An Efficient Analysis of the Long-wire antenna With Matched Loading Terminations

#Yanling.Guo, Jian.Wang ,Xinliang Wang College of Electronic Engineering

University of Electronic Science and Technology of ChinaUESTCNo.4, Section 2, North Jianshe Road, Chengdu, P.R.China(610054)

Email: gylingsky@yahoo.com.cn

Abstract In this paper, the method mixed potential integral equation (MPIE) combined with half rooftop basis function is used to analysis the long-wire antenna with matched loading terminations. MPIE is a kind of integral equations in Method of Moment (MOM) and can effectively deal with models with terminal loads. The contribution of this method is that we can take proper measures to the boundaries of the model by using the half basis function, which allows the current to flow across the edges. All of the elements in MoM impedance matrix, only the elements in the first and second rows need calculating for the structure of straight line according to the principle of reciprocity and symmetry. The results of current, voltage along the line and the radiation field calculated by this method agree well with the results of theoretical value and transmission-line theory(TL).

Keywords-long-wire antenna, Matched losding, MPIE MOM

I. INTRODUCTION

Although, many commercial software have been development very mature, it is difficult to deal with some electrical large size and complex structure models taking into account computer's memory and efficiency. Method of Moment (MOM) is one of the important numerical methods, and has been widely used in the electromagnetic field for its many advantages in analysis process. Such as high efficiency, flexible, fast, accurate, not limit geometric objects Shape, and solid theoretical foundation.

Usually, the problem of terminating with arbitrarily load at some ports will be encountered in the analysis of the multi ports circuits,. so analyzing this problem is the foundation of analyzing other more complex circuits.

So far, many scholars have put forth different methods. One is electromotive force (EMF) combined MOM in [1][2]. Another method is proposed in [3], which introduce special MOM half basis function to analysis micro-strip filters avoiding stub elements; Ulrich Jakobus [5] has made a similar analysis for micro-strip circuits with loaded monopole elements, but their analyzing method is complicated.

In this paper, an efficient analysis way will be presented to solve the long-wire antenna with a matched loading terminations using MPIE/MOM. The half basis and testing functions are selected as rooftop and pulse functions respectively. They are used at the excitation and loading ports. The numerical method proposed in this paper is very suitable for the analysis of loaded antennas. Moreover, it is also useful to analyze some other complex structures such as bend, cross, power divider , filter etc can easily be dealt with.

This paper is organized as below. Firstly, the detailed geometry of the antenna is described; In Section 3, the operational principle of selecting the basis and testing functions are briefly introduced; Methods of calculating impedance elements, current , voltage distributions and some conclusions are described in the last section. The effectiveness of this method is validated through the calculations of impedance and antenna pattern etc.

II. THE GEMOMETRY AND MPIE FOR THE LONGWIRE ANTENNA STRUCTURE

The geometry of the long-wire antenna in free space is shown in Figure 1.A traveling-wave wire antenna having a length 04L = and radius 00.001a = is placed

00.8H = above an infinite ground plane, where 0 is the free-space wavelength at frequency 0f , Vg is the excitation voltage of the antenna, Zg and LZ are the driving port impedance and terminal load respectively. According to the image theory, the geometry shown in figure1 is placed with an equivalent model composed of two driving long wire elements, two driving port impedances Zg and two matched terminal loads LZ . An integral equation for the currents and the charges can be set up according to the boundary conditions of the electric field on the surface S of the upper conductor

( ) ( ) ( )i st t s sZ + =E E J (1) where, is position vector of the observation point on S, sZis the impedance of the long-wire. itE is the excitation field and

stE is the scattered field which can be written as

Vjs = AE (2) where

)()/()( = ss

Asd JGA (3)

=s

sq qGsdV )()/()( (4)

1-4244-2424-5/08/$20.00 2008 IEEE ICCS 2008 388

Figure.1 the geometry of long-wire antenna

Figure.2 equivalent circuit and basis function on the line

The surface current and charge density sJ and sq are related through the continuity equation

0=+ ss qjJ (5) AG is the dyadic Greens function for the vector potential

and qG is the scalar potential, according to (1)-( 4), the MPIE can be written as

)()()/(

)()/(j

it

ss

qt

ss

A

sdqG

sd

E

JG

=

+

on S (6)

where, y

yx

xt

+

= . Substitution equation (5) into (6) we can obtained

)()(J)/(j1

)()/(j

it

ss

qt

ss

A

sdG

sd

E

JG

=

on S (7)

III. SELECTING THE BASIS FUNCTIONS AND TESTING FUNCTING

If the radius of the long wire a is much smaller than the wavelength in the free space, the current density of the y-component can be neglected. There is only s sxJ xJ=presence on the antenna.

