81
Analyses of Spherical antennas K. W. Leung State Key Laboratory of Millimeter Waves & Department of Electronic Engineering, City University of Hong Kong 1
Analyses of Spherical antennas
K. W. LeungState Key Laboratory of Millimeter Waves &
Department of Electronic Engineering,City University of Hong Kong
1
Characteristics of spherical array
Array elements distributed on a sphere surface.
Conformal, low profile, light weight, easy to install on aircraft surfaces
Stable antenna gain and radiation pattern for scanning
Wide angular coverage by activating different arrayelements
2
Spherical array application
Earth-orbiting scientific satellite system
The user satellite is using the spherical array
[Stockton and Hockensmith 1977]
Stockton & Hockensmith, "Application of spherical arrays -- A simple approach," Antennas and Propagation Society International Symposium, 1977 , vol.15, pp. 202- 205, June 1977. 3
Hemispherical coverage with a moderate gain for steered beams (~13 dBic).
A cluster of 12 patch elements out of a total 120 are activated at a time. Beam steering was accomplished by shifting the active part in small steps.
[Stockton 1978]
Hemispherical patch array for satellite data link:Electronic Switching Spherical Array (ESSA)
76 cm
2.0-2.3 GHz
551 pre-programedBeams
3-dB Beam width:26°
Max. VSWR: 1.5
R. Stockton, "Electronic switching spherical array antenna", NASA (Unspecified Center), Apr. 1, 1978.4
Elements of the ESSA: RHCP Microstrip Patch Antenna
[Stockton 1978]
Heigth :0.635 cm
Spacing :0.6 - 0.8 λ
5
Proposed by Mei
Radiate circularly polarizedfields over a wide beamwidth.
Suitable for systems requiringwide-beamwidths (e.g., GPS).
Spherical helical antenna
K. K. Mei and M. Meyer, “Solutions to spherical anisotropic antennas,” IEEE Trans. Antennas Progagat., vol. 12, pp.459-463, 1964.
A. Safaai-Jazi and J. C. Cardoso,"Radiation characteristics of a spherical helical antenna," IEE Proc. Microw. Antennas Propag., vol. 143, no. I, Feb. 1996
Radiation pattern of a 7-turn spherical helical antenna
3dB beamwidths: 60o
Radiation patterns of spherical helical antenna
7
Measured electric field patterns of the 6-turn cylindrical helical antenna
3dB Beamwidth: ~ 40o
Comparison:radiation pattern of the cylindrical helical antenna
J. D. Kraus,"Helical Beam Antennas for Wide-Band Applications," Proceedings of the IRE,vol. 36, no.10, pp1236-1242,1948. 8
9
Spherical Solutions
No edge-shaped boundaries as found in cylindrical and rectangular structures
Closed-form Green’s functions obtainable
Exact solutions used as references for checking the accuracy of numerical or approximation techniques
Helmholtz Equation in Spherical Cooridnates
Solution
where bn(kr) is the spherical Bessel function
is the Associated Legendre function of the first kind
is sinusoidal function.
)(cosmnP
jmnmnnm ePkrb )(cos)(,
jme
Remark: Since the associated Legendre function of the second kind, is singular at = 0 or , it is generally not used for engineering EM problems.
