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Ab(RineximcagrElrea InDi
wancoMthrasufeapanra
pa(MthcolesindiancaanEl
Za
Mne
Za
I
bstract— A RDRA) fed bynvestigated usxpansion in dmpedance matalculated. Alsoround plane alement Methoesults are compgood agreemen
ndex Terms—ielectric Image
N recent yeabeen developoffer several
weight, high rand no excitonventional an
Microstrip feedhin substrates adiation from urface wave exeed method thperture-couplenalyzed [6-7]adiation were o
In this paper articular DR MOMs). Onlyhe DIL. Theompared to thngth. The elenusoidal distifferent regiontenna impedarried out verntenna structulement Metho
F. Kazemi, Elahedan, Iran (e-m
M. H. Neshatashhad, Iran
[email protected], ) F. Mohanna, E
ahedan, Iran, (e-m
MDie
I
Rectangulary Dielectric I
sing Method different regitrix and retuo, the RDRA are numericalld (FEM) usinpared with thont using both m
—Rectangular e Line, Method
I. INT
ars Dielectric ped for using potential advadiation effication of surntennas such ad lines have h
are used fom the microstr
xcitation. A dhat has lower led microstrip . Good gain,observed.
r a RDRAs fedis investiga
y the dominan aperture whe DIL guidiectric field intribution. Us
ons by modadance matrix rsus frequenc
ure is investigod (FEM). Ret
lectrical Dept., mail: fatemeh.kazeti, Electrical De(corresponding
Electrical Dept., mail: f_mohanna@
Method electric
r Dielectric Image Line (of Moment
ions by modaurn loss on
structure witly investigatedg Ansoft HFS
ose obtained bymethods.
Dielectric Rd of Moments.
TRODUCTION Resonator An
g at microwavvantages such
ciency, wide brface waves as microstrip ahigh conduct
or circuit desrip line and
dielectric imagloss at higher patch antenn
, low return
d by DIL throuated using Mnt mode is assuwidth is assu
ing wavelengn the aperture ing field ex
al analysis ofis derived a
cy at the defgated using Hturn loss obtai
University of emi.ms@ gmail.cept., Ferdowsi
author, phone:
University of @hamoon.usb.ac.
of MomReson
F.
Resonator A(DIL) is num(MoM). Usinal analysis, athe definedth infinite an
d based on theSS and the simy MoM. Resul
Resonator A
ntennas (DRAve frequenciesh as small sizbandwidth, lo
compared antennas [1-4]ive losses, ansign to avoid
also to reduge line is an afrequencies [
na fed by a Dloss, and low
ugh an apertuMethod of M
umed propagamed to be
gth and the ais assumed t
xpansion metf field compand return lo
fined port. ThFSS based onined by two m
Sistan and Balucom). University of M: 05118815100
Sistan and Balu.ir).
ment (Mator An
Kazemi1, M
Antenna merically ng field antenna port is d finite e Finite
mulation lts show
Antenna,
As) has s. They ze, light ow loss to the
]. nd very d direct uce the lternate
[5]. The DIL was w back
ure for a Moments
ating in narrow
aperture to have thod in ponents, oss are hen the n Finite methods
uchestan,
Mashhad, , email:
uchestan,
arerea
A
recpeWaof trareldim
F
B
mea dwacoresattat frowh
MOM) ntenna
LineM. H. Neshati2
e compared asonably good
II.
A. Antenna SThe geometrctangular DRrmittivity of
Wa. A aperture of the metal pansmission melative permittimensions of th
Fig. 1. The geome
DRa b c
εrd
B. Analysis ofIn the analy
ethod is used.dielectric slaballs located atnducting walsonator are sttenuating in ythe walls of t
om the roots ohere 2
0yk =
Analysas Fed be
2, and F. Mo
with each od agreement is
MODAL ANA
Structure ry of the Rof length a, wεrd is placed of length L anplane to excedia, consist oivity εr is plahe structure ar
etry of the DRA fe
T Antenn
RA 6.2 mm6 mm6.1 mm
10 2
of the DRA ysis of the D
To simplify tb inside a wavt |x| = a/2 anll at z = 0. Ttanding wavesy-direction. Athe DRA we oof the followi
2 2 21 1 0x zk k k+ −
sis of Rby Diele
ohanna3
other and res obtained.
