RADIATION OF WAVES BY 2-D REFLECTOR ANTENNAS FED BY COMPLEX-SOURCE-POINT FEEDS
Alexander I. Alexander I. Alexander I. Alexander I. NosichNosichNosichNosich, , , ,
with inputs from with inputs from with inputs from with inputs from AyhanAyhanAyhanAyhan AltintasAltintasAltintasAltintas and YuriyYuriyYuriyYuriy V. V. V. V. GandelGandelGandelGandel
Institute of Radio-Physics and Electronics NASUwww.ire.kharkov.ua
Bilkent University, Ankarawww.ee.bilkent.edu.tr
Kharkiv National University, Kharkiv, Ukrainewww.univer.kharkov.ua
1. Motivation
>> Metallic quasioptical reflectors are durable and provide very high directivity and small losses
Most common simulation tools and their shortages:
>> GO and PO: fail to characterize resonances, which may appear if near-field environment is complicated
>> MoM: hits "numerical wall" if applied to a single
reflector larger than 20 λ λ λ λ
>> FDTD: not applicable to quasioptical-scale problems because of enormous resources needed
2. Aim of Research | Outline
1.Development of efficient discrete modelbased on the boundary IEs
fast and convergent numerical algorithm with controlled accuracy
2. Computer simulation of 2-D antennas:
Single-reflector: parabolic, elliptic, offset
Dual-reflector: Cassegrain, parabola-cone
3. Frequency analysis of EM field effects
3. 2-D Problem Geometry
2
1 2( , ) : ( ),L x y R x F y y y y= ∈ = < <= ∈ = < <= ∈ = < <= ∈ = < <
Total field in the presence of scatterer:
2-D reflector is a curved strip, assumed to be perfectly
electrically conducting (PEC) and zero-thickness:
0
totU U U= += += += +incident field
scattered field
4. 2-D Problem Formulation
(((( )))) (((( ))))2, 0k U x y∆ + =∆ + =∆ + =∆ + = 2-D Helmholtz equation (off L)
2 21( , )( , ) ,
U x yikU x y o r x y
r r
∂∂∂∂ − = = + → ∞− = = + → ∞− = = + → ∞− = = + → ∞ ∂∂∂∂
Sommerfeld radiation condition
2 22k U U dσσσσ+ ∇ < ∞+ ∇ < ∞+ ∇ < ∞+ ∇ < ∞∫∫∫∫Ω
Local integrability = Edge condition
E polarization
(((( )))) (((( )))) (((( ))))0, , , ,
L LU x y U x y x y L= − ∈= − ∈= − ∈= − ∈
Dirichletb.c.
H polarization
0 ( , )L L
UUx y
n n
∂∂∂∂∂∂∂∂= −= −= −= −
∂ ∂∂ ∂∂ ∂∂ ∂
Neumanb.c.
must satisfy boundary conditions:Uon L,
5. Log-Singular & Hyper-Singular Integral Equations
E-problem reduces to a logarithmic-SIE of the 1st kind, andH-problem reduces to a hyper-SIE of the 1st kind for the unknown current, j(s), induced on the reflector:
E polarization
H polarization
(((( )))) (((( )))) (((( )))) (((( ))))(((( ))))1
0 0 0 0 0( ) ( ) ,
4L
iH k r s r s j s ds U r s s L− = − ∈− = − ∈− = − ∈− = − ∈∫∫∫∫
single layer potential
(((( )))) (((( )))) (((( )))) (((( ))))(((( ))))1
0 0 0 0
1( ) ( )
2x x
Ly xL
j s H k r s r s ds U r sn n n
∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂− = −− = −− = −− = −
∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∂ ∂ ∂∫∫∫∫
π
double layer potential
6666. . . . Method of Analytical Regularization via SMethod of Analytical Regularization via SMethod of Analytical Regularization via SMethod of Analytical Regularization via Statictatictatictatic----Part Inversion: from SIE a Part Inversion: from SIE a Part Inversion: from SIE a Part Inversion: from SIE a FredholmFredholmFredholmFredholm 2222----nd Kind IEnd Kind IEnd Kind IEnd Kind IE
Numerical solution of a Fredholm 2-nd kind IE
can be obtained by any not pathologic
discretization scheme (collocation, Galerkin)
leading to the Fredholm 2-nd kind matrices.
