Immittance Matrix for Stratified Media Using Spectral Domain Dyadic Green’s Function (DGF)

Ghulam Kassem N.W.F.P University of Engineering and Technology

Peshawar PAKISTAN kasim_uet@yahoo.com

Siddique Akbar

Asif Zakriyya Institute of Computing and I.T (ICIT) Gomal University

Dera.Ismail Khan PAKISTAN asif_zakriyya@hotmail.com

Institute of Computing and I.T (ICIT) Gomal University Dera.Ismail Khan PAKISTAN ghtah200@yahoo.co.uk

Abstract— Dyadic Green’s function (DGF) find a good place in communication engineering in modelling microstrip lines and patch antennas at radio frequencies. The DGF relates electromagnetic fields with the source current in free space. Using simple geometry to define the dyadic Green’s function (DGF) in free space and then applied to a stratified media to find the components of Immittance matrix which is used in determining the resonant frequency, input impedance, radiation characteristics etc. of microstrip lines and phased antenna arrays at radio frequencies. This method can be easily applied to multi-layered printed structures in wireless networks as well. The derivation which introduces a method to model the printed structures without extra mathematical labour.

Keywords-Immittance matrix, stratified media, Antennas, Microstrip antennas, Green’s function, Dyadic green’s function, Resonant frequency, input impedance

I. INTRODUCTION By definition, the dyadic Green function which relates the electric field � �rE due to source current � �rJ in free spacei.e.for microstrip patch

� � � � � �''0 ., rJrrGjrE ���� (1)

���

���

�

���

���

�

����

���

�

z

y

x

zzzyzx

yzyyyx

xzxyxx

z

y

x

JJJ

GGGGGGGGG

EEE

(2)

where � � zyx EzEyExrE^^^

���

and � � zyx JzJyJxrJ^^^

���

ijG represent the electric field iE produced by the unitary

source current jJ . i.e. xxG means the field xE due to xJ via

� � � � � � dvrJrrGjrE ''0 .,����� �� (3)

If surface current is considered i.e. in case of microstrip patch antennas (for negligible thickness); third dimension is suppressed

� � � � � �dsrJrrGjrE ���� ''0 .,�� (4)

�

��

��

���

���

��

y

x

yyyx

xyxx

y

x

JJ

GGGG

jEE

0�� (5)

The dyadic Green’s function (DGF) for different shapes is of the same form..

II. DERIVATION The configuration and its equivalent circuits is shown in Fig.1(a) and (b) respectively [1-3]. The 2-D Fourier transform pair is defined as

� � � �� �

dydxezyxzyxj ��

�������

��

��

��� �� ,,,,~ (6)

� �� �

� � � � ������

� �� ddezzyx yxj� ���

��

��

��

��� ,,~2

1,, 2 (7)

��

���

��

���

��

��

���

y

x

yyyx

xyxx

y

x

J

J

GG

GGj

E

E~

~

~~

~~

~

~

0�� (8)

where the tilde over the letter denotes the Fourier transform of the letter. � �zyx ,,� is superposition of inhomogeneous

(in ^z ) plane waves in various �� ,

2011 UKSim 13th International Conference on Modelling and Simulation

978-0-7695-4376-5/11 $26.00 © 2011 IEEE

DOI 10.1109/UKSIM.2011.59

276

tt kvyxk^^^

��� �� (9)

Transform ���

��� ^^

, yx into ���

��� ^^

,vu by coordinate

transformation via

���

���

�

���

���

�

��

���

���

�

^

^

^

^

sincos

cossin

y

x

v

u

��

�� (10)

where � � 2/122 �� ��tk is the transverse wave number

with )/(cos 1tk�� �� .

Each plane wave is decomposed into TM (to^z )

i.e. � �uvz HEE ~,~,~ and TE (to

^z ) � �vuz HEH ~,~,~

and

generated by current components vJ~

and uJ~

respectively as shown in the equivalent circuits Fig. 2. The wave admittances in each region is calculated:

2,1,~~

0 ���

� ij

EH

Yi

i

v

uci �

��� (TM-wave) (11)

2,1,~~

0

���

� ijE

HY i

u

vci ��

� (TE-wave) (12)

where 20

222 krii ���� ��� , 000 ����k ,

,0�

�� ri � 11�r� , rr �� �2 , 0�� �i . 0k is free

space wave number, � the radian frequency 0� , 0� are the permittivity and permeability of the free space respectively. i� the propagation constants in region 1 and 2 respectively.

