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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

eshat@ieee.org, ) 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

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xpansion metf field compand return lo

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ment (Mator An

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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

applications,” Proceedings of 2003 IEEE Radar Conference, 258–264, May 2003.

[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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