffects).

Lacks physical picture for describing collective array radiation and surface field excitation mechanisms.

Integration to existing UTD codes for predicting the platform interaction effects via UTD is straightforward –but less efficient.

Comparison with Conventional Element-by-element UTD Field Summation Approach

85

• Asymptotic UTD ray solutions for the radiation andsurface fields produced by a single point currentsource on a smooth convex surface have beendeveloped previously by Pathak et al.

Pathak et al., IEEE Trans. AP, vol. AP-29, pp. 609-622, Jul. 1981.Pathak et al., IEEE Trans. AP, vol. AP-29, pp. 911-922, Nov. 1981.

nn

lt

lb

s

Q

LP

i

n

n

t

b

t

Q

SP

Qb

Conventional EBE Sum Utilizes UTD Solution for a Single Current Source

86

• Antenna array operated at 25 GHz (K-band); = 1.2 cm• Aperture size = 1 ft. × 1 ft. = 25.4 × 25.4 • If sampling at every /4, one needs 102×102 (10,404)

sample points and needs to trace such large number ofrays.

• Conventional UTD approach becomes less efficient thancollective UTD approach.

Hypothetical Example

Material Treatment

Tapered Absorber

Antenna Array Elements

Radome Cover Flush with Platform

PEC Platform PECPlatform

Example of a LARGE Array

87

Describes fieldsproduced by the wholearray aperture at once interms of only a few UTDrays arising from specificpoints in the interior, andon the boundary of thearray aperture – highlyefficient.

Provides physical picture for describing collectivearray radiation and surface field excitation mechanisms.

Integration to most existing UTD codes for predictingthe platform interaction effects via UTD is not directbecause it requires some code modifications to allowfor a new input description.

Observation point

Present Collective UTD Ray Approach

88

• A scanning phased array on a slowly varyingconvex platform can be modeled by a parametricsurface patch, such as a bi-quadratic surface, etc.

uv

x

yz

11r

12r13r

21r31r

32r33r

22r

23r

u

v

0 1

1

1/2

1/2

uv-plane

Curved Surface Modeling

89

( )

1

( ( , ))i iu v

Kj u v

ii

A r u v C e

* Fields produced by each TW can be represented in terms of a set of UTD rays.

Found numerically by matching to the actual distribution

• A traveling wave (TW) expansion for realistic aperturedistributions obtained from FEM, FE-BI, etc.

uv

uv-space

Traveling Wave Expansion

90

• DFT (Discrete Fourier Transform)

• CLEAN (or “Extract and Subtract”)

• Prony’s Method

• Other Available Methods

TW Extraction Methods

91

Observation point

uv

Each TW current radiates a small set of collective UTD rays.

1

( )K

lUTD

lE P E

4 4 4

1 1 1

GO ed cd GO edUTD GO i ei j i

i j iE E U E U E E E

Transition fields

Collective UTD Ray Fields

92

A

ssQie

je

x

yz

ˆssn

ssQ

P ˆsss

ˆ ( )u sst Q

ˆ ( )v sst Q

ssu

ssv

1 2

1 2( )( )ss

ss ssjksGO

ss ss ssss ss

E A es s

Astigmatic Ray Field

ˆ( | )ss L ss ssA T P Q L

( ' )( 2 ) ( )ssjk Qss

ssss

j J QL ek

Geometrical Optics orLocal Floquet Wave

93

A

ieje

,s eiQ,

ˆ ( )u s eit Q

,ˆs ein

,s eiQ

ˆeisP

,s eiu

ie

( )eijksd ei

ei eiei ei ei

E A es s

Conical WaveLit Region

, ˆ( | ) dei L s ei eiA T P Q D

,/ 4 ( ),

2

2 ( )ˆ[ ] | |

s eij jk Qei s eid

eiei v v ei u

j e J Q eDk s r k E

Edge Diffracted Fields

94

A

ieje

,s eiQ

( )dei

djksds s ei

ei ei d d dei ei ei

E A es s

Shadow Region

eiQ

SP

eiQ

deis

ie

1/ 6

,,,

, ,

( )( )ˆ( | )

( ) ( )ei g s eis eijkts ds

ei S s ei eis ei g s ei

Qd QA T P Q D e

d Q Q

,/ 4 ( )

,

2

2 ( )ˆ[ ] | |

s eij jk Qei s eids

eiei v v ei u

j e J Q eDk t r k E

Edge-Excited Surface Diffracted Fields

95

ˆcin

ciQ

P ˆcisSpherical Wave

ijjkscij ij

ij

eE As

Lit Region

A

ieje

ijQ

ˆ( | ) cij L ij ijA T P Q D

( )

2

( )ˆ ˆ[ ][ ]

ijjk Qei ej ijc

ijij u u ij v v

J Q eD

k s r s r

Corner Diffracted Fields

96

( )dij

djksijcs s

ij ij d d dij ij ij

E A es s

Shadow Region

ciQ

SP

ciQdcis

A

ieje

ijQ

1/ 6

0 ( )ˆ( | )

( ) ( )ijjkt g ijs cs

ij S s ij ijij g ij

QdA T P Q D ed Q Q

( )