ALYSIS FORMU
RDRA is showidth b, heighon the groun
nd width W is cite the resoof a rectangulaced under thre summarized
fed by DIL
TABLE I na Dimensions
DIL ad 4.2bd 4.0
m εr 10
DRA side, ththe analysis, thveguide with mnd z = −c/2, aThe compones, while outsid
Applying the bobtain kx1 = π/ing equation t and 2
1yk
Rectangectric I
esults show
ULATION
own in Fig. ht c with the rnd plane with etched at the
onator. DIL lar dielectric e ground pland in TABLE I
25 mm 03 mm
0.2
e modal exphe DRA is tremagnetic condand with an eent fields inside the DR, thboundary conπ/a, kz1 = π/c, atan(ky1b/2) =
2 21 1 0 1r zk kε= −
gular Image
that a
1. A relative h width center as the slab of ne. All I.
pansion ated as ducting electric ide the hey are nditions and ky1 ky0/ky1,
21 1xk− .
Page 48 /183
Considering the TEy111 mode, the far field radiation is similar to a magnetic dipole of moment Pm given by:
0 11
1 1 1
8 ( 1)sin( / 2)r
m yx y z
j AP k b
k k kωε ε− −
= (1)
The power radiated by a magnetic dipole is given by [8]: 4 2
r 010 | |mP k P= (2) From which the radiation Q factor (Qr) can be determined by Qr =ω0W/Pr, where W is the stored energy in the DRA. In addition to the radiated power, the dielectric dissipated power, Pd, and the conducting power loss, Pc, are computed using the following equations
2/2 /2 /2
0 1 0/2 /2 0
1 tan | | tan2
a b c
da b
P E dzdydx Wω ε δ ω δ− −
= =∫ ∫ ∫ur
(3)
/2 /2 /22 2 2
0 0/2 /2 /20 0/2 /2
2
/2 /2
| | | | | |12 2
| |
a b b
da b b
c w lcd
w l
H dy H dy H dy dx
P
H dydx
ω μσ
− ∞
− −∞ −
− −
⎡ ⎤⎡ ⎤⎢ ⎥+ + −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦= ⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
∫ ∫ ∫ ∫
∫ ∫
(4) where tan δ is the loss tangent of the dielectric used for the DRA and σc is the conductivity of the ground plane. H0 is the magnetic field outside the DRA, and Hd is the field inside the DRA. The total Q factor is determined by:
r0
1 1 1 1 1 (P )d cr d c
P PQ Q Q Q Wω
= + + = + + (5)
radiation efficiency η is : / rQ Qη = (6)
The field in the DRA is normalized such that:
( )2 2 2
20 111
02 2
| |
a b c
ya b
H dzdydxμ
− −
∫ ∫ ∫ (7)
C. Analysis of the DIL
The DIL supports the zmnTM and y
mnTM modes. However, the presence of the ground plane resolves the degeneracy because the strongest electric field component of the 11
yTM mode is shorted out, so only the 11
zTM mode is only propagates over the desirable frequency range. It is not possible to have a closed form solution for the DIL problem since it is an open structure that does not fit the rectangular coordinate system in a straight forward approach. However, it is possible to simplify the problem by considering infinitely long dielectric slabs having effective dielectric constant that can be obtained using the effective dielectric constant (EDC) method [9].
(a)
(b)
(c)
Fig. 2. Simplified geometry of the DIL Considering an infinite dielectric slab above a ground plane with height bd as shown in Fig. 2-a, the continuity of Ey and Hx at z = bd is enforced and the following equation is obtained:
0 2tan( ) ( / )z d z z rk b k k ε= (8)
Where [ ]2 2 20 0 2 1z r zk k kε= − − and 2 2 2 2 2
2 0x r z yk k k kβ ε= = − − . The other problem is a dielectric slab that extends infinitely in the z-direction in the upper half-space as shown in Fig. 2-b. When the continuity of Ez and Hx are applied at dy a= ± , one can obtain:
( )2 2
0 02 20
1tan y y x yy d
y rez yy x
k k k kk a
k kk k ε
+= =
− + (9)
Where 22 0( / )rez r zk kε ε= − , 2 2 2
0 0 1y rez yk k kε= − −⎡ ⎤⎣ ⎦ and 2 2 2
0y rezk kε β= − . kz, ky, kz0, and ky0 are the transverse propagation constants inside and outside the dielectric slab, respectively. The wave impedance of the image line is given by:
2 2 2 20
0 0
y x x yw
x rez x
k k k kEzZHy k kωε ωε ε
− + += − = = (10)
The characteristic impedance, dielectric attenuation constant, and conductor attenuation constant of the image line can be found in [10].