Hence, solutions are stable and convergent
kernel decomposition =static
singular part + regular part
H-pol (E-pol) IE for a PEC strip can be integrated (differentiated)
that leads to IE with Cauchy singularity plus auxiliary relation
This is a Fredholm IE of the 2-nd kind,
where
A X = B
X + CX = D
=>
( ) ( ) )(),(4/1
1
tfdttLtthtt
itq sss
s
s =
+
−∫
−
( ) ( ) )(,~)(~ 1
1
tfKdtttNtqtqtsss
−=+ ∫−
∫−
−
−−−=−=
1
1
2/122/12)()1(
4)(),,()1(),(
tx
dxxfx
itfKtthKtttN tstss
π
Singular operator is invertible - see Carleman 1928; Krein 1951; Erdogan&Gupta 1972
)()()(~ tLtqtq =
7. MAR via 7. MAR via 7. MAR via 7. MAR via diagonalizationdiagonalizationdiagonalizationdiagonalization with the aid with the aid with the aid with the aid of the staticof the staticof the staticof the static----part part part part eigenfunctionseigenfunctionseigenfunctionseigenfunctions
( )( ) ( ) ==−
−∫
−
)(,ln)1(
1
12/1
tTtTdtttt
tTnnnss
s
sn σ Chebyshev polynomials
of the 1-st kind
To transform SIE to the Fredholm
2nd kind matrix equation, take full
set of the corresponding Chebyshev
polynomials as a basis (I.e., make
analytical preconditioning)
Green’s function decomposition = static singular part + regular dynamic part
( ) |)|(||ln4
,0 sss
rrkRrri
rrG
−+−=
E-pol.
H-pol.
( )( )
==∂
−∂−∫
−
)(),(ln
)1(1
12
2
2/1 tUtUdtn
rrtUt
nnns
M
ssns
τ
Chebyshev polynomials
of the 2-nd kind
A X = B
X + CX = D
=>
8. Incident Field: Horn Modeling with CSP Beam
r0
x
y
β
2b
x0
y0
φcs
( ) ( )csin
rrkHru
−= 0In 2-D:
Advantages of the CPS field:
- Exact solution of the Helmholtz
equation at any space domain
- Variable beam width controlled by
the parameter kb
Dots and curvy line denote
the branch points and the
cut in the real space.
Branch cut plays the role of emitting aperture.
0,0 1,6 3,2 4,8 6,4 8,0
-4,8
-3,2
-1,6
0,0
1,6
3,2
4,8field intensity
r /λ0
kb = 5
1E-3
0,002796
0,007820
0,02187
0,06115
0,1710
0,4782
1,337
3,739
10,46
29,24
81,77
228,7
639,4
1788
5000
biryxr cscscs
+== 0,
, ββ bSinbCosb =
, 000 yxr =
-180 -120 -60 0 60 120 180
kb = 0.5
kb = 1
kb = 2
( )βϕ
π−−
∞→⋅⋅
−
cos
0
00
2~
kbrrik
ree
rrkiU
Factor, determ
ining the
directiv
e character of th
e CSP
field in
the fa
r zone
9. 2-D Parabolic Reflector Antenna in Free Space
Fed by a Complex-Source-Point Beam
(a) Directivity, and (b) Edge illumination as a function of the feed beamwidth parameter kb - aperture size of horn simulated with CSP
d=20λ, λ, λ, λ, f/d= = = = 0.5CSP feed in GO focus
10. 2-D Parabolic Reflector Antenna in Free Space
Fed by a CSP Beam
Dynamics of the far-field pattern variation with reflector size (left, kb=5), and
edge illumination given by the feed beamwidth parameter kb (right, d=10λλλλ)
11. 2-D Parabolic Reflector Antenna Fed by a CSP Beam
near a Flat Impedance Earth
Normalized radiation (a) and absorption (b) resistance, Efficiency (c), and Gain
(d), as a function of the antenna aiming angle. d=10λλλλ, f/d=0.5, kb=2 (E.I.=-8 dB)
12. 2-D Parabolic Reflector Antenna Fed by a CSP
Beam near a Flat Impedance Earth
Left = directivity as a function of the angle of inclination. Hump
corresponds to the spillover reflection matched with main beam.