)(cosmnQ
0sin1sin
sin11 2
2
2
2222
2
k
rrrr
rr
10
Spherical Bessel Functions
)(2
)( 2/1 krBkr
krb nn
where is the ordinary (cylindrical) Bessel function
)(2/1 krBn
)(),(~)( krykrjkrb nnn
11
Riccati-Bessel Functions (Schelkunoff-typeSpherical Bessel Function)
• All EM fields can be found from 2 potential functions
• Define and
• But Ar, Fr, are not solutions of Helmholtz equation
• Instead, Ar/r, Fr/r are solutions of Helmholtz equation
• Define Riccati-Bessel function
rAA r ˆ
rFF r ˆ
)(2
)()(ˆ2/1 krBkrkrkrbkrB nnn
12
General Solutions of Sphericl PotentialFunctions
mjmnnrr ePkrBFA )(cos)(ˆ~,,
13
Legendre Functions
m = 0 m = 1
14
E & H fields in Electromagnetics
15
16
Grounded Spherical Hemispherical Dielectric Resonator Antenna: Embedded Magnetic Source
17
(a) Side View (b) Top View
18
19
')(ˆ)'('ˆ')(ˆ)'('ˆ
)(cos)2(
)2(
0 rrkrJkrHrrkrHkrJePAG
nn
nnjmmn
n
n
nmnm
FM
rp
arrkHCarkrJBePG
nnm
nnm
n
n
nm
jmmn
FM
rh
)(ˆ)(ˆ
)(cos0
)2(0
')(ˆ)'(ˆ')(ˆ)'(ˆ
)(cos)2(
)2(
0 rrkrJkrHrrkrHkrJePDG
nn
nnjmmn
n
n
nmnm
AM
rp
arrkHFarkrJEePG
nnm
nnm
n
n
nm
jmmn
AM
rh
)(ˆ)(ˆ
)(cos0
)2(0
: Associated Legendre function of the first kind (order m,degree n)
: Spherical Bessel function of the first kind (Schelkunoff type)
: Spherical Hankel function of the second kind (Schelkunoff type)
0kk r
P xnm( ) ( )J xn
( )( )H xn2
Electric & Magnetic Green’s Functions Due to M
sMEE
Particular solutions obtained by matching theboundary conditions at the source point (r = r’)
0 EE
0 HH
0 HH
0
2
0)(cos
)!()!(
)1(12
4' ddePMm
mnmn
nnnrA jmm
nsnm
ddePddM
mnmn
nnn
krD jmm
nso
nm
)(cossin)!()!(
)1(12
4'
0
2
0
20
21
For a point source
sin')'()'(
2rM s
')(ˆ)'('ˆ')(ˆ)'('ˆ
)'(sin)(cos)'(cos'sin'
1)2(
)2(
1 1 rrkrJkrHrrkrHkrJmPPa
rG
nn
nnmn
mn
n
n
mnm
FM
rp
')(ˆ)'(ˆ')(ˆ)'(ˆ
)'(cos)(cos)'(cos''
1)2(
)2(
1 0 rrkrJkrHrrkrHkrJmPP
ddd
rG
nn
nnmn
mn
n
n
mnm
AM
rp
mmnmn
nnnjanm
)!()!(
)1(12
2
where
)!()!(
)1(12
2 mnmn
nnn
kd
m
onm
0 for 20 for 1
mm
m
,0 EE
Homogenous solutions obtained by matching theboundary conditions at the dielectric surface ( r = a)
,0 EE ,0
HH 0 HH
arrkHcarkrJbkrJmPP
rG
nnm
nnmn
n
n
m
mn
mn
FM
rh
)(ˆ)(ˆ
)'('ˆ)'(sin)(cos)'(cos'sin'
1
0)2(
1 1
arrkHfarkrJekrJmPP
dd
rG
nnm
nnmn
n
n
m
mn
mn
AM
rh
)(ˆ)(ˆ
)'(ˆ)'(cos)(cos)'(cos''
1
0)2(
1 0
)(ˆ)('ˆ)('ˆ)(ˆ0
)2()2()2()2( akHkaHkkakHkaHab nno