D. Analysis of the Aperture The aperture determines the amount of power coupled from
the DIL to the DRA. The slot aperture is centered at the origin
Page 49 /183
with length L and width W. The electric field on the aperture is assumed to have a sinusoidal distribution in y-direction and independent of x as described by equation (11):
[ ][ ]
sin ( / 2 | |sin / 2
Axap
E K l yE
Kl−
= (11)
where 0 1 2(2 / ) ( r e rK π λ ε ε= + The equivalent magnetic current on the aperture is given by:
$ $ [ ][ ]
$sin ( / 2 | |sin / 2
Axap y
E K l yM E x z y M y
Kl−
= × = =uur
$ (12)
The magnetic current excites only TEy modes of the DRA. The fields can be expressed in terms of these resonant modes. However, we only considered the TEy111 mode, hence the single mode approximation can be taken. The magnetic field (using modal expansion) is given by:
*2 2 .
( )i
ii i v
j HH M H dv
ω
ω ω=
−∑ ∫∫∫
uuuruur uur (13)
And for TEy111 mode we have: /2 /2
1111112 2
111 /2 /2( )
w ly
y y yw l
j HH M H dydx
ω
ω ω − −
=−
∫ ∫ (14)
where 2 2 2 2111 1 1 1 0 1
0
1 ( ), 1 ,x y z rjk k k
Qω ε ε ε
μ ε⎛ ⎞
= + + = −⎜ ⎟⎝ ⎠
1xkaπ
=
and 1zkcπ
= . Ky1 can be obtained from the roots of the
characteristic equation 1 0
1tan
2y y
y
k b kk
⎛ ⎞=⎜ ⎟⎜ ⎟
⎝ ⎠ where
2 2 20 1 0 1( 1)y r yk k kε= − − . The other field components can be derived from Hy using
the electric and magnetic vector potentials [11]. The power at the aperture is defined as
$ $( )/2 /2 2
2 2111/2 /2
.( )
w lxap y
w l
jP E x H y ds ωω ω− −
ℜ= × =
−∫ ∫
uur (15)
where
( )
1/2 /2 2 2
1 1111 2 2
/2 /2 1 10
1 1
2 sin8 ( ) 2
sin( )2
. sin sin1/ 4 1/ 4
xw l
x zy y
w l x y
y y
wkKj A k k
M H dydxKl k K k
K k K k
ωμ− −
⎛ ⎞⎜ ⎟− + ⎝ ⎠ℜ = =
−
⎡ ⎤+ −⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦
∫ ∫
(16) For the fundamental TMz11 mode of the DIL, the electric field in z-direction and the magnetic field in y-direction inside the DIL are given by:
0 cos( )cos( )dz y zE E k y k z= (17)
02 2 | |d d drez z
y z zx z
kH E N E
k kωε ε
= − = −+
(18)
The discontinuity in the voltage across the aperture is given by:
$( ).x d x dap ap y
slot slotV n E h ds E h dxdyΔ = × =∫∫ ∫∫
uur (19)
where dhuur
and dyh are the normalized magnetic fields in the
DIL. The power Px in the DIL is normalized such that:
$*
0. 1xP E H xdydz
∞ ∞
−∞
⎛ ⎞= × =⎜ ⎟⎝ ⎠∫ ∫
uuurur (20)
This determines the constant E0 in (11). The voltage variation across the aperture is expressed by:
( )( )
02 2
2 sin / 28. sin sin
1/ 4 1/ 4sin( )2
y yx
x y
K k K kK k wNEV
Kl k K k
⎡ ⎤+ −⎛ ⎞ ⎛ ⎞−Δ = ⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎢ ⎥− ⎝ ⎠ ⎝ ⎠⎣ ⎦
(21) The normalized impedance presented at the aperture is given as:
2( ) /apz V P= Δ (22) The normalized input impedance depends on the aperture size and DRA size. The reflection coefficient is also given by [12]:
1 ( , ) ( , )2
x dap y
slotR E x y h x y ds= ∫∫ (23)
III. RESULT AND DISCUSSION The DRA structure shown in Fig. 1 is analyzed with a = 6.2
mm, b = 6.0 mm, c= 6.1 mm, and 1 10.2rε = by method of moments using MATLAB software [13-14]. In this analysis field extension of DRA and DIL is calculated using modal expansion and potential vectors. Then, aperture is deleted and DRA is replaced with magnetic dipole and equivalent magnetic current is calculated. This current has sinusoidal distribution in equation (11) that use as entire-domain basis function in moment method [15]. Then impedance matrix and reflection coefficient is carried out with solving moment equations. The resonant frequency, radiation Q factor, and radiation efficiency for the DRA are verified with results available in literature. The calculated resonant frequency is 10.1. The radiation Q factor Qr is 7.02 using theory. The results obtained for the free space to guide wavelength ratio of the DIG as a function of the normalized height
0
4 1rbB ε
λ= − , are identical to the results obtained in [16].