Middle and right = far-field patterns for the optimally inclined and
in-zenith looking antennas. Note spillover lobes reflected from earth
d=20λ, λ, λ, λ, f/d= = = = 0.5CSP feed in
GO focus
13. 2-D Parabolic Reflector Antenna Fed by a CSP
Beam in a Circular Dielectric Radome
Directivity (top) and total radiated power (bottom), as a function of the radome
radius, for the matched and mismatched radomes. Note a chance to boost
either directivity or power due to resonances. d=5λλλλ, f/d=0.5, kb=2.6 (- 9 dB E.I.)
14. 2-D Parabolic Reflector Antenna with Resistive
Edges Fed by a CSP Beam in Free Space
Left column = far-field patterns for a uniformly resistive RA. d=20λλλλ , f/d=0.73, kb=5
Right column = far-field patterns for an RA with 1-λλλλ wide “linearly” resistive edge
E
2 2
2
( )( ) ( '( )) ( '( ))
1
v tj t x t y t
t+ =+ =+ =+ =
−−−−
(((( )))) (((( )))) (((( ))))(((( ))))
(((( )))) (((( )))) (((( ))))
1
0 0 02
01
1 1
0
0 02 2
1 1 0
1 1,
4 1
1
4 1 1
i dtK t t v t U t
t t t
dti dtM t v t U t
t t
π
π
−−−−
− −− −− −− −
′′′′+ = −+ = −+ = −+ = − −−−− −−−−
= −= −= −= − − −− −− −− −
∫∫∫∫
∫ ∫∫ ∫∫ ∫∫ ∫
15. MDS for Singular/Hyper-Singular Integral
Equations
H
(((( ))))
(((( ))))(((( ))))
(((( )))) (((( )))) (((( ))))
21 1
2 2
02
1 10
1
2
0 0 0
01
1 11 ln 1
8
1, 1
v t k Lt dt v t t t t dt
t t
v t N t t t dt L U tn
π π
π
− −− −− −− −
−−−−
− − − − +− − − − +− − − − +− − − − +
−−−−
∂∂∂∂+ − = −+ − = −+ − = −+ − = −
∂∂∂∂
∫ ∫∫ ∫∫ ∫∫ ∫
∫∫∫∫ (((( )))) (((( )))) 21j t v t t= −= −= −= −
Hyper-SIE contour parameterization
SIE contour parameterization x(t),y(t), -1≤t≤1
transformation of SIE into Cauchy-singular IE
note: K, M, N
–are
smooth functions
16. Method of Discrete Singularities
Discretization is done by using the quadrature formulas of interpolation type with the nodes in the nulls of the Chebyshev polynomials of the 1st and the 2nd kind:
2 1( ) 0, cos , 1,2, ...