onnTEn
nmnm
)('ˆ)(ˆ)(ˆ)('ˆ0
)2()2()2()2( akHkaHkkakHkaHde nno
onnTMn
nmnm
,TEn
nmonm
akkjc
where
TMn
nmnm
djf
22
Magnetic Green’s Function Due to Mr
sin)'()'()'()( 2
22
rrr
rrG
krp
r
FM
')(ˆ)'(ˆ')(ˆ)'(ˆ
)'(cos)(cos)'(cos'1
)2(
)2(
0 02 rrkrJkrH
rrkrHkrJmPPgr
Gnn
nn
n
n
m
mn
mnnm
FM
rp
r
Particular solution
)12()!()!(
2
nmnmn
kjg
mnm
Homogenous solution
arrkHiarkrJhkrJmPP
rG
nnm
nnmn
n
n
m
mn
mn
FM
rh
r )(ˆ)(ˆ
)'(ˆ)'(cos)(cos)'(cos'1
0)2(
0 02
,)(ˆ)('ˆ)('ˆ)(ˆ0
)2()2(
00
)2()2(
akHkaHkkakHkaHgh nnnnTE
n
nmnm
where
where
23
TEn
nmnm
gkkji
0
Define 3 Dyadic Green’s Function
rGrGrrGG HM
HM
HM
HM
rr
r
rr ˆˆˆˆˆˆ
rGrGrrG HM
HM
HM rr
r
rˆˆ,ˆˆ,ˆˆ
ˆˆˆˆˆˆ H
MHM
HM
HM GGrGG r
')]ˆ'()ˆ'([ dSMGrMGHHM
Sr
HM
o
r
Total H-field due to Mr & M is given by
'ˆ)''(ˆ)''(ˆ)''(
')ˆ'ˆ'ˆ'()ˆ'ˆ'ˆ'(
dSMGMGMGMGrMGMG
dSMGMGrMGMGMGrMGH
o
rr
rr
r
o
r
rr
r
r
S
HMr
HM
HMr
HM
HMr
HM
S
HM
HM
HMr
HMr
HMr
HM
24
cossin HHH ry Since
'sin'' yr MM
'cos'' yMM
o
y
y
o
r
rr
r
Sy
HM
Sy
HM
HM
HM
HM
dSMG
dSMGGGGH
''
'']cos)'cos'sin(sin)'cos'sin[(
cos)'cos'sin(sin)'cos'sin( H
MHM
HM
HM
HM GGGGG
r
rr
r
y
y
we have
Therefore, the required Green’s function is given by
25
HPHM GGG y
y
Expressing the Green’s function in GP & GH:
1 0
220
)()'()'(cos)(cos)'(cos)1('sin'sin1
nnn
mn
mn
n
mnmP krkrmPPgnn
rrjG
1 0
20
)()'(')'(sin)(cos)'(cos)1('
cos'cos1n
nnm
nm
n
n
mnm krkrmPPann
rrj
1 0
220
)(')'()'(sin)(cos)'(cos'
cos'sin-n
nnm
nm
n
n
mnm krkrmPPmg
rrjk
1 10
)()'()'(cos)(cos)'(cos''
cos'cos1-n
nnm
nm
n
n
mnm krkrmP
ddP
ddd
rr
1 00
)(')'(')'(cos)(cos)'(cos'cos'cos
nnn
mn
mn
n
mnm krkrmPPma
rrjk
where
'),'(ˆ'),'(ˆ
)'( )2( rrkrHrrkrJkr
n
nnin which
'),(ˆ'),(ˆ
)()2(
rrkrJrrkrHkr
n
nn
26
1 0
220
)(ˆ)'(ˆ)'(cos)(cos)'(cos)1('sin'sin1
nnn
mn
mn
n
mnmH krJkrJmPPhnn
rrjG
1 0
20
)(ˆ)'('ˆ)'(sin)(cos)'(cos)1('
cos'cos1n
nnm
nm
n
n
mnm krJkrJmPPbnn
rrj
1 0
20
)('ˆ)'(ˆ)'(sin)(cos)'(cos'cos'sin-
nnn
mn
mn
n
mnm krJkrJmPPmh
rrjk
1 10
)(ˆ)'(ˆ)'(cos)(cos)'(cos''
cos'cos1-n
nnm
nm
n
n
mnm krJkrJmP
ddP
dde
rr
1 00
)('ˆ)'('ˆ)'(cos)(cos)'(cos'cos'cos
nnn
mn
mn
n
mnm krJkrJmPPmb
rrjk
27
Converting double summations to single summation
By applying the addition theorem for Legendre polynomials
)(cos)(cos)(cos)!()!(2)(cos
om
mn
mn
mn mPP
mnmnP
where
)cos(sinsincoscoscos
28
HPHM GGG y
y
where we have used the fact that = =/2.
