radiation of waves by spherical-circular …phd.dii.unisi.it/corsi/matdid/47_4-spherical.pdf · and...

(1)A.I. NOSICH

with contributions of (1)V.V. RADCHENKO and (2) S. RONDINEAU

(1) Institute of Radiophysics and Electronics NASU

Ulitsa Proskury 12, Kharkov 61085 - Ukraine

(2) IETR - UMR CNRS 6164

Université de Rennes 1, 35042 Rennes Cedex - France

RADIATION OF WAVES BY

SPHERICAL-CIRCULAR

MICROSTRIP ANTENNAS

AND LUNEBURG LENSES FED BY

ELECTRICAL AND MAGNETIC

DIPOLES

Principle of operation of printed antennas

σ = ∞

ε µ0 0

ВЭД ТМД

ε µ0 0ε µ

Je

Jm

ИЗЛУЧАЮЩИЙ ЭЛЕМЕНТ

ПОДЛОЖКА

ОСНОВАНИЕ

СТОРОННИЕ ТОКИ

rad ПОЛЕ ИЗЛУЧЕНИЯ

РЕЗОНАТОР

• Metal radiating element above the metal ground forms an open

resonator near whose natural frequencies the power of radiation(equivalently, radiation resistance) grows up ~ Q-factors

• Dielectric substrate provides mechanical rigidity of the structure

however may support unwanted surface waves

• Excitation if performed by an open coaxial cable or a slot

Develop an efficient algorithm based on the

method of analytical regularization (MAR)

for the 3D scattering by axially-symmetric

spherical-circular conformal printed

antenna structures :

Analyze spherical-circular printed antenna

performances

Aims of study

To be fast because of small-size matrices to be inverted

To take full account of the patch and ground plane finite size

To be uniformly accurate including resonances

Expected MAR algorithm advantages

r1

r2 θ2

z

Disadvantages of other approaches not based on

analytical regularization

FDTD and MoM fail near

the natural frequencies

because of staircasing and

lack of convergence

MAR guarantees high

accuracy even near the

high-Q natural frequencies

Mathematical formulation for RED excitation

( )( ) [ ]

( ) ( ) ( )φδθδrθsinr

r̂pφθ,r,J

21 r,r2

ee 1 ⋅⋅⋅⋅=�

Given el. current:

( ) ( )dr

kr

krZp

4π

ηk12nb

sn

r

r

e2

s o0n

i 2

1∫⋅⋅+−= The other spherical coefficients are null.Spherical coefficients :

( ) ( ) ( ) ( ) ( ) ( ){ }rkKPrkrkZPnnrbrE isnnii

snn

isnsn θθθ coscos1 1,

4,11

�

��

�

++∑∑= =≥

( ) ( ) ( )rkZPbjrH isnn

isnsni θηφ cos1,

4,11 ∑∑−= =≥

�

�

�

Field representation

satisfies Maxwell eqs.

and radiation cond. :

Boundary

conditions

generate a single

pair of the

dual series

equations :

( ) ( ) 0cos1

1=⋅−⋅∑ ≥

θn

feedK

nn

K

nnPXXC

( ) ( ) 0cos1

1=⋅−⋅∑ ≥

θn

feedZ

nn

Z

nnPXXC

shellp Np …1,0 =<≤ θθ

shellp Np …1, =≤< πθθ

Excitation vectors

Unknown vectors

Non metallic structure description

Disk description

Power boundedness condition : +∞<⋅⋅∑ ≥

2

1 nKnn XCn

Uniqueness Theorem equation of the 1st kind C.X = Y

� ill-posed problem

Analytical RegularizationGeneral scheme :

Split C = C1 + C2, with C1-1 being a known operator

A is a compact operator, ||A||L² <

X + A.X = Y0, where A = C1-1.C2 , and Y0 = C1

-1 .Y

Result: Fredholm 2nd kind matrix equation

∞+

||Y0||L² < ∞+unique exact solution X = (I + A)-1Y0, I : the identity operator

� well-posed problem

Application :