0( ) ( )

N

sx n xnn

J x I T x=

= (8) According to equation (5)

0

( )1( )N

xns n

n

T xq x Ij x

=

=

(9)

where nI is the unknown expansion coefficients, the basis functions xnT are of rooftop type defined as

1 1

1 1

( ) / ,( ) ( ) / , , | |

0 ,

n n n

xn n n n

x x x x xT x x x x x x y a

elsewhere

+ +

=

(10)

The equivalent circuit and the basis functions on the line are shown in Fig.2. nxn = ( Nn ,...,2,1,0= ) is the partition node, NL /= is the partition size, L is the length of the long wire antenna. N is the number of nodes. Half basis functions should be considered for the half-subsections at the ends of the line ( Nn ,0= ). The first statement of (10) is used when Nn = , and the second statement is used when 0n = .

The testing functions are chosen to be NmyxWyx xmm ,...,2,1,0),,(),( == xW

(11)

where

0,,0,1

),( =

=

+

yelsewhere

xxxyxW mmxm

(12) in which 2/= mm xx and 2/+=

+mm xx .

IV. NUMERICAL EVALUTION

A. Calculating the impedance elements mnZAs described before, the half-testing functions should be

used for the half-subsections at the both ends of line also when 0=m and Nm = . With a suitable definition of inner

product, we can obtain

=

=

N

nmnmn VIZ

0

, Nm ,...,2,1,0= (13)

Where, qmnAmnmn ZZZ += (14)

AmnZ and

qmnZ can be given out( with 0== nm yy )

( )( )( )

00

00

j | |( , )(1 )4j4

A Amn x xx m n

A Ax mn mn

z xZ k h G x x x y dx

z k h

+

=

= +

(15)where xh n= .

0( , )(1 )A Amn xx m n

xG x x x y dx

= B (16)

( ) ( )0 m0j4q q q q qmn n m n m n m n

x

zZk h + + + +

= +

(17)

where / 2

/ 2( , )q q m nm n G x x x y dx

= (18) After introducing the half-basis functions and half-testing

functions at the ends of the long-wire antenna, the elements

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mnZ at the first and last rows and columns in matrix mnZ will relate to them. Only the elements in the first row need calculating because of the principle of reciprocity and symmetry. Except these impedance elements, the others satisfy the law of toeplitz.

In the first row m=0 and / 2m mx x+

= . The first element in this row (n=0) is

( ) ( )0 0

00 0 00 0 00

j4 j4

A qx g

x

z zZ k h Zk h + +

+= + + (19)

The last element in first row ( Nn = ) is

( ) ( )0 0

0 0 0 00

j4 j4

A qN x N N

x

z zZ k hk h +

= (20)

The other elements in first row ( 11 Nn ) are

( )

( )

00 0 0 0

00 0

0

j ( )4

( )j4

A An x n n

q qn n

x

zZ k h

zk h

+ + +

+= + +

(21)

The Impedance elements in the last row (m=N) can be expressed by the ones in the first row according to the principle of symmetry.

+=

==

LgNN

nNNn

ZZZZNnZZ

00

,0 1,...,2,1,0, (22)

The Impedance elements in the first and last columns can be obtained by the elements in the first row according to the principle of reciprocity and symmetry.

=

=

mNmN

mm

ZZZZ

,0

00, 1,...,2,1 = Nm (23)

The Impedance elements mnZ except the first and last rows and columns can be expressed by equation.(15) to (18). They construct a toeplitz type matrix. So only the elements in the frist row need calculating, Other remaining elements can be obtained by the rearrangement algorithm

= + 1111

,1||,1 NnNm

ZZ nmmn (24)

Of all the impedance elements, some need special processing. They are far fields and self-impedance elements. When the observer is located many cells away from the source, equations (16) and (18) can be written approximately as

( ,0)2

( ,0)

A Amn xx m n

q qm nm n

G x x

G x x

(25)

B. The voltage and current distributions along the line

The term mV on the right of (13) is the excitation voltage along the segment. The exciting field ixE can be derived by the boundary conditions.

