Current Density, Radiation Pattern, and Input Impedance

Calculation of Dipole Antenna using Integral Equation Method

SAMI UR REHMAN

Abstract: This brief discusses charge density, radiation pattern

and the input impedance of a dipole antenna (0.47λ long and with

a radius of 0.005λ) using Integral Equation(IE) approach and

Method of Moment (MoM) as numerical technique. A complete

mathematical analysis is presented in which MoM is used to

convert the electric field integral equation into a matrix equation.

MATLAB is used to simulate the resultant algebraic equation.

Those simulations are then compared with results obtained by

the commercial software.

Keywords: integral equation, method of moment, radiation

intensity, impedance, current distribution.

INTRODUCTION

Dipole antennas are categorized as straight wire antennas.

Modelling such antennas with MoM approach is well

established field of study. Two methods which have been

proved to be quite preeminent in the analysis of various

antenna problems are Integral Equation (IE) method and

Geometrical Theory of Diffraction (GTD) [1]. The integral

equation method provides the solution of the antenna problems

in the form of an integral where the unknown quantity is a part

of the integrand. Now various numerical techniques, Method

of Moment (MoM) to be the most prominent one can be used

to find the integrand. This method is most convenient for wire-

type antennas and efficient for structures that are small

electrically. The dipole antenna under consideration falls in

this category and therefore we shall use this approach.On the

other hand we use GTD method in situations where the

dimensions of the radiating system are many wavelengths.

High frequency asymptotic techniques can be used to analyze

such problems.

Sami urRehman is a researcher in Analog Mixed Signal Group (AMSG) at School of

Electrical Engineering and Computer Sciences, National University of Sciences and

Technology (NUST). He did his bachelors from NUST and is working towards his MS

degree in Electrical Engineering.

The first step in developingthe solution for the currents is

to derive the appropriate electric field integral equation.

The Method of Moments will then be used to convert the

integral equation into a system of linear equations which

can be solved by various techniques of linear algebra.

The electric field integral equation and Method of

Moments are general relationships, and I shall use these

relations under specific conditions.

ELECTRIC FIELD INTEGRAL EQUATION

The first step in developing the solution for the current on a

wire antenna is determining the appropriate integral equation.

The starting point in deriving the electric field integral

equation (EFIE) is Maxwell’s equations in the:

E jw H M (1)

H jw E J (2)

. eE

(3)

. mH

(4)

along with the continuity equations,

. eJ jw (5)

. mM jw (6)

The boundary conditions on the magnetic and electric

fields at the surface, S, are

2 1( ) sn H H J (7)

SAMI UR REHMAN 2011-NUST-MSEE-53

2 1( ) sn E E M (8)

The electric field outside of S,E2 , may be written as the sum

of an incident electric field, Ei, and a scattered electric field,

Es. The incident electric field induces the surface current

Jswhich in turn creates the scattered field Es. The Equivalence

Principle can be used to remove the PEC giving a

homogeneous free space problem. For the PEC case, the

equivalent current equals the induced current, Js.

Making use of the fact that the divergence of His zero when

m is zero, the magnetic field may be defined as the curl of an

auxillary vector,

H A

Where A is called the magnetic vector potential. Solving the

vector wave equation for the magnetic vector potential due to

Js , the free space solution is found to be

( ) ( )s

s

J r G R ds (9)

where the free space Green’s Function, G , is given by

'exp( )( ) ,

4

j RG R R r r

R

(10)

Now the scattered electric field, derived in [2] can be given by

1( . )s oE jw A j A

(11)

Final step in the derivation of Electric Field Integral Equation

is the application of boundary condition in the above scattered

field equation. Resultantly we get, what as explicitly be

derived in [3], as:

2 ' ' '[ ( ) ( ) ( . ( )) ( )]s S i

o s

jn k J r G R J r G R ds n E

(12)

The above derivation was mostly based on work presented by

Stutzman and Thiele (1987). [4] Provides a helpful insight into

the formulation of MoM over cylindrical structures.