NOTE: matrix elements are expressed via cylindrical and trigonometric functions

DSE = equation of the 1st kind, C.X = Y


∀(p,q) {1…Nshell}², (CnaK-1.Cn

aZ)p,q = O(kp+1rpn) + O((rp/rp+1)2n) + Ca

p δp,q, with Cap constant∈

∀(p,q) {1…Nshell}², (CnbZ-1.Cn

bK)p,q = O(kp+1rp/n) + O((rp/rp+1)2n) + Cb

p δp,q, with Cbp constant∈

set of coupled linear 2nd kind equations, feed

mnm,nnm YXAXm =⋅+≥∀ ∑ ≥1,1

Algorithm properties

Algorithm convergence rate is examined by plotting the relative error er(N) in the L2² norme :

( )²

N

²

1-NN

r

2

2

X

XXNe

L

L−

=

For a d-digit accuracy, take N = max|krmax| + Const d.(rmax/h) + 5

Typically, N < 120 for standard studies � Inversion of a small well-posed linear system

� Low CPU time & memory capacity consuming

Truncate at order N �

AAlimN

N

=+∞→

as ( ) 0NelimN

=+∞→�

Finite-size matrix equation � XN + AN.XN = Y0N

( ) ( )22 L

N

L

1

N

AAAIX

XXNe −⋅+≤

−=

−

,1 Nm ≤≤

Application to SCMA study, centered RED feeding

Very fast algorithm: far field computation time

- 1s with non optimized MAR-based program,

per frequency point

- 16h41m with HFSS 8 software (kr2=0.75 to 10,

fast sweep option with 3 frequency points, 41389 cells)

- problems of convergence with CST 3 µwave

Studio software

θ1 = 180°, θ2 = 18°, r1/r2 = 0.97

εr1 = εr3=1, εr2 = 1.3

µr1 = µr2 = µr3 = 1

0 50 100 150 20010

-6

10-4

10-2

100

102

k.r2=4

k.r2=10

k.r2=20

k.r2=30

k.r2=50

k.r2=75

k.r2=95

Order of truncation N

er(N)

Convergence, error(N)

r1

r2 θ2

z


Effect of éléments of spherical-ground SCMA on the input resistance


In-resonance far-field radiation patterns of SCMA with RED feed

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

KoC=0.8

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

KoC=10.3

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

KoC=18.9

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

KoC=27.4

010TM

patch

�

�

�

�030

TM

patch

sphere

ground

020TM

patch


Effect of each element of SCMA on the directivity

The ground improves the directivity�

θ2 = 18°,

r1/r2 = 0.97,

εr1 = εr3= 1.00,

εr2 = 1.30


Effect of each element of SCMA on the main beam direction

Excitation

by a

centered

RED

θ2 = 18°,

r1/r2 = 0.97,

εr1 = εr3= 1.00,

εr2 = 1.30

HF “keel-

over” of the

main beam

direction

due to WG

modes =>

“keel over”


Ground conductor size effect: from full sphere to small disk

θ2 = 18°,

r1/r2 = 0.97,

εr1 = εr3= 1.00,

εr2 = 1.30

Validation: SCMA, centered probe feeding

θ1 = 160°, θ2 = 16°

r1 = 35.1mm

r2 = 38.2mm

εr3 = 1

εr1 = εr2 = 1.225

µr1 = µr2 = µr3 = 1

r1

r2 θ2

z

Far field pattern at 9.0 GHz

Level shifts & maxima position shifts due to some difficulties

of positioning the sphere in the anechoic chamber & some

disturbances due to the feeding coaxial cable

θ1

Mathematical formulation for TMD excitation

( )( )

+

=

−∑ s

n

snsp

ni

sn

snsp

ni

sn M

Nb

N

Ma

rHj

rEσ

σσ

σ

σσ

ση 1

11

1

11

,,�

�

�

�

�

�

�

�

Field representation :

Uniqueness Theorem equation of the 1st kind C.X = Y


Given magn. current: ( ) ( ) ( ) ( )ppmm rr

rprJ ϕϕδθδδ

θ

θϕθ −−=

sin

ˆ,,

2

�

Spherical coefficients :( ) ( )

( ) ( ) ( )p

ppsn

ppsnm

pnpsp

ni

sp

ni

frkZ

rkKpck

nnj

b

aϕ

σπσ

σ

σ

11

2

1

1

12

2

−

+=

Boundary

conditions

generate two

pairs of the

dual series

equations :