The equivalent circuit of Fig.2 is a two-port network. The port 1 is driving source port with a internal impedance gZ , and the port 2 is loading port with load impedance LZ . The boundary conditions of the two ports are

0 0g g

N N L

V V I ZV I Z

=

= (26)

where 0V and NV are the voltages defined at the center of the two ends half-subsection respectively. In the delta-gap model, the port 1 and port 2 are assumed to be excited by a voltage source of magnitude 0V and NV . The delta-gap voltage source at each port provides an impressed excitation field described by the expression as flowing

)()()( 00+

+= NNix xxVxxVxE (27)

So, the mV can be written as

( , )mm

x ixm x mx

V E x y dx+

= (28) Substituting (18), (26) into (28), yields

=

=

=

= +

=

otherwiseNmZI

mZIV

dxxEyxWV

LN

gg

x

x yixxmm

m

m

,0,

0,

)(),(

0

0

(29)

The excitation voltage gV should be replaced by gV2according to the image principle. Substituting (30) into (13) yields

[ ] [ ] [ ]emnmn VIZ = (30) Where TNn IIII ),...,,,(][ 210=I

Tg

em V )0,...,0,0,2(][ =V , and the impedance element mnZ

in ][ mnZ is

+

==++

==++

=

otherwiseZZNnmZZZ

nmZZZ

Zqmn

Amn

Lqmn

Amn

gqmn

Amn

mn

,,

0,

(31)

The current distribution can be obtained by linear equation (30).

Substituting (9) into (4) yields

( ) ( )0 00( ) j4N

q qm n mn mn

nx

zV x Ik h +

=

= (32)

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where mx is the arbitrarily position on the line. The voltage is defined in the central line ( 0=y ) of the long-wire. The port voltage is defined at 2/0 += xxm and 2/= Nm xx .C. Result and discussing

For the purpose of verify the accuracy of analysis method described above, we can utilize the transmission line theory to compare with. The voltage and current wave on the transmission line terminated in load impedance LZ can be written as

[ ][ ] 0

0

0

0

0

,)(

)(

ZZZZ

eeZVxI

eeVxV

L

Lxjxj

xjxj

+

=

=

+=

+

+

(33)

where +0V is incident wave voltage, which equals gVdescribed in Fig.1. and 0Z are the propagation constant and characteristic impedance of definite transmission line respectively.

Fig.3 shows the current and voltage distributions along the long-wire antenna, which calculated by MOM as described in front and transmission line theory (TL). Theoretically, the current and voltage along the line with matched load should agree well with the result of TL .However, it is impossible without any reflect wave in practice because of electric loss on the line and complex environment. After debugging program, the load on the antenna best matched when ZL=0.52ZC.that is, the current on the line is traveling wave. As shown in Fig.3, smaller reflect wave exists when terminal matched (ZL=0.52ZC), which is in line with the actual situation.

Figure.4 shows the radiation patterns of the traveling-wave and standing wave long-wire antenna. From (a), We can observed that antenna pattern changes with the load changing. The antenna radiates to the both ends with standing-wave on the line (ZL=0ZC). In order to eliminate or suppress reflected wave, we add load resistances at the end port, good results can be obtained when ZL=0.52ZC, where ZC is the characteristic impedance. In this situation, the reflect-wave is the smallest gain is increased and the directional of antenna is strengthened. Obviously, radiation pattern agrees well with the theoretical value as well from (b).

V. CONCLUSIONIn this paper, an efficient analysis way has been presented

for the long wire antenna with a matched loading terminations using MPIE/MOM. This is achieved by introducing the half basis and testing functions, which allows the current to flow across the edges. The results of current and voltage along the line with matched load and the radiation field calculated by MoM agreed well with the results of theoretical values and TL.

Howevery, this method also need further improving in many aspects to get more accurate results.Having solved the problem of loading straight line, many complex and practical structures such as rhombic antenna, V-shaped antenna, power divider, filter etc can easily be dealt with [16-18].