)()'())(cos()12)(1('sin'sin
41- '
122
0
krkrPnnnrrk
G nnnn
P
)()'('))(cos('
)12('
sin'cos4
1- '
12
0
krkrPnrr nnn
n
)(')'())'(cos()12('cos'sin
41-
12
0
krkrPnrr nnn
n
)()'())'(cos(')1(
12'cos'cos
4-
1
krkrPnnn
rrk nnnn
)(')'('))'(cos('')1(
12'cos'cos
4-
2
1
krkrPnnn
rrk
nnnno
29
)(ˆ)'(ˆ))'(cos()12)(1('sin'sin
41-
122
0
krJkrJPnnnbrrk
G nnnn
nH
)(ˆ)'('ˆ))'(cos('
)12('
sin'cos4
1-1
20
krJkrJPnbrr nnn
nn
)('ˆ)'(ˆ))'(cos('
)12('cos'sin
41-
12
0
krJkrJPnbrr nnn
nn
)(ˆ)'(ˆ))'(cos(')1(
12'cos'cos
4-
1
krJkrJPnnne
rrk nnnn
n
)('ˆ)'('ˆ))'(cos('')1(
12'cos'cos
4-
2
1
krJkrJPnnnb
rrk
nnnn
no
)(ˆ)('ˆ)('ˆ)(ˆ1 )2()2()2()2( akHkaHkkakHkaHb onno
onnTEn
n
where
)('ˆ)(ˆ)(ˆ)('ˆ1 )2()2()2()2( akHkaHkkakHkaHe onno
onnTMn
n
30
Numerical Problem for GP
- GP is an infinite summation over n- Hankel functions of very high orders have very large
ampliudes- Difficult to handle numerically
Recall that physically Gp represents a z-directed electricfield excited by a z-directed point current, therefore
Solution
Rek
yjG
jkR
oP 4
22
2
22 )'sin'()'cos'cos( ryrrR
where
31
Mathematical Identity
Based on this fact, a mathematical identity can be established:
Rek
yj jkR
o 42
2
2
)()'())'(cos()12)(1('sin'sin
41-
122
0
krkrPnnnrrk nnn
n
)()'('))(cos('
)12('
sin'cos4
1- '
12
0
krkrPnrr nnn
n
)(')'())(cos()12('cos'sin
41- '
12
0
krkrPnrr nnn
n
)()'())'(cos(')1(
12'cos'cos
4-
1
krkrPnnn
rrk nnnn
)(')'('))'(cos(')1(
12'cos'cos
4-
2
1
krkrPnnn
rrk
nnnno
))'sin'()'cos'cos(( 22 ryrrR
32
Rek
yjG
jkR
o
HM
y
y 42
2
2
)(ˆ)'(ˆ))'(cos()12)(1('sin'sin
41
122
0
krJkrJPnnnbrrk nnn
nn
)(ˆ)'('ˆ))'(cos('
)12('
sin'cos4
11
20
krJkrJPnbrr nnn
nn
)('ˆ)'(ˆ))'(cos()12('cos'sin
41
12
0
krJkrJPnbrr nnn
nn
)(ˆ)'(ˆ))'(cos(')1(
12'cos'cos
4 1
krJkrJPnnne
rrk nnnn
n
)('ˆ)'('ˆ))'(cos('')1(
12'cos'cos
4
2
1
krJkrJPnnnb
rrk
nnnn
n
Finally, the required Green’s function is given as follows:
33
Method-of-Moments Solution
Expand the equivalent magnetic current of the slot:
)()(),( npun yyfxfyxf
2/0
2/1)(
Wx
WxWxfu
N
nnn yfVyM
1
)()(
where
hy
hyhk
yhkyf e
e
p
0sin
)(sin)(
34
Hmn
pmn
amn YYY
0 0
')','(),(2S S
npmP
mn dSdSyxfGyxfY
MoM Admittance of the Grounded HemisphericalDielectric Resonator Antenna
where
0 0
')','(),(2S S
nHmH
mn dSdSyxfGyxfY
However, GP is singular
Difficult to integrate numerically pmnY
35
Formulation of the Microstrip Feed network- Apply the reciprocity approach as done by David Pozar* - Not repeated here.