( ) 0cos1

1=⋅

−⋅∑ ≥

θαααn

feedK

nn

K

nn

PXXC

( ) ( ) 0cos1

1=⋅−⋅∑ ≥

θαααn

feedZ

nn

Z

nn

PXXC

shellp Np …1,0 =<≤ θθ

shellp Np …1, =≤< πθθ

Excitation vectors

Unknown vectors

Non metallic structure description

Disk description

ba ,=α

Power boundedness condition: +∞<⋅⋅∑ ≥

2

1n

aZ

na

nXCn +∞<⋅⋅∑ ≥

2

1n

bK

nb

nXCn

Application to SCMA study, centered TMD feeding

Effect of elements of SCMA on the input conductance

Thin substrate: h/c = 0.02, 0θ = 18°, ε = 1.3

0 20 40 60 80 100 0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

Log10

(G/Go)

Normalized frequency, k0c

no disk

1

2

3

4

5

6

7

8


In-resonance far-field radiation pattern of SCMA with TMD feed

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

KoC= 1 a/c= 0.98 To=18° ep=1.3

fi=0°

fi=90°

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

K oC=5 a/c=0.98 To= 18° ep= 1.3

fi= 0°

fi= 90°

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

KoC=14.2 a/c=0.98 To=18° ep=1.3

fi=0°

fi=90°

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

KoC=22.8 a/c=0.98 To=18° ep=1.3

fi=0°

fi=90°

120qTM

patch

130qTM

patch

sphere

ground

110qTM

patch

�

� �

�


Effect of elements of SCMA on the directivity of radiation

0 5 10 15 20 25 30 35 40 45 50 0

5

10

15

20

25

30

35

40


D - - - no disk


Effect of elements of SCMA on the main beam direction

0 5 10 15 20 25 30 35 40 45 50 0

20

40

60

80

100

120

140

160

180


Θmax - - - no disk

“keel over”

Conclusions

MAR-based simulation is a very fast and accuratemethod to study SCMA radiation and related problems.

This study :

- highlights effects of the ground conductor size and curvature,

- reveals several types of resonances connected with patch,

ground, and dielectric substrate,

- reveals the effect of the ”keel over” of the main radiation beam

at high frequencies

Operation principle and applications of Luneburg-lens antennas

( )2

2

−=

R

rrε

GO: parallel rays come to the focus on the surface of sphere of radius R if

• Wavelength range: cm – mm - THz

• Designs: spherical, cylindrical; discrete; effective-index

• Advantages: broad band, high radiation efficiency, multiple

beam capacity, easy mechanical scanning

• Applications: point-to-point communication, direction finding

in radar and radio astronomy

J. Sanford, Ph.D. thesis, EPFL, Lausanne, 1992D. Hilliard, D. Mensa, IEEExplore database

Huber & Zuchner Co., Zurich, 1990

Preliminaries: REDipole + Discrete Luneburg Lens

Nshell = 3 Nshell = 20

Nshell = 0 Nshell = 1 Nshell = 2

Luneburg

lens

somewhat

improves the

pattern

Influence of the

shells number on

the far field

pattern; WGM

effect

Lens structure :

Nshell : number of shells

θi = 0, ri = i/ Nshell

εri = 2 - [(2i-1)/ 2Nshell] ²

krmax = 10

"small lens"

Preliminaries: TMDipole + Discrete Luneburg Lens

Influence of the

shells number on

the far field

pattern; WGM

effect

Lens structure :

Nshell : number of

shells

krmax = 10

"small lens"

εri = 2- [(2i-1)/ 2Nshell] ²

N=3N=0

N=1

N=2

N=20

Luneburg lens

improves pattern,

however not

dramatically

� Develop an efficient algorithm based on the method of analytical regularization

(MAR) for the 3D scattering by axially-symmetric spherical structures :

� Analyze beamforming by a spherically stratified lens fed by a double-disk

spherical-circular printed antenna in two modes:

- axially-symmetric excitation by RED resulting in a conical pattern

- excitation by a centrally located TMD resulting in a broadside pattern

Aims of study

To be fast because of small-size matrices to be inverted

To take full account of PEC patch feed finite size and backward radiation

To be uniformly accurate including resonances

Expected MAR algorithm advantages

Analyzed geometry: Dipole + SCMA + DLL

1

Nshell

Nshell +1

rNshell

θin

2

...