2

( )ˆ ˆ[ ][ ]

ijjk Qei ej ijcs

ijij u u ij v v

J Q eD

k t r t r

Corner-Excited Surface Diffracted Fields

97

2( )2 4 4

, ,21 3

( , , , )4

ijj kGO GO

ci j ei cj i ej ei eji j ij

eE E W k k k kk

2( )2 42

1

1 ( )2

eij k

ei ei eii ei

e F k Uk

2( )4 4

2

3

1 ( )2

ejj k

ej ej ejj ej

e F k Uk

2,( )4 4

2,

3 ,

1 ( )2

ci jj kd dei ei ci j ej

j ci j

eE E F kk

2 2 2 21 1 2 22 21 2

1 1 2 2 1 21 2 1 1 2 2

( , , , ) ( ) ( )x y x yx xW x y x y F x F x

y y x y x y

1 21 1 2 2

1 2

( , , , )y y T x y x yx x

Fields in Transition Regions

98

Taylor distribution

vL

uW

0n

0u

0v

O

0k

f = 24 GHza = 150.0 cm (120.0 )Wu = 37.5 cm (30.0 )Lv = 50.0 cm (40.0 )Broadside scanCurrent polarization: 45 w.r.t. axial direction

105 TWs were used !!

Aperture on a PEC Circular Cylinder

99

vL

uW

0n

0u

0v

r

O

Circumferential plane cutr = 40 (near zone)REF: 25.36 sec.TW-UTD: 3.78 sec.

Aperture on a PEC Circular Cylinder (cont.)

100

Oblique plane cut: = 40r = 40 (near zone)REF: 24.24 sec.TW-UTD: 3.76 sec.

vL

uW

0n

0u

0vr

O


101

Circumferential plane cutr = 9600 (far zone)REF: 31.25 sec.TW-UTD: 4.18 sec.

0v

0u

0n

vLuW

a O

r


102

0v

0u

0n

a

0k

0

0O

0 0Scan direction: 30 , 90

0.55 , 0.244s sL W

9.0 GHzf 100.0a

0.65u vd d 101 101 elements

93 TWs (0.9%) were used !!

sW

vd

ud

sL

Slot array on a PEC Circular Cylinder

103

Axial plane cut (scan plane)r = 100 (near zone)REF: 34.57 sec.TW-UTD: 5.47 sec.

0n

0u

0vrO

Slot array on a PEC Circular Cylinder (cont.)

104

Oblique plane cut: = 60r = 100 (near zone)REF: 38.40 sec.TW-UTD: 6.12 sec.

0n

0u

0vr

O


105

0v

0u

0nr

a

O

Axial plane cut (scan plane)far zoneREF: 33.48 sec.TW-UTD: 4.39 sec.


106

Oblique plane cut: = 45far zoneREF: 51.39 sec.TW-UTD: 6.08 sec.

0v

0u

0n

a

O

r


107

Interface between Full-Wave solver and High Frequency method

J-F. Lee’s DD FEBI code.

108

Solution of 7 by 7 Vivaldi array mounted on F16 Platform

109

Solution of 7 by 7 Vivaldi array mounted on F16 Platform

110

20 by 20 Vibaldi array mounted on larger platform

111

20 by 20 Vibaldi array mounted on larger platform

112

Conclusions• An asymptotic UTD ray solution has been developed fordescribing, in a collective fashion, the fields radiated bylarge conformal antenna arrays on a doubly curved,smooth convex surface.

• The present solution will provide an efficient linkbetween the local array part to be analyzed numericallyand the full external platform part to be analyzed by UTD,in a hybrid method for analyzing large complexantenna phased arrays integrated into a realistic complexplatform.

• The present collective UTD ray solution shows a goodagreement with the conventional element-by-elementUTD field summation solution.

113

The Ohio State Univ. (OSU), ElectroScience Lab.(ESL), Columbus, Ohio, USA

(1)Prof. Robert J.Burkholder(2)Dr. Youngchel Kim(3)Dr. His-Tseng Chou (now Prof. at Yuan-Ze Univ. Taiwan ROC)(4)Dr. Titipong Lertwriyaprapa (now at King Mongkut university)(5)Dr. Pawuwat Janpugdee (Was member of research staff at

TEMASEK Labs. NUS, Singapore; Now at Chulalongkorn univ.)(6)Dr. Koray Tap (Now at ASELSAN, Ankara, Turkey)(7)Prof. W.D. Burnside(8)Dr. R.J. Marhefka(9)Dr. Nan Wang(10)Prof. Jin-Fa Lee

The work presented is done in conjunction with the following researchers

Univ. of Siena, Italy(1)Giorgio Carluccio(2)Prof. Matteo Albani(3)Prof. Stefano Maci

Applied EM(1)Dr. Cagatay Tokgoz(2)Dr. C. J. Reddy

Date post:	27-May-2018
Category:	Documents
Upload:	dinhdien
View:	213 times
Download:	0 times

UTD Ray and Beam Methods for Analysis of Large EM … · UTD Ray and Beam Methods for Analysis of...

Documents