D. M. Pozar, “A reciprocity method of analysis for printed slot and slot-coupled microstrip antennas,” IEEE Trans. Antennas Progagat., vol.34, pp. 1439-1446, Dec. 1986.*
Solution: Use the reduced kernel (Richmond form)
2
2
2
2
2222252 ')'()32)(1()(
42 L
L
L
L npeeeeee
jk
mpp
mn dydyyyfkaajkeyyfk
jYe
where
4/Wae
22)'( ee ayy
with being the equivalent radius of the slot.
36
Results
At the reference plane (slot position): a = 12.5 mm, ra = 9.5, xd = yd = 0.0, W = 0.9 mm, Wf= 1.45 mm, d = 0.635 mm, rs = 2.96, Ls = 13.6 mm .
37
At the reference plane (slot position): a = 12.5 mm, ra = 9.5, L = 13.5mm, yd = 0.0, W = 1.3mm, Wf = 1.45 mm, d = 0.635 mm, rs = 2.96, Ls = 13.6 mm .38
At the reference plane (slot position): a = 12.5 mm, ra = 9.5, L = 13.5mm, xd = 0.0, W = 1.3mm, Wf = 1.45 mm, d = 0.635 mm, rs = 2.96, Ls = 13.6 mm .39
40
Grounded Spherical Hemispherical Dielectric Resonator Antenna: Surface Electric Source
Ground Plane
HemisphericalDRA
Conducting Conformal Strip
la
W
zx
y
r CoaxialAperture
41
Advantages of Conformal-Strip Method
Compatible with the probe-feed method Drilling hole not required Very convenient post manufacturing trimmings
- Cut shorter without leaving an air hole- Extended longer without the need for
deepening the hole It is conformal
42
Hemispherical DRA for Demonstration
Analytical closed-form Green function can be obtained
Efficient in numerical implementation Excited in the fundamental broadside TE111
mode
43
Co-ordinate system in the analysis
z
W2l
x y
a
Spherical DRA
Conformalr
k k 0
strip
44
DRA Green’s FunctionInside the DRA (r < a)
0
1 )(cos)(ˆn
n
nm
jmmnnnmr ePkrJAF
0
1 )(cos)(ˆn
n
nm
jmmnnnmr ePkrJBA
Outside the DRA (r > a)
0
0)2(
2 )(cos)(ˆn
n
nm
jmmnnnmr ePrkHCF
0
0)2(
2 )(cos)(ˆn
n
nm
jmmnnnmr ePrkHDA
(1)
(2)
(3)
(4)
45
Boundary Conditions
At the DRA-air interface:
0)(ˆ 12 EEr
ˆˆ 12 sJHHr
where Js is the conformal strip current.
(5)
(6)
46
On the DRA surface, we have r = r = a and G1 = G2 = G,which is given by:
)'(cos'sin
)'(cossin
)(cos)!()!(2
)('ˆ)('ˆ)1(12
4
)'(cos'
)'(cos)(cos)!()!(2
)(ˆ)(ˆ)1(12
4
2
1 10
)2(TM2
0
1 00
)2(TE2
0
mPP
mnmnm
akHkaJnn
na
j
md
dPd
dPmnmn
akHkaJnn
na
jG
mn
mn
n
n
mnn
n
mn
mn
m
n
n
mnn
n
)(ˆ)('ˆ)('ˆ)(ˆ0
)2(
00
)2(TE akHkaJkkakHkaJ nnnnn
)('ˆ)(ˆ)(ˆ)('ˆ0
)2(
00
)2(TM akHkaJkkakHkaJ nnnnn
0,20,1
mm
m and 0 is the wave impedance in vacuum.