0222

020

221 zjkkjk ������� ����� (13)

2222

020

222 zrr jkkjk ������� �������

(14) Now at 0�z , the impedance is found by the formula

� ��� �

�hehe

he YYZ

,,,

1,~ �� (15)

1, che YY �� (16)

)coth( 11, hYY che ��� (17) �Y and �Y are the input admittances at 0�z , looking up

and down into the equivalent circuits. Tangential electric field E is continuous and equal at the interface ( 0�z ); and discontinuity of the magnetic field H on both sides of

the strip is represented by the equivalent currents uJ~

and

vJ~

. The voltages uE~

and vE~

are related to the current

sources uJ~

and vJ~

via

),(~),(~)0,,(~ ������ uohu JZE � (18)

277

),(~),(~)0,,(~ ������ voev JZE � (19)

��

���

��

���

�

��

���

v

u

oe

oh

v

u

J

J

Z

Z

E

E~

~

~0

0~

~

~ (20)

Now using Equation (10), the coordinate transformation

��

���

�

��

�

�

��

���

�

��

�

���

���

y

x

oe

oh

y

x

J

J

Z

ZE

E

~

~

sincoscossin

~0

0~

sincoscossin

~

~

����

����

(21)

��

���

��

�

�

���

�

��

��

���

y

x

ohoe

ohoe

ohoe

ohoe

y

x

J

J

ZZ

ZZ

ZZ

ZZE

E

~

~

cos~sin~cossin)~~(

cossin)~~(

sin~cos~~

~

22

22

��

��

��

��

(22)

Comparing with Eq. (8)

���� 220 sin~cos~~

ohoexx ZZGj ��� (23)

���� cossin)~~(~0 ohoexy ZZGj ��� (24)

���� 220 cos~sin~~

ohoeyy ZZGj ��� (25) where

)/(cos 2222 ���� �� , )/(sin 2222 ���� ��

and )/(cossin 22 ������ ��

xyG~

and yxG~

are equal due to symmetric property of Green’s functions

)tanh(()tanh(~

2210

221

hjhZ

roe ������

����

� (26)

)coth(~

221

0

hj

Zoh �����

�� (27)

Using Eq. (26-27) and substituting in Eq. (23-25) the relationships are obtained as:

� )tanh()()(

1~

2220

21

20

20

0

hkkTTj

Gj

r

HExx

������

����

���

��� (28)

� )tanh(

1~~

221

000

hTTj

GjGjHE

yxxy

�����

����

�

����� (29)

� )tanh()()(

1~

2220

21

20

20

0

hkkTTj

Gj

r

HEyy

������

����

���

��� (30)

Using Eq. (13-14) in conjunction with Eq. (28-30) via the trignometric identities

)tan()tanh( 22 hkjhjk zz �

and )cot()coth( 22 hkjhjk zz �� yield the components of the Green’s function in spectral domain:

� )tanh()()(

1~

2222

0022

0

0

hkkkkkjTTj

G

zzzr

HHEExx

���

��

����

�� (31)

� )tanh((

1~~

220

0

hkkjkTTj

GG

zzz

HHEEyxxy

��

����� (32)

� )tanh()()(

1~

2222

0022

0

0

hkkkkkjTTj

G

zzzr

HHEEyy

���

��

����

�� (33)

Poles corresponding to TM and TE surface wave are extracted by Muller’s Method by 0�EET and 0�HHT . By inverse Fourier transform of Equations (31-33), the Green’s function in space domain are obtained.

��

��� dde

GG

GGGGGG

G

yxj

yyyx

xyxx

yyyx

xyxx

)(

2 ~~

~~

)2(1

��

��

��

��

��� � �

�

���

�

��

���

�

(34)

278

III. APPLICATION TO STRATIFIED MEDIA The structure and its equivalent circuits are shown in Fig.2(a) and (b) respectively. Proceeding in way as discussed above, the admittance for TE-waves at sz � :

�

��

��

��

)tanh()tanh(

343

3343 dYY

dYYYY

TETE

TETETETE �

�

�

��

��

��

)tanh()tanh(

343

334

0

3

dd

jYTE ���

������

At 0�z :

�

��

��

��

)tanh()tanh(

2

22 sYY

sYYYY sTE

sTE

sTE

sTE

TETE ��

��TEY �

��

��

)tanh()tanh(

21312

21213

0

2

sABsBA

j ������

���

(35)

The admittance at hz �� :

)coth( 11 hYY TETE ���

)coth( 10

hj

YTE ����

�� (36)