ACKNOWLEDGMENT

This research work has been supported by Natinonal Natural Science Foundation of China, NSFC-60632020

(a) current distribution

(b) voltage distribution Figure.3 The current and voltage distributions with loading

ZL=0.52ZC and ZL=0 calculated by MoM and TL(traveling-wave)

391

(a) rectangular coordinate system

(b) polar system Figure.4 radiation pattern in rectangular and polar coordinate

system

REFERENCES

[1] P. J. Papakanellosand G. Fikioris, A POSSIBLE REMEDY FOR THE OSCILLATIONS OCCURRING IN THIN- WIRE MOM ANALYSIS OF CYLINDRICAL ANTENNAS Progress In Electromagnetics Research, PIER 69, 7792, 2007

[2] M. LashabANALYSIS OF ELECTROMAGNETICS SCATTERING FROM REFLECTOR AND YLINDRICAL ANTENNAS USING WAVELET-BASED MOMENT METHOD Progress In Electromagnetics Research, PIER 76, 357368, 2007

[3] M. N. Abdulla and M. B. Steer, Extraction of network parameters in the electromagnetic analysis of planar structures using the method of moments, IEEE Transactions on Microwave Theory and Techniques, vol. 49, Jan. 2001, pp. 94103.

[4] Lashab, M., F. Benabdelaziz, and C. Zeberi, On the use of wavelet-based moment method for analysis of two dimensional electromagnetic scattering, applied for circular and square contour antennas, SETIT 2007, 256, Tunisia, March 2529, 2007."

[5] M. Khalaj-Amirhosseini ANALYSIS OF LONGITUDINALLY INHOMOGENEOUS WAVEGUIDES USING THE METHOD OF MOMENTSProgress In Electromagnetics Research, PIER 74, 5767, 2007

[6] A. T. M. Sayem and M. Ali CHARACTERISTICS OF A MICROSTRIP-FED MINIATURE PRINTED HILBERT SLOT ANTENNAProgress In Electromagnetics Research, PIER 56, 118, 2006

[7] A. G. Tyzhnenko and Y. V. Ryeznik ESTIMATES OF ACCURACY AND EFFICIENCY OF A MOM ALGORITHM IN L2 FOR 2-D SCREENSProgress In Electromagnetics Research, PIER 71, 295316, 2007

[8] J. A. Ansari, R. B. Ram, and P. Singh ANALYSIS OF A GAP-COUPLED STACKED ANNULAR RING MICROSTRIP ANTENNAProgress In Electromagnetics Research B, Vol. 4, 147158, 2008

[9] R. Mittra and K. DuCHARACTERISTIC BASIS FUNCTION METHOD FOR ITERATION-FREE SOLUTION OF LARGE METHOD OF MOMENTS PROBLEMS , Progress In Electromagnetics Research B, Vol. 6, 307336, 2008

[10] S. V. A. Makki DETERMINING THE SPECIFIC GROUND CONDUCTIVITY AIDED BY THE HORIZONTAL ELECTRIC DIPOLE ANTENNA NEAR THE GROUND SURFACE, Progress In Electromagnetics Research B, Vol. 1, 4365, 2008

[11] Kazushi NISHIZAWA, Hiroaki MIYASHITA, higeru MAKINO, and Kunio ,SAWAYA, Broadening Beam width of E-Plane Radiation Pattern of a Dipole Antenna With Loaded Monopole Elements, IEEE Transactions on Antennas and Propagation, 2004, pp.39843987.

[12] Kazushi Nishizawa Hiroaki Miyashita, Shigeru Makino, and Kunio Sawaya Broad Beam width and Cross Polarization Free Dipole Antennas With Reactive Loaded Monopoles IEEE Transactions on Antennas and Propagation, vol 55, May 2007, pp. 12301238.

[13] A. Samet and A. Bouallegue, Analysis of microstrip filters using efficient MOM approach, electronics letters,Vol. 36 No. 15. July 2000,pp.1290-1291.

[14] J. J. van Tonder and U. Jakobus, Fullwave analysis of arbitrarily shaped geometries in multilayered media, in 14th International Zurich Symposium on Electromagnetic Compatibility, Feb. 2001, pp. 459464.

[15] K. W. Leung, Efficient Computation for the General Solution of a Slot Loaded by a Hemispherical Dielectric and/or Backing Cavity, IEEE Transactions on Antennas and Propagation,vol. 50, December 2002, pp. 18591862.

[16] R. C. Hall and J. R. Mosig, The analysis of arbitrarily shaped aperturecoupled patch antennas via a mixedpotential integral equation, IEEE Transactions on Antennas and Propagation, vol. 44, May 1996, pp. 608614.

[17] Y. Suh and K. Chang, A new millimeter-wave printed dipole phased array antenna using microstrip-fed coplanar stripline tee junctions, IEEE Trans. Antennas Propagation., vol. 52, Aug. 2004, pp. 20192026.

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