where
(7)
(8)
(9)
(10)
47
MoM Solution for the Strip Current
Using the MoM:
N
qqq fII
1
)()(
ha
hahk
ahkf
q
qe
qe
q
,0
,sin
)(sin)(
where
in which
,1
N
Lh ,2
1
qhLaq 02)1( kk re
(14)
(15)
(16-18)
48
MoM Solution for the Strip Current (Cont’)
By Galerkin’s procedure, the following matrix equation is obtained:
)]0([][][ pqpq fIZ where
dSdSfGfW
ZS S
qppq ')'()',';,()(1
0 0
2
The input impedance is given by
N
q qq fIZ
1
in)0(
1
(19)
(20)
(21)
49
Problems in Evaluating Zpq
The Green function G is singular as Excessive no. of modal terms required Higher-order Hankel functions difficult to
handle numerically Considerable computation time required
'rr
50
Solution
),,(),()!()!(2)('ˆ)('ˆ
),,(),()!()!(2)(ˆ)(ˆ
)1(12
4
12
21
20)2(
1 01
21
0)2(
20
2
mqpmnmnmnmakHkaJ
mqpmnmnmnakHkaJ
nnn
WjaZ
n
mTMn
nn
n
n
m mTEn
nnpq
Integral Zpq evaluated using novel recurrence formulas.
First express Zpq in the following form:
where
,sin)(cos),( 2
11
d
ddPmn
mn
dPmn m
n2
1
)(cos),(2
,')'()'(cos)(),,('1
hp
hp
hq
hq
ddfmfmqp qp
h = h/a.
51
Recurrence formulas for 2(n,m)(A) Recurrence formula recursive in n
Initial values : 2(0,0) = 2 sin-1x12(1,0) = 0
2(n,0) can be found for all n.
(B) Recurrence formula recursive in m
),())(1()(])1(1[2)2,( 211
2 mnmnmnxPmn mn
mn
),1()()(1)1(1)12()1)(1(
1),1( 21212 mnmnnxPxn
mnnmn m
nmn
Initial values : 2(n,0) [from (A) above]2(n, 1) = [(-1)n - 1] Pn(x1)
2(n, m) can be found for all m and n.
52
Evaluation of 2(n,m) and 1(p,q,m)
),1()1(),1()(12
1)(11)1(),( 221211 mnnmmnmn
nxPxmn m
nmn
1(n, m) are found in terms of 2(n, m):
Evaluation of 1(n,m)
Analytical evaluation of 1(p,q,m)
]cos[sin
coscos2
')'()'(cos)(),,(
2
'1
qpeee
hee
qp
makmakmhk
mhkak
ddfmfmqp hp
hp
hq
hq
53
Radiation Fields
),(4
),,(1
00
N
qqq
rjkEI
re
WajrE
),(4
),,(1
00
N
qqq
rjk
EIr
eWarE
),( qE),( qEwhere and are found in a similar fashion.
54
Convergence Check
2.8 3 3.2 3.4 3.6 3.8 4 4.2-100
-50
0
50
100
150
200
Frequency (GHz)
Input impedance ( )
n=1 to 10
n=1 to 20
n=1 to 40
n=1 to 60
n=1 to 80
a = 12.5 mm, r = 9.5, l = 12.0 mm, and W = 1.2 mm.
55
Measured and Calculated Input Impedances
2.8 3 3.2 3.4 3.6 3.8 4 4.2
-50
0
50
100
150
200
Frequency (GHz)
Input impedance ( )
TheoryExperiment
a = 12.5 mm, r = 9.5, l = 12.0 mm, and W = 1.2 mm
•Measured resonant frequency (zero reactance):3.60 GHz
•Calculated resonant frequency (zero reactance):3.56 GHz
•Error :1.1 %
56
Results
l = 12 mm
10.9 mm
9 mm
2.8 3 3.2 3.4 3.6 3.8 4 4.2-50
0
50
100
150
200
Frequency (GHz)
Input resistance ( )
2.8 3 3.2 3.4 3.6 3.8 4 4.2-50
0
50
100
150
200
Frequency (GHz)
Input reactance ( )
l = 12 mm
10.9 mm
9 mm
a = 12.5 mm, r = 9.5, and W = 1.2 mm
57
Modified Configuration with a Parasitic Patch
l2
a
o
W1
Grounded parasitic patch
Hemispherical DRA
Conformal excitation strip
Coaxial apertureGround plane
l1
z
x
y
W2
58
Modified Configuration with a Parasitic PatchSuperscript A : excitation strip Superscript B : parasitic patch
E-field vanishes on the excitation strip
0 iAJ
BJ
BJ
A EEEE
In terms of Green functions
0 iA
S
BEJ
S S
BEJ
AEJ ESdJGSdJGSdJG
BA B
59
Current expansions of Excitation & Parasitic Patches
2 4
1 1)()(),(
N
l
N
mm
Bl
Blm
B gfII
3 5
1 1
)()(),(N
v
N
n
Bvn
Bvn
B fgII
1
1
)()(N
p
Ap
Ap
A fII
elsewhere,0
||,sin
|)|sin(~)(),( hagf p
h
ph
θnB
l
where and are of the following form: )(Blf )(ng
60
New Integrals of Pnm(cos )
hp
hp
dfPmnp pm
n
)()(cos),,(1
hp
hp
dfd
dPmnp p
mn
)(sin)(cos),,(2
Their recurrence formulas have also been found [A] but are not included here for brevity.