Conformal patches

Lens: concentric

spherical layers

of dielectrics

Patches: PEC,

zero-thickness,

co-axially placed

spherical disks,

0≤θ ≤π

θoutGiven driving

current:

centered

RED or

TMD,

as a probe or

a slot model

RED+SCMA+Discrete Luneburg Lens

Influence of number of lens shells on the resonance

SCMA structure :

θθθθin = 18°, θθθθout = 36°

rin/rout = 0.97

µµµµr = εεεεr = 1.0

"small lens"

Lens => small resonance frequency shift of SCMA

Lens structure :



εri = 2 - [(2i-1)/ 2Nshell] ²

µri = 1.0

shellNi1i, ≤≤∀

RED+SCMA+Discrete Luneburg Lens study

Influence of the lens shells number on the far field pattern

SCMA structure :

θin = 18°, θout = 36°,

rin/rout = 0.97

µr = εr = 1.0

Frequencies tuned to the

1-st SCMA resonance

krmax~11

"small lens"

Nshell = 3, k0rmax = 11.45 Nshell = 20, k0rmax= 11.6

Nshell = 0, k0rmax = 11.65 Nshell = 1, k0rmax = 11.10 Nshell = 2, k0rmax = 11.35

Luneburg

lens

compresses

the main

beam

Lens structure :



εri = 2 - [(2i-1)/ 2Nshell] ²



Influence of number of shell constituting the lens on the directivity

SCMA structure :

θin = 18°, θout = 36°,

rin/rout = 0.97

µr = εr = 1.0

Lens structure :

discrete Luneburg

distribution

Nshell : shell number,


εri = 2 - [(2i-1)/ 2Nshell] ²

µri = 1.0


Centered RED excitation


Influence of number of lens shells on the resonance; WGM effect

SCMA structure :

θin = 0.02°,

θθθθout = 0.04°

rin/rout = 0.999

"large lens"

Lens => small shift in SCMA resonance frequencies

Lens structure :



εri = 2 - [(2i-1)/ 2Nshell] ²


TM010 TM011

REDipole+Discrete Luneburg Lens Radiation

N=0

N=1

Uniform

dielectric

sphere

significantly

improves

far-field

radiation

pattern,

however

WGM

sidelobes

appear

TM010: krmax=53 TM011: krmax=97.6

RED + SCMA+ Discrete Luneburg Lens

N=2

N=3

Adding

several layers

to discrete

Luneburg lens

still improves

far-field

radiation

pattern

TM010: krmax=53 TM011: krmax=97.6

RED + SCMA + Discrete Luneburg Lens

N=5

N=10

Multilayer

dielectric

Luneburg lens

dramatically

improves

far-field

radiation

pattern, WGM

sidelobes are

negligible

TM011: krmax=97.6TM010: krmax=53

Validation: RED + SCMA + DLL

centered coaxial probe feeding

θin = 2.0°, θout = 4.0°

rNshell-1 = 99% rNshell

εrNshell = 1

k0 rNshell = 20.94 (f = 5.0GHz)

Very good agreement between theory and

measurements

Nshell

rNshell

θin

...

θout

Eccostok Luneberg lens-P16©

(Emerson & Cuming)

theorymeasurements

TMD+SCMA+Discrete Luneburg Lens study

Influence of the lens shells number on the far field pattern

Lens

compresses

the beam

Lens structure :

discrete Luneburg

distribution

Nshell = 40 shells


εri = 2 - [(2i-1)/ 2Nshell] ²

µri = 1.0


θθθθin = 1.3°, θθθθout = 3°,

rin/rout = 0.97

µµµµr = εεεεr = 1.0

k0rmax = 10

"small lens"

TMD+SCMA+Discrete Luneburg Lens study

Influence of number of lens shells on the resonance; WGM effect

SCMA structure :

θθθθin = 0.02°,

θθθθout = 0.04°

rin/rout = 0.999

"large lens"

Lens => small resonance frequency shift of SCMA

Lens structure :



εri = 2 - [(2i-1)/ 2Nshell] ²


qTM110

TMD + SCMA + Discrete Luneburg Lens

N=0

N=1

Uniform

dielectric

sphere

significantly

improves

far-field

radiation

pattern

krmax=69

"large lens"

TMD + SCMA + Discrete Luneburg Lens

N=2

N=3

Several-layer

dielectric

Luneburg lens

still improves

far-field

radiation

pattern,

however not

much

krmax=69

"large lens"

Validation: TMD + SCMA + DLL

centered slot feedingNshell

rNshell

θin

...