In order to further verify the results obtained by the present method, the reflection coefficient is computed using HFSS [21]. Results are shown in Fig. 3. It can be seen that the results obtained using HFSS are oscillating as compared with matching transition used in the numerical model as compared to the infinite size of the analytical results. One may relate that to the finite size of the structure and the matching transition used in the numerical model as compared to the infinite size of the analytical model.
Page 50 /183
Fig
Flen
Fig
apinin
batrath
veinFoplnudiplscth
g. 3. Reflection c
ig. 4. Real part ngth.
g. 5. Normalized Figure 4 sho
perture lengthncreased by innput admittanc
In order to eand rectangulansition in sim
he RDRA is shFig. 7-a sho
ersus frequencnfinite ground or practical aplanes. Finallyumerically inimension is 1lane reasonablcattering paramhe antenna rad
coefficient at the a
of the normaliz
input admittance
ows the real ph. It can bencreasing aperce versus frequexcite the DILlar waveguidmulation [17]hown in Fig. 6ws radiation cy is also shoplane, maxim
pplications, DR, antenna stru
nvestigated in134.2×100 mmle radiation chmeters is showdiation patter
aperture for the R
zed impedance a
e as a function of
part of antenne seen that rture length. uency is also sL, the DIL isde by means. The transmi
6. pattern at 10
own in Fig. 7-mum gain is 6.
RAs are mounucture with fin HFSS softwm2. With thisharacteristics wn in Fig. 8. Arn at 10 GHz
RDRA geometry.
as a function of
f frequency
na impedanceinput impedaThe variationshown in Fig.
s connected to of a three-ission coeffic
GHz. Antenn-b. For antenn4 dB at 10 GHnted on finite nite ground pware. Grounds size of the is obtained. AAlso, Fig. 9-az. Results sho
f aperture
e versus ance is n of the 5.
o an X--section ient for
na gain na with Hz. ground
plane is d plane ground
Antenna a shows ow that
DRpebuduMoDIvsMathe
Fig
FigRD
(RinvregretstrbaHF
RA structurrpendicular to
ut finite groundue to diffractoreover backwIL radiates in. frequency foaximum gain e structure wit
g. 6. Transmission
g. 7. a) RadiationDRA with infinite
In this paperRDRA) fed byvestigated usigions by moturn loss are ructure with ased on the FFSS commerc
re provides o the ground d plane have ation from thward radiation
n both directioor the RDRA s
is 5.6 dB at th infinite gro
n coefficient of th
n pattern at 10 Ge ground plane
IV. CO
r a Rectanguly dielectric iing MoM. Usdal analysis, calculated foinfinite and
Finite Elemencial software
broadside plane and hasan effect on th
he edges of n is high becaons. Fig. 9-b structure with10 GHz whicund plane [18
he RDRA with in
(a)
(b) GHz, b) DRA ga
ONCLUSION lar Dielectric image line (Dsing field expantenna imp
or the definedfinite ground
nt Method (Fe and results
radiation ps a good retuhe radiation pthe ground
ause the apertushows antenn
h finite groundch is 1 dB les8-20].
nfinite ground pla
ain versus frequ
Resonator ADIL) is numepansion in dipedance matrd port. Then Rd plane are sFEM) using
is compared
pattern urn loss patterns
plane. ure and na gain d plane. ss than
ane
ency for
Antenna erically ifferent ix and RDRA studied Ansoft d with
Page 51 /183
theory results by Method of Moments. Results have good agreement using both methods.