[A] K. W. Leung, and H. K. Ng, “Theory and experiment of circularly polarized dielectricresonator antenna with a parasitic patch,” IEEE Trans. Antennas Propagat., vol. 51, pp.405-412, Mar. 2003.
Measured and Calculated Results
61
3 3.1 3.2 3.3 3.4 3.5 3.6
Frequency (GHz)
Inpu
t Im
peda
nce Rigorous Theory
Experiment
Resistance
Reactance
-40
-20
0
20
40
60
80
100
120
140
160
Simplified Thoery
a=12.5mm, =9.5, l1=14mm, l2=7.9mm, W1=1.2mm, W2=2.2mm, and o=157.4o.r
Measured and Calculated Results (Cont’d)
62
a=12.5mm, =9.5, l1=14mm, l2=7.9mm, W1=1.2mm, W2=2.2mm, and o=157.4o.r
Measured and Calculated Results (Cont’d)
63
f = 3.52 GHzo
(a) x-z plane (b) y-z Plane
30
60
90-40 -30 -20 -10 0
030
60
90o o
oo
oo
o
-40 -30 -20 -10 0
30300
6060
9090 oo
oo
oo
o
Right Hand
Left Hand
Right Hand
Left Hand
TheoryExperiment
The method can be used to formulatemicrostrip antenna problems.
64
Analysis of Spherical Slot Antenna
Perspective View Top View65
ceH
ceP
ce GGG ,,,
1 1
2,
)2(,
,2
1 0,
)2(,
,2
,
)'(cos)'(cos)(cos)!()!(2)('ˆ)('ˆ
)1(12
sinsin1
41
)'(cos'
)'(cos)(cos)!()!(2)(ˆ)(ˆ
)1(12
41
n
n
m
mn
mncencen
ce
n
n
m
mn
mn
mcencen
ce
ceP
mPPmnmnmakHakJ
nnn
a
md
dPd
dPmnmnakHakJ
nnn
aG
)'(cos)'(cos)(cos)!()!(2)('ˆ
)1(12
sinsin1
41
)'(cos'
)'(cos)(cos)!()!(2)(ˆ
)1(12
41
1 1
22)2(2
1 0
2)2(2
mPPmnmnmakH
nnn
a
md
dPd
dPmnmnakH
nnn
aG
mn
mn
n
n
men
TEn
e
n
n
m
mn
mn
men
TMn
e
eH
)'(cos)'(cos)(cos)!()!(2
)1(12)('ˆ)(ˆ)('ˆ)(ˆ
sinsin1
41
)'(cos'
)'(cos)(cos)!()!(2
)1(12)(ˆ)('ˆ)(ˆ)('ˆ
41
2
1 1
)2()2(2
1 0
)2()2(2
mPPmnmnm
nnnakHakJcakJakHb
a
md
dPd
dPmnmn
nnnakHakJfakJakHe
aG
mn
mn
n
n
mcncnncncnn
c
mn
mn
m
n
n
mcncnncncnn
c
cH
Green’s Function of the Antenna
66
MoM Admittances
dSdSfGfW
YS S
qceHPp
ceHpqP ')'()(1
0 0
,,2
,,
dSdSfGGfW
YS S
qce
ppq ')'()(1
0 0
2
where
ha
hahk
ahkf
q
qe
qe
q
,0
,'sin
)('sin)(
,1
N
Lh ,2
1
qhLaq
02)1(' kk re
Let
where
67
cpqH
epqH
cpqP
epqPpq YYYYY
Method A
1
1
1
1
,
')'(
)]'(sin)33(
)'cos()1([4)(1 2
2,
22,
2
,22
,2
5,
,
2,
,
ddaf
RjkRka
RjkRkRRk
ej
fY q
cece
cecece
Rjkce
pce
cepqP
ce
),,(),()!()!(2)(
),,(),()!()!(2)(
)1(12
4
22
1
2
1 0
212
2
mqpmnmnmnmnB
mqpmnmnmnnA
nnn
WaY
n
m
n
n
m mpqH
where2
122 ]2/)'[(sin4 raR
)(ˆ)('ˆ)(ˆ)('ˆ1)(ˆ)( )2()2(2)2( akHbkJfakJakHeakHnA cncnncncnnc
ene
TMn