Using Equations (35) and (36)

�� ��

TETEoh YYZ 1),(~ �� (37)

)coth()tanh()tanh(

1),(~

10

1

23121

22131

0

2 hjsAB

sBAj

Zoh�

���

������

���

����

��

��

�

� � )tanh()coth()tanh(

)tanh(),(~

23121112221321

231210

sABhsBAsABj

Zoh ��������������

�����

��

(38)

Similarly for TM-waves, at sz � :

�

��

��

�)tanh()tanh(

343

3343 dYY

dYYYY

TMTM

TMTMTMTM �

�

�

��

��

��

)tanh()tanh(

34334

33443

3

30

ddj

Yrr

rrrTM �����

������

���

At z=0:

�

��

��

��

)tanh()tanh(

2

22 sYY

sYYYY sTM

sTM

sTM

sTM

TMTM ��

�

��

��

��

)tanh()tanh(

2223232

2223232

2

20

sBAsBAj

Yrr

rrrTM �����

������

��� (39)

The admittance at hz �� :

)coth( 11 hYY TMTM ���

)coth( 11

10 hj

Y rTM �

����

�� (40)

�� ��

TMTMoe YYZ 1),(~ ��

� )coth(),(~

11232130

213

hABjA

Zrr

oe ���������

���

� (41)

�������� 22 sin),(~cos),(~),(~ohoexx ZZG �� (42)

� �������� cossin),(~),(~),(~ohoexy ZZG �� (43)

��

�

��

���

��

��

��

��

��

��!��

221

2120

211220

211220

221

2120

210

0

sincos

sincos)(

sincos)(

cossin),(~

DNDNkDNDNk

DNDNkDNDNk

DDkj

G

(44)

279

Where

2131 ��AN �

)tanh( 231212 sABN ��� ��

)tanh( 221 hT rE ���� ��

)coth( 221 hTH ��� ��

)tanh( 220 hkkjkT zzrzEE �� �)coth( 220 hkkjkT zzzHH �� ,

)tanh( 3341 dA ��� ��

)tanh( 334432 dA rr ����� ��)tanh( 23222323 sABA rr ����� ��

)tanh( 3431 dB ��� ��

)tanh( 343342 dB rr ����� ��

)tanh( 22323223 sBAB rr ����� ��

)coth( 12232131 hABD rr ����� ��

� )tanh()coth()tanh(

23121

1122213212

sABhsBAD

���������

����

The Green’s function derived is applied to the microstrip patch antennas and other printed structures very frequently [3-5] and produced excellent results being in natural domain.

IV. CONCLUSION

The derived Immittance matrix in the spectral domainhasbeen successfully used to the array of rectangular

patchmicrostrip patch antenna for the calculation of resonant frequency, input impedance and the radiation characteristics of any planar configuration with co-planar feeds [4,5]. The results agreed very well with the available data in literature [1,2] for both simple and stacked i.e. stratified printed structures in 1-D and 2-D as well.

REFERENCES: [1]. Tatsuo Itoh, “Spectral Domain Immittance

Approachfor Dispersion Characteristics of Generalised Printed Transmission Lines”, IEEE Trans. on Microwave Theory and Techniques, Vol. MTT-28, No.7, July 1980.

[2] Tomoki Uwano and T. Itoh, “Spectral Domain Approach”, in T. Itoh (Edited) Numerical Techniques for Microwave and Millimeter-waves Passive Structures, John Wiley and Sons (1989), Chapter 5, pp.334-380.

[3] G. Qasim, "A Simplified Full-wave Analysis for Microstrip Patch Antennas ", J. of Shanghai Univ. Vol. 4, No. 1, pp. 27-30, March 2000, China (P.R).

[4]. Gang Liu, S.S.Zhong and G.Qasim, " Closed FormExpressions for Rectangular Patch Antennas with Multiple Dielectric Layers", IEEE Trans. on Antennas and Propagation, Vol. 42, no. 9, pp. 1360 1364, U.S.A.

[5]. G. Qasim, S.S.Zhong, “Resonant Frequency of a Rectangular Microstrip Antennas Covered with Dielectric Layers”, J. of Shanghai Univ. of Science and Tech., Vol. 14, No. 4, pp. 77-84, Dec.1991, China (P.R).

[6]. G.Qasim, “Ph.D Thesis, Shanghai University of Science and Technology Shanghai, China (P.R), Chapter-II, pp. 32-46, July (1989-92).

.

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