2
n n
n i i
iT t t i n
nπ
−−−−= = == = == = == = =
1 0( ) 0, cos , 1, 2, ... 1
n n
n oj j
jU t t j n
nπ−−−− = = = −= = = −= = = −= = = −
17. Method of Discrete Singularities
E
H
Resulting matrix equations are uniquely solvable for each polarization case and give us the required current functions, v(t). Accuracy is controlled by the number of nodes "n":
(((( )))) (((( ))))
(((( ))))
0 0 0
1 0
1
0
0 02
1 1 0
1 1 1( , ) ( ) ( ) , 1,2, ... 1
4
1 1( ) ( ) ( ) ,
4 1
nn n n n
q j q jn nq q j
nn n
q q
q
iK t t v t U t j n
n t t
dtiM t v t U t j n
n t
π
π
====
==== −−−−
′′′′+ = − = −+ = − = −+ = − = −+ = − = −
−−−−
= − == − == − == − =−−−−
∑∑∑∑
∑∑∑∑ ∫∫∫∫
(((( )))) (((( ))))(((( ))))
(((( ))))(((( )))) (((( ))))
(((( ))))(((( ))))
(((( )))) (((( ))))(((( )))) (((( )))) (((( ))))
12 2
2 0 0 0 0 0 02
10 0
2
0 2
2 0 0 0 0 0 0
1 111 , ,
1, , , 1, ..., 1
2
j qn
n q q n j q j q
qj qq j
j
n j n j j j j j
v t t S t t N t tn t t
tnv t S t t N t t f t j n
n
λ
λ
++++−−−−
−−−−====≠≠≠≠
−−−−
− −− −− −− −
− + + +− + + +− + + +− + + + −−−−
−−−− + − + + = = −+ − + + = = −+ − + + = = −+ − + + = = −
∑∑∑∑
here (((( )))) (((( )))) (((( ))))(((( ))))
21
2
0 0 0 0
1
11, ln 2 2 ,
2 8
j qn
n j q l j l q
l
k LS t t T t T t
l nλ
++++−−−−
====
−−−−= + + == + + == + + == + + =
∑∑∑∑
18. Convergence and Accuracy in MDS Modelling
of 2-D Parabolic Reflector Antennas
f/d=0.5, kb=2.6
MDS mean-square errors in the
surface current functions of
symmetric parabolic antennas
versus the interpolation order, n
MDS mean-square errors in the far-
field patterns of the same antennas
versus interpolation order, n
19. Complex-Source-Point Beam Field
Normalized radiation patternsof CSP feed for different kb
Near fields of several CSP feeds
Advantages of CSP beam field:• Exact solution of the Helmholtz
equation at any space domain• The greater the kb, the narrower the CSP beam
r0x
y
ββββ2
b x
0
y0
Branch cut plays the role of emitting aperture
20. Front20. Front20. Front20. Front----Fed Parabolic ReflectorFed Parabolic ReflectorFed Parabolic ReflectorFed Parabolic Reflector
d/λλλλ = 10
kb = 2.5
d/λλλλ is fixed, kb is varied:
main beam width – no change
side lobe level - decreasing
kb is fixed, d/λλλλ is varied:
main beam width – decreasing
side lobe level - no change
::: Symmetric dish :::
Focal point
21. Front21. Front21. Front21. Front----Fed Parabolic Reflector Near FieldFed Parabolic Reflector Near FieldFed Parabolic Reflector Near FieldFed Parabolic Reflector Near Field
d/λλλλ = 50
kb = 2.5
d/λλλλ = 5
kb = 2.5
d/λλλλ =10
kb = 2.5
22. Front22. Front22. Front22. Front----Fed Parabolic Reflector Focal ShiftFed Parabolic Reflector Focal ShiftFed Parabolic Reflector Focal ShiftFed Parabolic Reflector Focal Shift
Total radiated power and main beam directivity:
(((( )))) (((( ))))2
22
0 0 0
0
2| | , max
π
πϕ ϕ ϕΦ ΦP d D
P= == == == =∫∫∫∫
= total field Radiation Pattern
= Observation Angle
(((( ))))0ϕΦ
0ϕ
Directivity as a function of kb
grows up with reflector size d/λ λ λ λ , and reaches maximum if kb≈2.5
reaches maximum if the source is slightly shiftedfrom the reflector focus
D as a function of x0 (source location)