METHOD OF MOMENTS

The Method of Moments (MoM) is a well known technique for

solving linearequations. In antenna analysis, the MoM is used

to convert the electric field integral equation into a matrix

equation or system of linear equations. The basic form of the

equation to be solved by the Method of Moments is

( )L u f (13)

where L is the linear operator, u is the unknown function, and f

is the source or forcing function. In order to create the matrix

equation, the unknown function is defined to be the sum of a

set of known independent functions, un, called basis or

expansion functions with unknown amplitudes, αn,

n n

n

u u (14)

Using the linearity of the operator, L, the unknown amplitudes

can be brought out of the operator giving

( )n n

n

L u f (15)

The unknown amplitudes cannot yet be determined because

there are n unknowns, but one functional equation. A fixed set

of equations are found by defining independent weighting or

testing functions, wm , which are integrated to give m different

linear equations.

, ( ) ,n m n m

n

w L u w f (16)

For antenna problems, the matrix form of above equation is

usually written in a form similar to Ohm’s Law:

,m n n mZ I V (17)

The generalized impedance matrix is given by

, , ( )m n m nZ w L u , the generalized current matrix is

given by n nI , and the generalized voltage matrix is

given by ,m mV w f , .

POCKLINGTON’S IE FOR DIPOLE ANTENNA

One of the most well-known forms of the EFIE was developed

by Pocklington (1897). Let us consider the geometry of Fig. 1

for derivation of Pocklington IE for the case of dipole antenna.

Fig. 1. Simple structure of a dipole antenna

For this geometry, the Lorentz gauge along the wire length, l,

reduces to:

1l

o

Aj l

(18)

Now using this Lorentz gauge the above calculated scattered

field becomes:

l o lE j Al

(19)

where the l subscripts indicate the component of the vectors in

the l direction. After taking the derivative of (18) and

substituting the result into (19), the electric field may be

written as:

22

2

1l l l

o

E k A Aj l

(20)

Making use of the thin wire approximation, which states that

the current around the circumference of the wire is uniform

and the axially directed electric field is to be estimated along

the axis of the current in the l direction, the vector potential, Al,

is given by:

/ 2

' ' ' 2 2

/ 2

( ) ( ) , ( )

L

l

L

A I l G R dl R l l a

(21)

Combining the last two equations and the electric field

boundary condition at the surface of the wire, Pocklington’s

electric field integral equation is given by:

/ 2 2

' 2 '

2

/ 2

( ) ( ) ( )

L

li

o L

jI l G R k G R dl E

l

(22)

Although linear basis functions can be used, curved sinusoidal

basis functions provide better approximation and decrease the

computation time as validated in [5]Now using the Method of

Moments to solve the problem, the current is expanded using

piece-wise sinusoidal basis functions giving:

'

' '11

1

sin ( )( ) ,

sin ( )

nn n n

n n n

k z zI z I z z z

k z z

'' '1

1

1

sin ( )( ) ,

sin ( )

nn n n

n n n

k z zI z I z z z

k z z

(23)

and the weighting functions are pulses one segment wide The

incident electric field is approximated by the delta gap source

model which assumes that the incident electric field is due to

the applied voltage across a small gap in the antenna, of width

d approaching zero, and is confined to that gap. Incorporating

the piece-wise sinusoidal basis functions, the pulse weighting

functions, and the delta gap source model, EFIE becomes:

2'

1

2

[ ( )sin

mn

nm

zz

n n

n o zz

j kI z z

k

' ' '

1( ) 2 ( )cos ] ( )n nz z z z k G R dz dz

2

2

( )

m

m

z

msms

z

Vz z dz

(24)

Where is one segment length, nz is n from the origin from

the z axis, msz is the location of thm source and msV is the

voltage of the thm source.

Lets compare this approach with the approach followed by

Ballanis in “Antenna Theory”. The resulting equation

apparently seems different but conveys almost the same

meaning as conveyed by equation (24).

Pocklingtons IE derived by Ballanis in [1] is given by:

/ 2

' 2 2

2

/ 2

exp( )( ) [(1 )(2 3 )

4

L

L

jkRI z jkR R a

R

2 '( ) ] ( 0)izkaR dz j E (25)

Similarly in [6] authors have derived the following expression

for a generic entry of impedance matrix from the IE:

22 2

exp( 2( ) ( )

8

omn

jZ j rG r

r

22 2{(1 2 )[2 3( ) ] 4 }

aj r a

r

(26)

Both equations (24 and 25) cab be used to calculate the current

distribution and the radiation pattern of the dipole antenna. A

mathematically rigorous approach to work out problems

concerning current distribution on wire antennas is reported in

[7].Current distribution and Impedance obtained from the

above equations is plotted in Fig. 2 and Fig. 3 respectively for

N=79.