θout

Eccostok Luneberg lens-P16©

(Emerson & Cuming)

θin = 2.0°, θout = 4.0°

rNshell-1 = 99% rNshell

εrNshell = 1

k0 rNshell = 25.5 (f = 6 GHz)

E-plane

H-plane

Conclusions

• MAR-based simulation of SCMA+DLL is accurate,

fast, and powerful

• It highlights key effects of beam compression, printed-feed

resonances, WGM resonances, and patch feed backward radiation

Characteristic comparison of algorithms:

- 1sec with MAR algorithm for a 0.3 to 30-λ lens - per frequency point

- 16h41m with the HFSS 8 software for a 0.3 to 3-λ lens in the fast

sweep option with 3 frequency points, 41389 cells

-problems of convergence with the CST 3 µwave Studio software

- both HFSS and µwave Studio fail for a 30-λ lens

Microwave power absorption in a partially screened

two-layer lossy dielectric sphere

Boundary-value problem:

1. Maxwell equations

2. Boundary conditions

3. Edge condition

4. Radiation condition

(1)

0

1

' (1) (1)

2 0 0 0 0 0

1 1

( )

1(1 ) (2 1) ( ) ( ) ( ) ( 1) ( )

2

m n nm n m

n

m

m n n nm n n nm

n n

X Q X V

V k c n k b k c Q n n Q

ε θ

ε ζ ψ θ α δ θ

∞

=

∞ ∞

= =

= +

= + + + + −

∑

∑ ∑

Final result of analytical

regularization:

Geometry: layered sphere axisymmetrically

excited by a Radial Electric Dipole (RED)

θ

a

0

c

b

2ε1ε

RED as antenna

Bone:

thickness 2 mm

Brain:

radius 100 mm

Skin:

thickness 1 mm

f = 900 MHz f = 1500 MHz

52.7rε =

9.67rε =

59.1rε =

0.0508 /Сим мσ =

7.75rε =

0.105 /Сим мσ =

1.05 /Сим мσ = 1.65 /Сим мσ =

46.0rε =

1.26 /Сим мσ =

45.6rε =

1.93 /Сим мσ =

02

r

i

f

σε ε ε

π= +

Averaged electrophysical parameters of the

human head tissues

Dipole antenna characteristics in the presence of a head

phantom capped with a PEC helmet

2x109

3x109

4x109

5x109

6x109

7x109

8x109

1

10

100 Rrad

Rabs

Rinput

Соп

роти

влен

ие, Ом

Частота, ГГц

1,00E+0092,00E+0093,00E+0094,00E+0095,00E+0096,00E+0097,00E+0098,00E+009

1E-4

1E-3

0,01

0,1

Пог

лощен

ная мощ

ность,

Вт

Частота, ГГц

00

450

900

1350

531

=ε

232

=ε

A1.00 =I cm2=l

S/m5.11 =σ

S/m6.02 =σ

mm100=c

mm3=h

0

090=θ

Averaged parameters of the outer layer (skin and bone):

Parameters of the inner domain (brain):

Dipole antenna parameters:

0 20 40 60 80 100 120 140 160 18010

-10

10-8

10-6

10-4

10-2

100

Угловой размер, градусы

Поглощенная мощность, Вт

Microwave absorption in a head phantom

covered with a helmet

0.003 W

0.3 W

Angular half-width of PEC spherical screen (“helmet”)

f = 900 MHz

Ab

so

rbe

d p

ow

er,

W

Conclusions

It has been shown that

• Metal hemispherical helmet can reduce EM absorption in

the head phantom in 100 times at the mobile-

communication frequencies

• No absorption resonances in the head phantom take place

in the presence of helmet, due to high conductivity values

of tissues

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