(a)
(b)
Fig. 8. Scattering parameters for RDRA with finite ground plane. a) S11, b) S21
(a)
(b)
Fig. 9.a) Radiation pattern at 10 GHz, b) Gain vs. frequency for RDRA with finite ground plane
REFERENCES [1] A. Petosa, “Dielectric Resonator Antennas Handbook”, Artech House
Inc, 2007. [2] A. A. Kishk, “Dielectric resonator antenna, a candidate for radar
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[3] J. Shin, A. Kishk, and A. W. Glisson, “Analysis of rectangular dielectric resonator antennas excited through a slot over a finite ground plane,” IEEE AP-S International Symposium, Vol. 4, 2076–2079, July 2000.
[4] Y. Coulibaly, and T. A. Denidni, “Design of a broadband hybrid dielectric resonator antenna for X-band applications,” J. of Electromagnetism. Waves and Appl., Vol. 20, No. 12, pp. 1629–1642, 2006.
[5] D. Yau, and N. V. Shuley, “Numerical analysis of coupling between Dielectric image guide and microstrip,” J. of Electromagn. Waves and Appl., Vol. 20, No. 15, 2215–2230, 2006.
[6] S. Kanamaluru, M. Y. Li, and K. Chang, “Aperture coupled microstrip antenna with image line feed,” IEEE AP-S Symp. Dig., Vol. 2, 1186–1189, 1994.
[7] S. Kanamaluru, M. Li, and K. Chang, “Analysis and design of aperture coupled microstrip patch antennas and arrays fed by dielectric image line,” IEEE Transactions on Antennas and Propagation, Vol. 44, No. 7, 964–974, July 1996.
[8] R. K. Mongia, and A. Ittipiboon, “Theoretical and experimental investigations on rectangular dielectric resonator antennas,” IEEE Transactions on Antennas and Propagation, Vol. 45, No. 9, 1348–1356, Sept. 1997.
[9] R. M. Knox and P. P. Toulios, “Integrated circuits for the millimeter through optical frequency range,” Proc. Symp. Submillimeter Waves, 497–516, 1970.
[10] P. Bhartia and I. J. Bahl, Millimeter-Wave Engineering and Applications, Wiley, New York.
[11] C. A. Balanis, Advanced Engineering Electromagnetics, Wiley, New York, 1989.
[12] D. M. Pozar, “A reciprocity method of analysis for printed slot and slot-coupled microstrip antennas,” IEEE Transactions on Antennas and Propagation, Vol. 34, No. 12, 1439–1446, 1986.
[13] S. N. Makarov, “Antenna and EM Modeling with MATLAB.”, Wiley, New York, 2002.
[14] "MATLAB: The Language of Technical Computing, "V. 7.8.0.347 ed: MathWork Corporation.
[15] C. A. Balanis, Antenna Theory, Analysis and Design., third ed., Wiley, New York, 2005.
[16] R. M. Knox and P. P. Toulios , "Integrated circuits for the millimeter through optical frequency range," Proc. Symp. Submillimeter Waves, pp. 497-516,1970.
[17] A. S. Al-Zoubi, A. A. Kishk, and A. W. Glisson, “Analysis and Design of A Rectangular Dielectric Resonator Antenna Fed by Dielectric Image Line Through Narrow Slots, ” Progress In Electromagnetics Research , PIER 77, pp.379-390, 2007.
[18] H. Dashti, M. H. Neshati, F. Mohanna, “Numerical investigation of rectangular dielectric resonator antennas (DRAs) fed by dielectric image line”, Progress In Electromagnetic Research, PIER 2009, Moscow.
[19] H. Dashti, M. H. Neshati, F. Mohanna, F. Kazemi “ Numerical Investigation of Rectangular Dielectric Resonator Antenna Array Fed by Dielectric electric Image Line”, IEEE CEM'09 Computational Electromagnetics International Workshop, Izmir, Turkey,2009.
[20] F. Kazemi, M. H. Neshati, F. Mohanna, “Design of Rectangular Dielectric Resonator Antenna Fed by Dielectric Image Line With a Finite Ground Plane”, International Journal of Communications, Network and System Sciences, Vol. 3, No. 7, pp. 620-624, July 2010.
[21] " HFSS: High Frequency Structure Simulator Based on Finite Element Method," V.11 ed: Ansoft Corporation.
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