grows up with d/λλλλ
kb = 2.5
focal point
::: Shallow dish :::
23. Offset Parabolic Reflector Simulation23. Offset Parabolic Reflector Simulation23. Offset Parabolic Reflector Simulation23. Offset Parabolic Reflector Simulation
kb = 11
d/λλλλ = 47
ββββ = 1400
Phase: main beam is locally close to the plane wave
Near field perfectly illustrates the wave effects and the interference in near zone
The larger the reflector => the
narrower the main beam width
Focal point
24. Offset Parabolic Reflector 24. Offset Parabolic Reflector 24. Offset Parabolic Reflector 24. Offset Parabolic Reflector ““““Quiet ZoneQuiet ZoneQuiet ZoneQuiet Zone””””
Phase Pattern: main beam is locally close to the plane wave in the so-called “quiet zone” of a quasioptical-size offset reflector
d/λλλλ=47, f/d=0.5, kb=11 (-10 dB edge illumination)
::: Deep dish :::
d/λλλλ = 30
kb = 3
ββββ = 1230
d/λλλλ = 5
kb = 3
ββββ = 1230
25. Offset Parabolic Reflector Simulation25. Offset Parabolic Reflector Simulation25. Offset Parabolic Reflector Simulation25. Offset Parabolic Reflector Simulation
26. Elliptic Reflector Focuser Near Field26. Elliptic Reflector Focuser Near Field26. Elliptic Reflector Focuser Near Field26. Elliptic Reflector Focuser Near Field
kb = 9
d/λλλλ = 30
kb = 2
d/λλλλ = 30
kb = 0.5
d/λλλλ = 14
27. Elliptic Cross27. Elliptic Cross27. Elliptic Cross27. Elliptic Cross----Section Focuser Near FieldSection Focuser Near FieldSection Focuser Near FieldSection Focuser Near Field
kb = 2
d/λλλλ = 12
kb = 2
d/λλλλ = 9
kb = 10
d/λλλλ = 46
kb = 10
d/λλλλ = 20
28. 28. 28. 28. CassegrainCassegrainCassegrainCassegrain Antenna SimulationAntenna SimulationAntenna SimulationAntenna Simulation
kb = 9 d1 = 30λ λ λ λ d2 = 7λλλλ
29. Offset 29. Offset 29. Offset 29. Offset CassegrainCassegrainCassegrainCassegrain Antenna SimulationAntenna SimulationAntenna SimulationAntenna Simulation
kb = 25 β β β β = 22= 22= 22= 220000
d1 = 10λ λ λ λ d2 = 4λλλλ
30. 230. 230. 230. 2----D Model of "D Model of "D Model of "D Model of "OmnidirectionalOmnidirectionalOmnidirectionalOmnidirectional" PACO Antenna " PACO Antenna " PACO Antenna " PACO Antenna
single parabola directivity
PACO directivity
Edge Illumination
= 600
single-feed design provides higher directivity if kb is small.
however, if kb gets larger, double source is able to
increase the PACO directivity
31. 231. 231. 231. 2----D PACOD PACOD PACOD PACO----type Antenna Simulationtype Antenna Simulationtype Antenna Simulationtype Antenna Simulation
single source
double source
kb = 9.2 provides maximum to directivity of double-source PACO
in case of 2 CSP the main beams are narrower
single source double source
32. Conclusions32. Conclusions32. Conclusions32. Conclusions
>> Complex-Source-Point field is a very good model of typical small-aperture feed
>> MAR & MDS possess fast convergence and controlled accuracy
>> MDS implementation is simpler than MAR
>> MAR & MDS compute 2-D models of real-life multi-reflector antennas of arbitrary shape mostly in seconds
>> MAR or MDS can be used as a core of the computer optimization and synthesis software