)('ˆ)(ˆ)('ˆ)(ˆ1)('ˆ)( )2()2(2)2( akHbkJcakJakHbakHnB cncnncncnnc
ene
TEn
68
Numerical integration
Recurrence formulas
69
Method B
• Combine the particular and homogeneous solutions
• Only recurrence formulas are used
• Advantage: Easy to implement and fast (no numerical integration required)
• Disadvantage: ??
70
Convergence Check for Method A
a = 6.25 cm, b = 3 cm, L = 12.46 cm, W = 2.4 mm, and r = 1.
71
Convergence Check for Method B
a = 6.25 cm, b = 3 cm, L = 12.46 cm, W = 2.4 mm, and r = 1.
72
Comparison between Methods A & BAdvantages of Method A
• Method A converges much faster than Method B• More suitable for problems of larger spherical sizesAdvantages of Method B
• Numerical integrations not required• Easier to implement• Much faster than Method A
Method A Method BNumerical
integration for both Gp, GH
32 sec. 112 sec. 77 sec.
73
Measured and calculated input impedances of the rectangular slot for a = 6.25 cm, b= 4.0 cm, L = 12.46 cm, W = 2.4 mm, and r = 1.
Measured & Calculated (Method B) Results
74
Recurrence Formulas for Integral of )(ˆ krJ n
Side View Top View
MoM Admittance
1
,
0
, )()()12)(1(2
1n
nnCD
nCD
pqH qApAnnnk
Y
1
1 sin)(sin)(ˆ
)( 2i
i
y
ye
ienn dy
hkyyhk
ykyJiA
where
1
122
2
2
2
2
2
)(ˆ)(
sin
)()12)(12(
)2)(1()(])32)(12(
)322([)()32)(12(
)3)(2(
j ji
jin
e
e
nne
n
ykyJ
jBhkk
k
iAnn
nniAnn
nnkkiA
nnnn
Recurrence formula for An(i)
Define YD & YC as the dielectric & cavity admittances, respectively.
Backward recurrence for An(i), needless to know A0, A1, A2, & A3.
a = 12.5 mm, r1 = 9.5, L = 20.0 mm, W = 1.0 mm, = 0, and N = 5
Measured and Calculated Results
Input resistance Input reactance
Solving Planar annular problem usingspherical solution
Top View Side View
MoM Solution
• First obtain the spherical Green’s function of Hdue to a -directed magnetic point current
• Substitute = = /2 (both source & field one the ground plane)
2
2
1
1212
2
2
1
12
112
21
)()()0()1(
)()()0(1)!1()!1(
)1(12
nnPq
nnPqnqn
nnn
WY
TETEq
n
TMTMq
nqn
qqq
where
odd0
even)(642)1(531)1()0(
2)(
mn
mnmn
mnP
mnm
n
MoM Solution (Cont’d)
N
q qqYZ
1in
1
in which
otherwise112
1
a = 19.25 mm, b = 10.75 mm, W = 1.5 mm, r1 = 4, and r2 = 1
Measured and Calculated Results
Q & A
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