Fig. 2 Current distribution on dipole antenna

Fig. 3 Input Impedance of dipole antenna

Now let’s compute the value of impedance by varying the

number of segments N. Fig 4 shows impedance for N=50 and

Fig. 5 for N=100, and we conclude that as we increase the

number of segments the accuracy of the results increases.

The dipole under discussion is a finite length dipole antenna

whose radiation pattern can be plotted once the current

distribution is known since this distribution is used in the E

and H field equations which can be computed to find out the

radiation pattern.

Fig. 4 Impedance for N=50

Fig. 5 Impedance for N=50

[8] Provides the numerical solution for calculating near and far

field Consider the geometry as shown in Fig. 6. Using this

figure we can work out a mathematical expression for the

radiated electric field as shown in (27).

sinexp( ( ))

2

o

o

jI lE j t kr

cr

(27)

Fig. 6 Figure for calculating E far field

Where c is the speed of light and r is the distance from the

dipole to the observation point.For the given parameters, the

radiation pattern is as shown in Fig. 7.

Fig. 7 Radiation pattern of dipole antenna

For the same dipole the 3-D plot is shown in Fig. 6.

Fig.8 3-D radiation pattern of the given dipole

The directivity of such an antenna is shown in Fig. 7.

Fig. 9 Directivity of dipole antenna

But there could be many other ways in which we can find

current distribution over straight wire or cylindrical structures.

Like [9] uses Fourier analysis to approximate the solution of

the Integral Equation to find the current distribution.

SIMULATIONS USING CST

The given dipole antenna is simulated using CST tool and the

results are shown as below:

Fig. 10 Dipole implemented in CST environment

Fig. 11 2-D Radiation pattern of dipole antenna

Fig. 12 Impedance plot of the given dipole

Fig. 13 S11 Parameter of the dipole

Fig. 14 Surface Current Density of the given dipole

CONCLUSION

This paper discusses current distribution, radiation pattern and

the impedance computation of a simple center fed dipole

antenna. A rigorous mathematical modeling has been

performed to compute the aforementioned parameters using

Integral Equation approach. Then MoM was used as a

analytical technique to solve the integral. MATLAB is used to

compute the integral and results are then compared with the

commercial CST tool.

REFERENCES

[1] “Antenna Theory” 2nd Edition by Constantine A. Balanis

[2] “Advanced Engineering Electromagnetics” by Constantine A.

Balanis

[3] Huffman, J.A.; Werner, D.H.; , "Modeling of a cylindrical wire

antenna with flat end caps using a rigorous moment method technique,"

Antennas and Propagation Society International Symposium, 1995. AP-S. Digest , vol.2, no., pp.1258-1261 vol.2, 18-23 Jun 1995

[4] Inagaki, M.; Sawaya, K.; Adachi, S.; , "Numerical analysis of a dipole antenna in the vicinity of conducting circular cylinder with finite

length moment method analysis by using the interior Green's function,"

Antennas and Propagation Society International Symposium, 1998. IEEE , vol.4, no., pp.1918-1921 vol.4, 21-26 Jun 1998

[5] Kubiak, I.; , "The analysis of distribution of the current in the slender cylindrical antenna," MILCOM 97 Proceedings , vol.1, no., pp.247-

251 vol.1, 2-5 Nov 1997

[6] “Antennas for All Applications, 3rd ed. by J. D. Kraus and R. J.

Marhefka

[7] Tsai, L.; , "A numerical solution for the near and far fields of an

annular ring of magnetic current," Antennas and Propagation, IEEE

Transactions on , vol.20, no.5, pp. 569- 576, Sep 1972

[8] Hsieh, H. C.; , "Current Distribution of a Cylindrical Antenna in a

Warm Plasma," Plasma Science, IEEE Transactions on , vol.9, no.2, pp.52-

57, June 1981

[9] Khamas, S.K.; Cook, G.G.; , "Moment-method analysis of printed wire spirals using curved piecewise sinusoidal subdomain basis

and testing functions," Antennas and Propagation, IEEE Transactions on ,

vol.45, no.6, pp.1016-1022, Jun 1997